Coverage of quantile confidence intervals

Andrey Akinshin · 2020-12-22

There is a common misunderstanding that a 95% confidence interval is an interval that covers the true parameter value with 95% probability. Meanwhile, the correct definition assumes that the true parameter value will be covered by 95% of 95% confidence intervals in the long run. These two statements sound similar, but there is a huge difference between them. 95% in this context is not a property of a single confidence interval. Once you get a calculated interval, it may cover the true value (100% probability) or it may don’t cover it (0% probability). In fact, 95% is a prediction about the percentage of future confidence intervals that cover the true value in the long run.

However, even if you know the correct definition, you still may experience some troubles. The first thing people usually forgot is the “long run” part. For example, if we collected 100 samples and calculated a 95% confidence interval of a parameter for each of them, we shouldn’t expect that 95 of these intervals cover the true parameter value. In fact, we can observe a situation when none of these intervals covers the true value. Of course, this is an unlikely event, but if you automatically perform thousands of different experiments, you will definitely get some extreme situations.

The second thing that may create trouble is the “prediction” part. If weather forecasters predicted that it will rain tomorrow, this does not mean that it will rain tomorrow. The same works for statistical predictions. The actual prediction reliability may depend on many factors. If you estimate confidence intervals around the mean for the normal distribution, you are most likely safe. However, if you estimate confidence intervals around quantiles for non-parametric distributions, you should care about the following things:

The used approach to estimate confidence intervals
The underlying distribution
The sample size
The position of the target quantile

I have already showed how to estimate the confidence interval around the given quantile using the Maritz-Jarrett method. It’s time to verify the reliability of this approach. In this post, I’m going to show some Monte-Carlo simulations that evaluate the coverage percentage in different situations.

Preparation

For this simulation, I decided to use synthetic latency distribution by Brendan Gregg:

0   uniform narrow                           Uniform(500,1500)
1   uniform wide                             Uniform(0,3000)
2   uniform outliers                         Mix(Uniform(500,1500)|0.99;Uniform(1500,10000)|0.01)
100 unimodal normal narrow                   Normal(1000,100^2)
101 unimodal normal medium                   Normal(1000,200^2)
102 unimodal normal wide                     Normal(1000,300^2)
103 unimodal normal with tail                Mix(Normal(1000,200^2)|0.96;Uniform(1000,2250)|0.04)
104 unimodal normal wide                     Normal(1120,700^2)
111 uniform normal spike                     Mix(Normal(1000,200^2)|0.98;Normal(750,1^2)|0.02)
130 unimodal pareto narrow                   Pareto(1000,3
131 unimodal pareto wide                     Pareto(1000,10
140 unimodal normal outliers 1% medium       Mix(Normal(1000,200^2)|0.99;Uniform(1000,5000)|0.01)
141 unimodal normal outliers 1% far          Mix(Normal(1000,200^2)|0.99;Uniform(1000,10000)|0.01)
142 unimodal normal outliers 1% very far     Mix(Normal(1000,200^2)|0.99;Uniform(1000,50000)|0.01)
143 unimodal normal outliers 2%              Mix(Normal(1000,200^2)|0.98;Uniform(1000,5000)|0.02)
144 unimodal normal outliers 4%              Mix(Normal(1000,200^2)|0.96;Uniform(1000,5000)|0.04)
145 unimodal normal outliers 2% clustered    Mix(Normal(1000,200^2)|0.98;Normal(3000,35^2)|0.02)
146 unimodal normal outliers 4% close 1      Mix(Normal(1000,200^2)|0.96;Uniform(1000,2700)|0.04)
147 unimodal normal outliers 4% close 2      Mix(Normal(1000,200^2)|0.96;Uniform(1000,2900)|0.04)
148 unimodal normal outliers 4% close 3      Mix(Normal(1000,200^2)|0.96;Uniform(1000,3100)|0.04)
149 unimodal normal outliers 4% close 4      Mix(Normal(1000,200^2)|0.96;Uniform(1000,3300)|0.04)
150 unimodal normal outliers 4% close 5      Mix(Normal(1000,200^2)|0.96;Uniform(1000,3500)|0.04)
151 unimodal normal outliers 4% close 6      Mix(Normal(1000,200^2)|0.96;Uniform(1000,3700)|0.04)
152 unimodal normal outliers 4% close 7      Mix(Normal(1000,200^2)|0.96;Uniform(1000,3900)|0.04)
153 unimodal normal outliers 0.5%            Mix(Normal(1000,200^2)|0.995;Uniform(1000,5000)|0.005)
154 unimodal normal outliers 0.2%            Mix(Normal(1000,200^2)|0.998;Uniform(1000,5000)|0.002)
155 unimodal normal outliers 0.1%            Mix(Normal(1000,200^2)|0.999;Uniform(1000,5000)|0.001)
200 bimodal normal very close                Mix(Normal(850,110^2)|0.5;Normal(1150,110^2)|0.5)
201 bimodal normal close                     Mix(Normal(825,110^2)|0.5;Normal(1175,110^2)|0.5)
202 bimodal normal medium                    Mix(Normal(750,110^2)|0.5;Normal(1250,110^2)|0.5)
203 bimodal normal far                       Mix(Normal(600,110^2)|0.5;Normal(1400,110^2)|0.5)
204 bimodal normal outliers 1%               Mix(Normal(750,110^2)|0.495;Normal(1250,110^2)|0.495;Uniform(1000,5000)|0.01)
205 bimodal normal outliers 2%               Mix(Normal(750,110^2)|0.49;Normal(1250,110^2)|0.49;Uniform(1000,5000)|0.02)
206 bimodal normal outliers 4%               Mix(Normal(750,110^2)|0.48;Normal(1250,110^2)|0.48;Uniform(1000,5000)|0.04)
210 bimodal normal major minor               Mix(Normal(750,110^2)|0.7;Normal(1250,110^2)|0.3)
211 bimodal normal minor major               Mix(Normal(750,110^2)|0.3;Normal(1250,110^2)|0.7)
212 bimodal normal major minor outliers      Mix(Normal(750,110^2)|0.695;Normal(1250,110^2)|0.295;Uniform(1000,5000)|0.01)
213 bimodal normal major minor outliers      Mix(Normal(750,110^2)|0.295;Normal(1250,110^2)|0.695;Uniform(1000,5000)|0.01)
214 bimodal far normal far outliers 1%       Mix(Normal(500,150^2)|0.499;Normal(2000,300^2)|0.499;Uniform(1000,180000)|0.002)
215 bimodal very far normal far outliers 1%  Mix(Normal(500,100^2)|0.499;Normal(4000,500^2)|0.499;Uniform(1000,180000)|0.002)
216 bimodal very far major minor outliers 1% Mix(Normal(500,100^2)|0.667;Normal(4000,100^2)|0.333;Uniform(1000,180000)|0.002)
300 trimodal normal close                    Mix(Normal(750,90^2)|0.333;Normal(1000,90^2)|0.334;Normal(1250,90^2)|0.333)
301 trimodal normal medium                   Mix(Normal(500,100^2)|0.333;Normal(1000,100^2)|0.334;Normal(1500,100^2)|0.333)
302 trimodal normal far                      Mix(Normal(500,65^2)|0.333;Normal(1000,65^2)|0.334;Normal(1500,65^2)|0.333)
302 trimodal normal outliers                 Mix(Normal(500,100^2)|0.333;Normal(1000,100^2)|0.334;Normal(1500,100^2)|0.333;Uniform(1000,5000)|0.01)
304 trimodal normal major medium minor       Mix(Normal(500,100^2)|0.5;Normal(1000,100^2)|0.33;Normal(1500,100^2)|0.17)
305 trimodal normal minor major minor        Mix(Normal(500,100^2)|0.25;Normal(1000,100^2)|0.5;Normal(1500,100^2)|0.25)
306 trimodal normal minor major medium       Mix(Normal(500,100^2)|0.17;Normal(1000,100^2)|0.5;Normal(1500,100^2)|0.33)
307 trimodal normal major minor medium       Mix(Normal(500,100^2)|0.5;Normal(1000,100^2)|0.17;Normal(1500,100^2)|0.33)
400 quad normal close                        Mix(Normal(700,75^2)|0.25;Normal(900,75^2)|0.25;Normal(1100,75^2)|0.25;Normal(1300,75^2)|0.25)
401 quad normal medium                       Mix(Normal(700,50^2)|0.25;Normal(900,50^2)|0.25;Normal(1100,50^2)|0.25;Normal(1300,50^2)|0.25)
402 quad normal far                          Mix(Normal(400,60^2)|0.25;Normal(800,60^2)|0.25;Normal(1200,60^2)|0.25;Normal(1600,60^2)|0.25)
403 quad normal outliers                     Mix(Normal(700,50^2)|0.25;Normal(900,50^2)|0.25;Normal(1100,50^2)|0.25;Normal(1300,50^2)|0.25;Uniform(1000,5000)|0.01)

Next, we will do the following:

For each distribution, enumerate different percentiles values: $50^\textrm{th}$, $70^\textrm{th}$, $90^\textrm{th}$, $95^\textrm{th}$, $99^\textrm{th}$.
For each percentile, enumerate sample sizes from 2 to 50.
For each sample size, generate 10000 samples of the given sample size for the given distribution.
For each sample, estimate a 95% confidence interval for the given quantile.
For each combination of distribution/percentile/sample size, calculate the percentage of confidence intervals that cover the true percentile value.

The source code of this simulation can be found here. You can observe the simulation results in the next section.

The coverage percentage of the 95% confidence interval

Let’s start with the $50^\textrm{th}$ percentile (the median):

                              Quantile = 0.50, ConfidenceLevel = 95%
Dist    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50
0    0.924 0.946 0.958 0.963 0.962 0.956 0.956 0.953 0.952 0.956 0.951 0.952 0.950 0.949 0.951 0.952 0.946 0.949 0.950 0.945 0.948 0.948 0.951 0.947 0.953 0.953 0.949 0.950 0.948 0.947 0.949 0.948 0.943 0.946 0.949 0.945 0.946 0.948 0.948 0.948 0.946 0.950 0.950 0.947 0.948 0.946 0.946 0.946 0.951 
1    0.928 0.946 0.958 0.965 0.960 0.958 0.954 0.952 0.953 0.954 0.951 0.949 0.949 0.950 0.950 0.948 0.951 0.950 0.952 0.952 0.946 0.944 0.950 0.948 0.953 0.949 0.951 0.948 0.946 0.947 0.948 0.950 0.946 0.949 0.948 0.949 0.947 0.950 0.947 0.950 0.946 0.946 0.948 0.946 0.954 0.950 0.946 0.950 0.946 
2    0.927 0.945 0.959 0.964 0.965 0.956 0.959 0.958 0.956 0.955 0.954 0.956 0.951 0.952 0.953 0.949 0.952 0.954 0.946 0.951 0.952 0.954 0.951 0.953 0.948 0.945 0.948 0.948 0.946 0.949 0.947 0.951 0.946 0.950 0.949 0.946 0.949 0.947 0.947 0.946 0.945 0.947 0.946 0.945 0.948 0.942 0.949 0.945 0.949 
100  0.948 0.969 0.976 0.980 0.978 0.975 0.971 0.967 0.968 0.967 0.969 0.960 0.963 0.965 0.962 0.959 0.961 0.961 0.957 0.961 0.958 0.961 0.955 0.960 0.960 0.956 0.954 0.955 0.955 0.955 0.956 0.959 0.956 0.957 0.951 0.954 0.953 0.955 0.955 0.955 0.952 0.957 0.953 0.951 0.952 0.950 0.952 0.955 0.953 
101  0.952 0.968 0.978 0.978 0.977 0.975 0.972 0.969 0.966 0.966 0.964 0.963 0.965 0.964 0.960 0.962 0.960 0.959 0.958 0.960 0.959 0.959 0.960 0.955 0.958 0.958 0.955 0.956 0.958 0.952 0.956 0.955 0.958 0.950 0.956 0.955 0.954 0.949 0.955 0.953 0.954 0.956 0.955 0.950 0.953 0.956 0.948 0.955 0.954 
102  0.948 0.968 0.976 0.979 0.977 0.971 0.970 0.969 0.970 0.968 0.960 0.965 0.961 0.962 0.963 0.962 0.961 0.959 0.958 0.956 0.959 0.960 0.958 0.957 0.955 0.957 0.954 0.955 0.955 0.954 0.954 0.958 0.956 0.956 0.952 0.954 0.955 0.954 0.953 0.954 0.956 0.954 0.955 0.953 0.950 0.955 0.953 0.952 0.955 
103  0.953 0.971 0.978 0.980 0.980 0.977 0.976 0.972 0.972 0.968 0.965 0.967 0.964 0.961 0.963 0.964 0.962 0.964 0.961 0.960 0.957 0.958 0.958 0.958 0.956 0.958 0.956 0.955 0.957 0.958 0.956 0.954 0.955 0.956 0.956 0.953 0.957 0.957 0.957 0.956 0.955 0.952 0.955 0.950 0.958 0.950 0.957 0.953 0.953 
104  0.948 0.968 0.977 0.981 0.975 0.977 0.971 0.970 0.968 0.965 0.963 0.965 0.963 0.962 0.961 0.962 0.961 0.958 0.960 0.959 0.958 0.955 0.959 0.956 0.955 0.961 0.956 0.953 0.955 0.958 0.955 0.954 0.953 0.955 0.957 0.953 0.955 0.955 0.956 0.950 0.957 0.954 0.950 0.956 0.949 0.953 0.956 0.953 0.950 
111  0.950 0.967 0.975 0.976 0.973 0.973 0.970 0.967 0.971 0.964 0.966 0.964 0.964 0.958 0.962 0.961 0.959 0.961 0.959 0.960 0.958 0.959 0.961 0.954 0.959 0.956 0.955 0.955 0.957 0.955 0.959 0.956 0.955 0.957 0.954 0.954 0.951 0.955 0.955 0.954 0.953 0.953 0.955 0.955 0.948 0.953 0.949 0.956 0.953 
130  0.935 0.954 0.968 0.972 0.975 0.975 0.974 0.973 0.973 0.970 0.967 0.967 0.967 0.966 0.969 0.965 0.965 0.963 0.959 0.961 0.960 0.960 0.963 0.961 0.961 0.958 0.961 0.961 0.958 0.960 0.958 0.959 0.958 0.956 0.958 0.956 0.953 0.958 0.957 0.955 0.959 0.955 0.960 0.957 0.959 0.959 0.953 0.955 0.953 
131  0.937 0.955 0.967 0.974 0.974 0.972 0.969 0.971 0.970 0.969 0.965 0.964 0.961 0.962 0.962 0.963 0.962 0.963 0.960 0.958 0.958 0.958 0.953 0.958 0.958 0.959 0.961 0.959 0.957 0.960 0.955 0.956 0.956 0.960 0.955 0.951 0.954 0.957 0.955 0.957 0.953 0.954 0.956 0.951 0.952 0.954 0.951 0.952 0.955 
140  0.953 0.969 0.979 0.978 0.979 0.975 0.974 0.973 0.969 0.966 0.968 0.967 0.963 0.966 0.962 0.961 0.965 0.958 0.961 0.958 0.958 0.958 0.958 0.958 0.959 0.958 0.956 0.959 0.953 0.957 0.955 0.956 0.955 0.952 0.950 0.953 0.956 0.952 0.952 0.954 0.956 0.951 0.954 0.955 0.955 0.955 0.953 0.950 0.950 
141  0.954 0.967 0.978 0.978 0.978 0.974 0.971 0.973 0.970 0.968 0.968 0.968 0.961 0.964 0.963 0.959 0.963 0.955 0.960 0.960 0.960 0.959 0.958 0.956 0.959 0.958 0.958 0.958 0.956 0.956 0.954 0.959 0.950 0.956 0.956 0.952 0.953 0.954 0.952 0.949 0.950 0.952 0.952 0.953 0.954 0.953 0.957 0.954 0.954 
142  0.956 0.970 0.976 0.981 0.978 0.975 0.972 0.974 0.970 0.971 0.969 0.970 0.967 0.968 0.962 0.962 0.960 0.962 0.959 0.962 0.957 0.958 0.960 0.956 0.955 0.954 0.955 0.959 0.958 0.956 0.959 0.954 0.956 0.957 0.954 0.952 0.953 0.954 0.953 0.957 0.955 0.956 0.955 0.951 0.956 0.951 0.950 0.956 0.951 
143  0.951 0.970 0.974 0.979 0.980 0.977 0.975 0.972 0.973 0.970 0.970 0.968 0.962 0.966 0.963 0.960 0.960 0.961 0.959 0.960 0.960 0.956 0.961 0.961 0.959 0.955 0.957 0.958 0.953 0.955 0.956 0.957 0.959 0.956 0.955 0.959 0.957 0.952 0.954 0.954 0.954 0.954 0.952 0.952 0.955 0.953 0.953 0.950 0.947 
144  0.953 0.971 0.980 0.981 0.979 0.978 0.978 0.977 0.974 0.969 0.970 0.966 0.966 0.966 0.968 0.962 0.959 0.963 0.960 0.960 0.960 0.958 0.957 0.960 0.956 0.957 0.958 0.954 0.960 0.960 0.957 0.959 0.959 0.959 0.954 0.957 0.954 0.956 0.953 0.953 0.954 0.954 0.955 0.956 0.952 0.952 0.952 0.953 0.952 
145  0.953 0.967 0.977 0.981 0.979 0.980 0.979 0.973 0.970 0.969 0.969 0.966 0.964 0.966 0.966 0.961 0.959 0.961 0.962 0.963 0.959 0.960 0.959 0.957 0.958 0.958 0.957 0.958 0.960 0.954 0.951 0.956 0.957 0.956 0.958 0.958 0.953 0.954 0.954 0.958 0.958 0.959 0.953 0.956 0.949 0.954 0.952 0.952 0.951 
146  0.951 0.969 0.979 0.978 0.980 0.979 0.975 0.977 0.971 0.971 0.966 0.966 0.966 0.965 0.964 0.959 0.962 0.959 0.961 0.957 0.959 0.957 0.957 0.959 0.959 0.956 0.961 0.958 0.954 0.958 0.953 0.956 0.959 0.958 0.956 0.956 0.955 0.955 0.955 0.956 0.950 0.954 0.953 0.949 0.956 0.956 0.954 0.952 0.951 
147  0.949 0.972 0.979 0.982 0.981 0.979 0.979 0.974 0.970 0.969 0.970 0.968 0.964 0.965 0.963 0.961 0.959 0.965 0.958 0.960 0.961 0.961 0.958 0.960 0.960 0.961 0.958 0.957 0.956 0.957 0.955 0.954 0.954 0.961 0.955 0.955 0.951 0.955 0.952 0.953 0.955 0.955 0.951 0.953 0.954 0.950 0.953 0.955 0.955 
148  0.953 0.969 0.980 0.980 0.980 0.979 0.977 0.972 0.972 0.969 0.969 0.963 0.966 0.966 0.968 0.965 0.965 0.962 0.961 0.963 0.955 0.954 0.958 0.958 0.960 0.954 0.959 0.956 0.956 0.957 0.958 0.959 0.955 0.957 0.957 0.957 0.951 0.956 0.956 0.954 0.952 0.952 0.956 0.954 0.951 0.955 0.952 0.953 0.953 
149  0.954 0.970 0.978 0.980 0.981 0.978 0.974 0.976 0.970 0.970 0.969 0.968 0.967 0.962 0.960 0.961 0.961 0.962 0.961 0.962 0.957 0.959 0.959 0.957 0.955 0.958 0.957 0.960 0.958 0.956 0.958 0.957 0.955 0.955 0.955 0.958 0.952 0.955 0.956 0.949 0.951 0.955 0.956 0.953 0.954 0.953 0.953 0.949 0.955 
150  0.950 0.971 0.977 0.979 0.979 0.978 0.975 0.976 0.971 0.969 0.971 0.967 0.964 0.964 0.964 0.962 0.956 0.959 0.961 0.957 0.961 0.958 0.962 0.955 0.957 0.955 0.956 0.958 0.956 0.958 0.956 0.954 0.957 0.954 0.955 0.955 0.952 0.957 0.953 0.956 0.954 0.956 0.956 0.950 0.955 0.953 0.956 0.956 0.956 
151  0.953 0.974 0.981 0.982 0.984 0.981 0.978 0.978 0.971 0.972 0.973 0.967 0.967 0.964 0.967 0.960 0.962 0.962 0.963 0.956 0.957 0.956 0.955 0.958 0.954 0.957 0.960 0.958 0.958 0.957 0.953 0.953 0.953 0.956 0.954 0.954 0.952 0.955 0.957 0.953 0.952 0.954 0.951 0.954 0.950 0.952 0.953 0.949 0.956 
152  0.955 0.972 0.983 0.981 0.983 0.979 0.975 0.974 0.972 0.972 0.968 0.964 0.966 0.966 0.966 0.963 0.959 0.963 0.961 0.959 0.959 0.958 0.960 0.959 0.957 0.960 0.960 0.959 0.956 0.956 0.953 0.957 0.957 0.955 0.955 0.958 0.954 0.954 0.954 0.955 0.953 0.954 0.956 0.953 0.956 0.954 0.955 0.952 0.952 
153  0.952 0.969 0.976 0.977 0.977 0.977 0.969 0.968 0.969 0.968 0.967 0.967 0.962 0.963 0.961 0.962 0.960 0.961 0.961 0.959 0.957 0.956 0.957 0.956 0.956 0.955 0.959 0.954 0.957 0.955 0.959 0.955 0.955 0.958 0.955 0.959 0.953 0.955 0.951 0.953 0.949 0.952 0.952 0.951 0.954 0.955 0.954 0.954 0.951 
154  0.954 0.968 0.976 0.979 0.976 0.973 0.971 0.969 0.969 0.968 0.969 0.961 0.962 0.963 0.961 0.961 0.963 0.962 0.957 0.959 0.957 0.958 0.958 0.956 0.953 0.953 0.956 0.956 0.955 0.954 0.956 0.956 0.956 0.958 0.955 0.952 0.955 0.955 0.953 0.950 0.956 0.955 0.952 0.953 0.955 0.950 0.955 0.953 0.953 
155  0.946 0.968 0.978 0.980 0.979 0.973 0.970 0.975 0.965 0.968 0.962 0.962 0.964 0.961 0.962 0.960 0.962 0.958 0.958 0.957 0.960 0.957 0.960 0.955 0.957 0.957 0.956 0.954 0.955 0.958 0.955 0.956 0.956 0.951 0.953 0.954 0.951 0.953 0.955 0.951 0.957 0.951 0.952 0.954 0.955 0.954 0.953 0.950 0.949 
200  0.936 0.950 0.956 0.961 0.960 0.958 0.956 0.946 0.950 0.945 0.944 0.946 0.942 0.946 0.940 0.944 0.943 0.939 0.945 0.939 0.941 0.937 0.940 0.941 0.940 0.941 0.939 0.939 0.936 0.938 0.938 0.935 0.934 0.935 0.942 0.942 0.940 0.939 0.936 0.940 0.938 0.935 0.942 0.939 0.938 0.938 0.943 0.937 0.936 
201  0.925 0.938 0.953 0.959 0.955 0.948 0.946 0.936 0.939 0.935 0.936 0.933 0.937 0.935 0.933 0.930 0.933 0.933 0.931 0.927 0.930 0.934 0.929 0.928 0.929 0.926 0.932 0.931 0.926 0.924 0.928 0.926 0.925 0.925 0.923 0.930 0.928 0.928 0.930 0.923 0.931 0.931 0.924 0.928 0.930 0.926 0.927 0.930 0.926 
202  0.900 0.897 0.921 0.946 0.926 0.914 0.928 0.913 0.909 0.914 0.908 0.907 0.907 0.903 0.911 0.905 0.905 0.908 0.901 0.904 0.904 0.898 0.901 0.902 0.895 0.890 0.898 0.892 0.896 0.903 0.898 0.897 0.897 0.897 0.891 0.895 0.897 0.894 0.896 0.896 0.901 0.898 0.896 0.894 0.898 0.895 0.892 0.896 0.890 
203  0.839 0.817 0.880 0.938 0.877 0.882 0.921 0.870 0.889 0.907 0.865 0.899 0.878 0.883 0.898 0.870 0.895 0.878 0.887 0.894 0.873 0.893 0.874 0.888 0.880 0.886 0.893 0.879 0.887 0.877 0.886 0.877 0.874 0.885 0.874 0.882 0.872 0.883 0.880 0.878 0.878 0.877 0.883 0.881 0.882 0.875 0.878 0.877 0.887 
204  0.900 0.902 0.927 0.945 0.927 0.917 0.927 0.914 0.918 0.912 0.902 0.908 0.907 0.904 0.906 0.903 0.903 0.901 0.903 0.899 0.902 0.907 0.898 0.900 0.898 0.898 0.904 0.899 0.898 0.900 0.907 0.892 0.895 0.890 0.896 0.895 0.904 0.892 0.896 0.895 0.893 0.895 0.896 0.893 0.901 0.890 0.891 0.891 0.891 
205  0.903 0.905 0.929 0.946 0.931 0.918 0.927 0.924 0.919 0.912 0.909 0.913 0.909 0.905 0.911 0.902 0.902 0.906 0.902 0.901 0.901 0.898 0.898 0.898 0.898 0.901 0.895 0.901 0.899 0.906 0.898 0.901 0.899 0.897 0.897 0.898 0.897 0.894 0.894 0.889 0.898 0.897 0.898 0.898 0.900 0.900 0.891 0.897 0.898 
206  0.903 0.909 0.933 0.947 0.930 0.929 0.932 0.927 0.926 0.916 0.918 0.917 0.908 0.910 0.911 0.908 0.906 0.903 0.908 0.900 0.908 0.902 0.906 0.909 0.894 0.900 0.902 0.897 0.901 0.899 0.903 0.902 0.898 0.900 0.899 0.900 0.896 0.897 0.900 0.899 0.897 0.894 0.899 0.896 0.897 0.898 0.894 0.894 0.901 
210  0.936 0.958 0.972 0.967 0.971 0.973 0.974 0.973 0.969 0.971 0.974 0.975 0.972 0.973 0.976 0.975 0.975 0.977 0.973 0.974 0.975 0.972 0.973 0.975 0.972 0.971 0.973 0.974 0.973 0.976 0.973 0.974 0.973 0.975 0.976 0.972 0.975 0.973 0.973 0.970 0.975 0.972 0.972 0.973 0.973 0.973 0.972 0.973 0.970 
211  0.941 0.959 0.973 0.969 0.970 0.972 0.973 0.969 0.970 0.970 0.972 0.972 0.974 0.975 0.976 0.973 0.972 0.972 0.973 0.971 0.972 0.974 0.975 0.973 0.974 0.975 0.975 0.974 0.974 0.973 0.973 0.974 0.974 0.971 0.971 0.975 0.974 0.973 0.972 0.972 0.972 0.973 0.970 0.972 0.971 0.972 0.974 0.971 0.976 
212  0.938 0.956 0.975 0.968 0.971 0.973 0.973 0.972 0.971 0.975 0.973 0.974 0.975 0.971 0.971 0.972 0.970 0.973 0.971 0.975 0.974 0.975 0.973 0.974 0.973 0.976 0.975 0.974 0.972 0.971 0.975 0.974 0.976 0.973 0.972 0.973 0.975 0.973 0.973 0.972 0.975 0.972 0.976 0.974 0.976 0.971 0.972 0.970 0.973 
213  0.935 0.961 0.976 0.970 0.971 0.975 0.975 0.974 0.971 0.975 0.976 0.972 0.974 0.972 0.970 0.971 0.973 0.975 0.976 0.975 0.972 0.973 0.975 0.976 0.975 0.973 0.972 0.977 0.973 0.974 0.974 0.970 0.972 0.976 0.972 0.975 0.974 0.974 0.972 0.973 0.974 0.976 0.973 0.972 0.971 0.972 0.974 0.972 0.972 
214  0.847 0.834 0.886 0.936 0.898 0.882 0.914 0.881 0.895 0.901 0.884 0.906 0.884 0.881 0.897 0.883 0.898 0.881 0.890 0.887 0.886 0.887 0.877 0.894 0.880 0.887 0.889 0.878 0.883 0.881 0.889 0.883 0.881 0.880 0.881 0.883 0.875 0.888 0.887 0.880 0.885 0.881 0.878 0.880 0.884 0.881 0.882 0.878 0.879 
215  0.694 0.805 0.887 0.924 0.873 0.884 0.879 0.891 0.888 0.861 0.896 0.858 0.884 0.886 0.854 0.888 0.853 0.886 0.872 0.877 0.879 0.867 0.879 0.864 0.878 0.865 0.877 0.862 0.869 0.864 0.870 0.870 0.858 0.874 0.858 0.876 0.860 0.872 0.862 0.858 0.870 0.861 0.866 0.865 0.864 0.866 0.870 0.862 0.866 
216  0.856 0.941 0.971 0.939 0.969 0.945 0.970 0.953 0.976 0.980 0.976 0.984 0.974 0.984 0.980 0.985 0.979 0.986 0.978 0.985 0.984 0.987 0.986 0.986 0.987 0.987 0.990 0.986 0.988 0.986 0.985 0.988 0.988 0.987 0.988 0.987 0.987 0.986 0.986 0.986 0.986 0.986 0.984 0.988 0.985 0.985 0.983 0.986 0.984 
300  0.932 0.949 0.964 0.964 0.961 0.956 0.955 0.956 0.952 0.955 0.957 0.955 0.956 0.955 0.954 0.955 0.955 0.953 0.955 0.958 0.952 0.958 0.959 0.959 0.956 0.955 0.955 0.960 0.957 0.956 0.957 0.960 0.958 0.960 0.959 0.957 0.957 0.960 0.962 0.960 0.961 0.958 0.959 0.961 0.960 0.958 0.959 0.959 0.961 
301  0.899 0.932 0.962 0.952 0.960 0.953 0.961 0.962 0.962 0.965 0.966 0.964 0.967 0.970 0.970 0.970 0.973 0.974 0.976 0.976 0.977 0.978 0.981 0.980 0.982 0.980 0.981 0.983 0.985 0.985 0.987 0.987 0.987 0.987 0.990 0.989 0.990 0.990 0.992 0.991 0.991 0.991 0.991 0.994 0.993 0.992 0.993 0.994 0.994 
302  0.862 0.924 0.965 0.950 0.964 0.953 0.966 0.956 0.962 0.967 0.970 0.970 0.972 0.972 0.977 0.978 0.976 0.981 0.980 0.984 0.982 0.985 0.985 0.988 0.987 0.989 0.989 0.990 0.989 0.992 0.992 0.992 0.993 0.993 0.994 0.993 0.995 0.994 0.997 0.995 0.996 0.996 0.996 0.996 0.997 0.996 0.997 0.997 0.997 
302  0.902 0.929 0.960 0.953 0.961 0.958 0.959 0.956 0.963 0.961 0.963 0.965 0.967 0.972 0.969 0.971 0.972 0.973 0.975 0.977 0.979 0.979 0.979 0.979 0.980 0.982 0.984 0.985 0.987 0.985 0.984 0.986 0.988 0.988 0.989 0.989 0.989 0.990 0.991 0.992 0.992 0.992 0.992 0.991 0.993 0.993 0.994 0.993 0.994 
304  0.903 0.920 0.936 0.959 0.953 0.937 0.945 0.935 0.929 0.936 0.925 0.929 0.925 0.917 0.921 0.914 0.917 0.911 0.911 0.909 0.902 0.906 0.906 0.903 0.904 0.898 0.903 0.905 0.899 0.896 0.900 0.894 0.898 0.897 0.895 0.894 0.890 0.892 0.893 0.897 0.898 0.894 0.887 0.892 0.895 0.892 0.895 0.885 0.888 
305  0.938 0.967 0.980 0.979 0.985 0.986 0.987 0.986 0.990 0.987 0.991 0.990 0.993 0.992 0.993 0.994 0.993 0.995 0.994 0.994 0.995 0.996 0.995 0.993 0.994 0.995 0.995 0.994 0.994 0.995 0.995 0.994 0.994 0.993 0.993 0.991 0.993 0.993 0.991 0.992 0.993 0.992 0.989 0.991 0.988 0.987 0.988 0.989 0.988 
306  0.927 0.957 0.977 0.971 0.979 0.974 0.975 0.974 0.977 0.976 0.978 0.977 0.980 0.980 0.982 0.983 0.979 0.981 0.983 0.981 0.980 0.984 0.982 0.982 0.982 0.980 0.982 0.982 0.984 0.982 0.985 0.981 0.983 0.981 0.981 0.983 0.977 0.978 0.981 0.978 0.981 0.979 0.979 0.981 0.976 0.978 0.978 0.980 0.979 
307  0.870 0.895 0.925 0.935 0.930 0.941 0.925 0.929 0.932 0.926 0.927 0.928 0.925 0.927 0.926 0.930 0.924 0.928 0.934 0.928 0.927 0.928 0.925 0.921 0.930 0.927 0.924 0.926 0.923 0.922 0.923 0.926 0.926 0.925 0.927 0.927 0.921 0.921 0.919 0.922 0.925 0.921 0.925 0.920 0.921 0.917 0.919 0.922 0.918 
400  0.930 0.949 0.960 0.964 0.962 0.954 0.958 0.957 0.955 0.948 0.954 0.952 0.952 0.954 0.949 0.949 0.946 0.952 0.950 0.948 0.949 0.946 0.949 0.949 0.947 0.949 0.946 0.942 0.944 0.943 0.942 0.942 0.945 0.945 0.944 0.944 0.947 0.944 0.943 0.943 0.939 0.940 0.939 0.938 0.947 0.940 0.944 0.942 0.940 
401  0.910 0.949 0.954 0.966 0.964 0.959 0.958 0.952 0.956 0.953 0.952 0.949 0.951 0.947 0.950 0.950 0.946 0.949 0.945 0.945 0.939 0.943 0.942 0.942 0.937 0.934 0.935 0.935 0.936 0.932 0.931 0.931 0.929 0.929 0.931 0.925 0.929 0.922 0.922 0.931 0.923 0.925 0.921 0.919 0.919 0.920 0.918 0.919 0.922 
402  0.873 0.944 0.955 0.969 0.966 0.965 0.965 0.960 0.958 0.961 0.957 0.954 0.958 0.952 0.950 0.949 0.950 0.947 0.947 0.945 0.941 0.945 0.939 0.940 0.936 0.931 0.933 0.930 0.928 0.922 0.920 0.919 0.918 0.918 0.917 0.912 0.909 0.911 0.903 0.908 0.913 0.904 0.902 0.899 0.899 0.901 0.899 0.904 0.894 
403  0.915 0.950 0.957 0.965 0.961 0.954 0.961 0.953 0.954 0.955 0.951 0.950 0.950 0.948 0.949 0.947 0.947 0.947 0.950 0.943 0.943 0.943 0.940 0.938 0.939 0.940 0.938 0.939 0.935 0.933 0.932 0.934 0.933 0.926 0.927 0.927 0.928 0.921 0.929 0.924 0.921 0.924 0.917 0.915 0.917 0.922 0.918 0.920 0.918

As we can see, for most simple distributions, the Maritz-Jarrett method produces very accurate confidence intervals even for extremely small samples. However, in some tricky cases (e.g., 202, 203, 204, 205, 206, 214, 215, 304 307), the prediction about coverage percentage of confidence intervals is not valid even for big sample sizes.

Next, let’s look at the $70^\textrm{th}$ percentile:

                              Quantile = 0.70, ConfidenceLevel = 95%
Dist    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50
0    0.910 0.908 0.929 0.943 0.951 0.951 0.957 0.956 0.957 0.953 0.954 0.951 0.953 0.951 0.952 0.953 0.950 0.952 0.953 0.951 0.949 0.952 0.950 0.952 0.951 0.951 0.947 0.951 0.954 0.952 0.949 0.949 0.951 0.950 0.948 0.948 0.946 0.950 0.947 0.950 0.952 0.945 0.951 0.951 0.945 0.950 0.948 0.946 0.948 
1    0.902 0.910 0.926 0.942 0.948 0.950 0.955 0.954 0.953 0.952 0.949 0.952 0.947 0.948 0.949 0.953 0.954 0.949 0.949 0.952 0.947 0.951 0.952 0.949 0.949 0.952 0.949 0.950 0.949 0.950 0.950 0.950 0.946 0.950 0.954 0.946 0.951 0.946 0.948 0.944 0.943 0.945 0.946 0.947 0.949 0.950 0.949 0.950 0.948 
2    0.897 0.915 0.933 0.944 0.953 0.957 0.957 0.960 0.956 0.955 0.958 0.957 0.960 0.958 0.960 0.957 0.953 0.960 0.951 0.951 0.954 0.950 0.951 0.952 0.952 0.957 0.951 0.950 0.946 0.954 0.952 0.947 0.949 0.947 0.944 0.951 0.953 0.944 0.947 0.945 0.947 0.947 0.950 0.951 0.950 0.946 0.949 0.946 0.944 
100  0.923 0.934 0.940 0.949 0.956 0.959 0.963 0.966 0.964 0.960 0.966 0.966 0.964 0.959 0.959 0.958 0.959 0.957 0.960 0.961 0.955 0.958 0.958 0.955 0.957 0.954 0.957 0.956 0.954 0.957 0.954 0.956 0.954 0.956 0.955 0.954 0.954 0.950 0.953 0.958 0.954 0.953 0.957 0.953 0.951 0.949 0.952 0.952 0.954 
101  0.923 0.931 0.942 0.951 0.960 0.961 0.967 0.967 0.967 0.963 0.965 0.959 0.963 0.961 0.964 0.960 0.960 0.958 0.957 0.958 0.957 0.956 0.959 0.956 0.957 0.957 0.958 0.955 0.957 0.956 0.952 0.954 0.956 0.957 0.954 0.956 0.957 0.954 0.955 0.952 0.952 0.952 0.956 0.950 0.955 0.955 0.957 0.955 0.953 
102  0.925 0.937 0.943 0.952 0.958 0.962 0.964 0.967 0.968 0.967 0.965 0.964 0.962 0.963 0.963 0.960 0.959 0.961 0.959 0.957 0.958 0.955 0.957 0.958 0.956 0.959 0.960 0.956 0.957 0.958 0.955 0.958 0.957 0.951 0.955 0.958 0.953 0.952 0.956 0.955 0.954 0.954 0.955 0.955 0.954 0.952 0.956 0.953 0.954 
103  0.925 0.935 0.943 0.950 0.961 0.967 0.969 0.967 0.970 0.970 0.972 0.971 0.970 0.967 0.968 0.967 0.966 0.965 0.965 0.963 0.967 0.963 0.960 0.960 0.964 0.962 0.958 0.959 0.960 0.959 0.956 0.958 0.958 0.957 0.960 0.962 0.956 0.960 0.954 0.957 0.958 0.959 0.958 0.956 0.956 0.953 0.957 0.960 0.953 
104  0.923 0.936 0.946 0.944 0.958 0.960 0.964 0.966 0.968 0.964 0.966 0.964 0.962 0.961 0.956 0.959 0.959 0.957 0.960 0.960 0.957 0.958 0.957 0.961 0.959 0.956 0.956 0.954 0.955 0.953 0.956 0.953 0.955 0.957 0.953 0.954 0.955 0.952 0.954 0.956 0.957 0.955 0.957 0.955 0.955 0.956 0.954 0.956 0.957 
111  0.926 0.931 0.946 0.951 0.958 0.960 0.964 0.966 0.968 0.962 0.966 0.964 0.962 0.962 0.963 0.960 0.963 0.960 0.960 0.962 0.961 0.960 0.955 0.958 0.957 0.954 0.956 0.956 0.957 0.957 0.956 0.951 0.953 0.957 0.955 0.955 0.953 0.953 0.956 0.952 0.958 0.953 0.955 0.952 0.952 0.951 0.953 0.954 0.953 
130  0.864 0.859 0.888 0.921 0.935 0.947 0.957 0.959 0.963 0.968 0.964 0.969 0.969 0.968 0.969 0.965 0.962 0.966 0.965 0.966 0.962 0.964 0.965 0.961 0.961 0.959 0.960 0.962 0.959 0.956 0.960 0.960 0.960 0.962 0.959 0.957 0.957 0.961 0.957 0.956 0.958 0.958 0.959 0.955 0.953 0.955 0.955 0.956 0.958 
131  0.874 0.868 0.897 0.922 0.938 0.947 0.956 0.962 0.965 0.963 0.964 0.964 0.964 0.963 0.959 0.959 0.961 0.963 0.960 0.960 0.957 0.958 0.960 0.958 0.956 0.956 0.957 0.955 0.959 0.957 0.961 0.957 0.957 0.959 0.954 0.958 0.958 0.955 0.958 0.955 0.953 0.957 0.956 0.950 0.956 0.957 0.953 0.955 0.955 
140  0.925 0.933 0.946 0.957 0.958 0.965 0.967 0.964 0.968 0.968 0.968 0.961 0.966 0.964 0.963 0.963 0.964 0.965 0.965 0.960 0.963 0.961 0.955 0.960 0.958 0.955 0.961 0.958 0.956 0.955 0.958 0.952 0.955 0.954 0.956 0.955 0.952 0.955 0.954 0.951 0.955 0.953 0.956 0.957 0.958 0.950 0.957 0.948 0.955 
141  0.924 0.934 0.939 0.952 0.962 0.966 0.966 0.965 0.964 0.970 0.969 0.967 0.964 0.964 0.966 0.967 0.967 0.965 0.961 0.967 0.959 0.962 0.960 0.960 0.960 0.958 0.955 0.957 0.958 0.956 0.957 0.957 0.954 0.952 0.954 0.952 0.957 0.956 0.959 0.959 0.956 0.956 0.953 0.958 0.955 0.953 0.955 0.952 0.953 
142  0.927 0.934 0.946 0.954 0.960 0.963 0.965 0.969 0.970 0.974 0.967 0.966 0.966 0.967 0.964 0.970 0.969 0.964 0.968 0.966 0.964 0.967 0.961 0.969 0.963 0.964 0.963 0.957 0.958 0.956 0.956 0.955 0.960 0.956 0.955 0.956 0.952 0.956 0.958 0.953 0.952 0.955 0.954 0.951 0.951 0.955 0.959 0.955 0.950 
143  0.923 0.931 0.942 0.950 0.959 0.961 0.966 0.970 0.969 0.971 0.972 0.967 0.971 0.969 0.968 0.969 0.967 0.968 0.967 0.961 0.966 0.961 0.965 0.958 0.961 0.957 0.960 0.958 0.961 0.960 0.957 0.957 0.960 0.957 0.958 0.958 0.956 0.956 0.951 0.957 0.952 0.958 0.957 0.954 0.958 0.955 0.954 0.953 0.957 
144  0.929 0.931 0.943 0.953 0.961 0.965 0.970 0.970 0.971 0.973 0.973 0.973 0.972 0.977 0.971 0.971 0.970 0.971 0.968 0.970 0.966 0.964 0.967 0.964 0.965 0.964 0.964 0.963 0.957 0.964 0.959 0.961 0.961 0.957 0.959 0.959 0.957 0.958 0.960 0.959 0.960 0.957 0.955 0.956 0.955 0.950 0.955 0.955 0.957 
145  0.926 0.933 0.942 0.953 0.962 0.966 0.968 0.968 0.968 0.967 0.967 0.969 0.969 0.969 0.969 0.968 0.966 0.969 0.964 0.964 0.962 0.964 0.962 0.964 0.960 0.962 0.956 0.960 0.959 0.960 0.959 0.959 0.958 0.956 0.958 0.958 0.954 0.958 0.957 0.956 0.953 0.953 0.953 0.955 0.954 0.952 0.956 0.954 0.952 
146  0.922 0.929 0.942 0.951 0.958 0.965 0.969 0.976 0.972 0.970 0.973 0.971 0.969 0.971 0.968 0.971 0.966 0.967 0.965 0.969 0.965 0.966 0.961 0.965 0.964 0.960 0.960 0.960 0.961 0.960 0.961 0.956 0.958 0.961 0.956 0.961 0.956 0.956 0.959 0.957 0.955 0.955 0.956 0.957 0.958 0.956 0.959 0.954 0.954 
147  0.927 0.933 0.947 0.957 0.959 0.964 0.966 0.970 0.971 0.971 0.972 0.970 0.969 0.975 0.972 0.965 0.968 0.967 0.965 0.962 0.966 0.963 0.967 0.962 0.962 0.962 0.961 0.960 0.962 0.963 0.959 0.958 0.957 0.959 0.958 0.957 0.957 0.960 0.956 0.960 0.956 0.955 0.954 0.953 0.952 0.959 0.957 0.952 0.957 
148  0.927 0.934 0.945 0.953 0.959 0.963 0.970 0.971 0.972 0.974 0.972 0.974 0.971 0.970 0.969 0.970 0.970 0.969 0.966 0.964 0.967 0.963 0.963 0.961 0.963 0.963 0.961 0.962 0.962 0.958 0.959 0.957 0.958 0.955 0.958 0.957 0.955 0.957 0.957 0.954 0.954 0.960 0.956 0.958 0.953 0.957 0.957 0.955 0.956 
149  0.927 0.928 0.945 0.955 0.965 0.966 0.969 0.970 0.970 0.973 0.972 0.971 0.971 0.975 0.970 0.971 0.969 0.966 0.965 0.968 0.965 0.964 0.964 0.966 0.968 0.964 0.962 0.961 0.959 0.959 0.957 0.959 0.958 0.958 0.956 0.956 0.957 0.956 0.955 0.957 0.956 0.957 0.953 0.954 0.956 0.954 0.957 0.956 0.954 
150  0.926 0.933 0.946 0.953 0.962 0.966 0.971 0.971 0.971 0.971 0.975 0.972 0.974 0.973 0.971 0.972 0.969 0.967 0.969 0.965 0.966 0.965 0.965 0.960 0.967 0.967 0.961 0.964 0.965 0.961 0.962 0.959 0.962 0.958 0.958 0.955 0.954 0.957 0.956 0.956 0.954 0.956 0.956 0.956 0.957 0.955 0.956 0.952 0.953 
151  0.928 0.936 0.942 0.953 0.961 0.963 0.970 0.972 0.972 0.972 0.974 0.973 0.973 0.970 0.972 0.968 0.971 0.969 0.967 0.967 0.967 0.966 0.964 0.965 0.963 0.962 0.960 0.961 0.962 0.959 0.958 0.958 0.964 0.960 0.958 0.960 0.958 0.956 0.957 0.956 0.955 0.955 0.956 0.954 0.954 0.956 0.952 0.955 0.955 
152  0.924 0.935 0.945 0.952 0.962 0.965 0.970 0.970 0.972 0.973 0.974 0.970 0.973 0.972 0.971 0.972 0.972 0.971 0.969 0.967 0.964 0.968 0.964 0.965 0.965 0.959 0.963 0.960 0.962 0.963 0.960 0.959 0.962 0.962 0.954 0.960 0.960 0.960 0.956 0.955 0.956 0.958 0.954 0.957 0.957 0.957 0.956 0.955 0.957 
153  0.928 0.937 0.947 0.949 0.957 0.964 0.965 0.966 0.968 0.963 0.964 0.966 0.963 0.962 0.961 0.964 0.966 0.965 0.963 0.958 0.960 0.959 0.958 0.959 0.960 0.961 0.954 0.956 0.954 0.955 0.954 0.953 0.955 0.956 0.958 0.955 0.956 0.956 0.956 0.959 0.953 0.955 0.956 0.956 0.953 0.953 0.957 0.954 0.950 
154  0.921 0.933 0.943 0.952 0.956 0.962 0.965 0.966 0.967 0.966 0.967 0.961 0.963 0.961 0.962 0.961 0.962 0.961 0.959 0.959 0.959 0.958 0.960 0.958 0.959 0.958 0.957 0.957 0.953 0.956 0.956 0.954 0.952 0.952 0.953 0.954 0.953 0.953 0.954 0.956 0.955 0.953 0.955 0.957 0.955 0.957 0.955 0.956 0.957 
155  0.930 0.934 0.943 0.952 0.956 0.959 0.962 0.966 0.965 0.968 0.964 0.962 0.963 0.961 0.961 0.961 0.959 0.957 0.961 0.956 0.960 0.957 0.956 0.957 0.958 0.960 0.952 0.951 0.958 0.956 0.953 0.955 0.956 0.954 0.958 0.954 0.954 0.953 0.956 0.955 0.955 0.953 0.955 0.957 0.954 0.951 0.953 0.951 0.954 
200  0.912 0.913 0.935 0.947 0.955 0.961 0.963 0.960 0.963 0.961 0.965 0.962 0.963 0.960 0.963 0.959 0.962 0.961 0.959 0.960 0.959 0.958 0.956 0.957 0.959 0.958 0.959 0.959 0.958 0.959 0.960 0.960 0.960 0.959 0.958 0.956 0.958 0.957 0.957 0.957 0.956 0.958 0.960 0.957 0.958 0.954 0.957 0.956 0.957 
201  0.902 0.905 0.934 0.949 0.961 0.960 0.967 0.965 0.967 0.962 0.963 0.963 0.968 0.964 0.962 0.962 0.962 0.966 0.963 0.961 0.965 0.965 0.963 0.963 0.962 0.961 0.964 0.964 0.963 0.963 0.967 0.962 0.963 0.961 0.965 0.963 0.964 0.957 0.964 0.961 0.963 0.964 0.961 0.958 0.963 0.965 0.962 0.960 0.960 
202  0.878 0.888 0.932 0.955 0.964 0.970 0.966 0.963 0.968 0.971 0.971 0.973 0.975 0.972 0.975 0.974 0.976 0.975 0.974 0.974 0.975 0.971 0.976 0.977 0.975 0.974 0.978 0.976 0.974 0.976 0.973 0.976 0.974 0.976 0.972 0.975 0.977 0.975 0.974 0.975 0.974 0.975 0.973 0.978 0.974 0.974 0.972 0.973 0.973 
203  0.826 0.873 0.929 0.961 0.973 0.974 0.964 0.975 0.978 0.976 0.975 0.980 0.980 0.979 0.979 0.980 0.979 0.981 0.980 0.982 0.982 0.980 0.981 0.981 0.981 0.983 0.981 0.980 0.981 0.981 0.980 0.980 0.977 0.980 0.979 0.982 0.978 0.979 0.980 0.979 0.978 0.978 0.980 0.978 0.979 0.975 0.979 0.980 0.976 
204  0.877 0.888 0.933 0.956 0.968 0.970 0.969 0.972 0.973 0.977 0.975 0.976 0.977 0.976 0.977 0.974 0.976 0.974 0.978 0.979 0.975 0.976 0.977 0.978 0.975 0.974 0.976 0.980 0.978 0.975 0.976 0.976 0.972 0.974 0.974 0.974 0.975 0.977 0.974 0.975 0.973 0.974 0.975 0.975 0.978 0.973 0.972 0.975 0.977 
205  0.879 0.889 0.931 0.961 0.966 0.974 0.972 0.972 0.978 0.977 0.979 0.976 0.980 0.980 0.981 0.982 0.982 0.981 0.981 0.981 0.980 0.979 0.979 0.979 0.978 0.981 0.979 0.979 0.978 0.978 0.977 0.978 0.975 0.976 0.977 0.974 0.976 0.976 0.976 0.977 0.976 0.975 0.976 0.976 0.973 0.973 0.974 0.976 0.974 
206  0.880 0.899 0.946 0.961 0.973 0.976 0.975 0.978 0.982 0.981 0.982 0.982 0.985 0.986 0.985 0.987 0.984 0.985 0.986 0.987 0.985 0.985 0.984 0.985 0.984 0.986 0.983 0.983 0.984 0.981 0.980 0.981 0.981 0.979 0.980 0.977 0.979 0.977 0.979 0.978 0.974 0.981 0.978 0.977 0.978 0.975 0.973 0.975 0.976 
210  0.864 0.823 0.829 0.862 0.873 0.904 0.911 0.916 0.912 0.901 0.900 0.901 0.907 0.907 0.904 0.903 0.906 0.904 0.907 0.897 0.898 0.900 0.904 0.907 0.899 0.899 0.902 0.899 0.897 0.893 0.898 0.903 0.896 0.898 0.899 0.896 0.900 0.903 0.895 0.902 0.899 0.894 0.899 0.894 0.898 0.898 0.895 0.892 0.895 
211  0.928 0.958 0.975 0.983 0.983 0.983 0.985 0.983 0.984 0.980 0.980 0.981 0.978 0.975 0.975 0.977 0.971 0.973 0.968 0.971 0.969 0.967 0.969 0.967 0.967 0.969 0.966 0.965 0.966 0.968 0.964 0.964 0.964 0.960 0.962 0.963 0.961 0.959 0.962 0.958 0.959 0.959 0.960 0.962 0.958 0.959 0.961 0.956 0.958 
212  0.844 0.812 0.833 0.863 0.886 0.908 0.913 0.926 0.923 0.910 0.908 0.916 0.920 0.917 0.915 0.907 0.902 0.907 0.905 0.903 0.906 0.899 0.906 0.904 0.897 0.904 0.907 0.899 0.902 0.902 0.896 0.897 0.897 0.900 0.898 0.900 0.904 0.898 0.897 0.894 0.895 0.896 0.899 0.900 0.895 0.898 0.892 0.902 0.899 
213  0.924 0.961 0.977 0.982 0.985 0.987 0.985 0.983 0.986 0.983 0.984 0.980 0.982 0.979 0.976 0.978 0.978 0.978 0.974 0.976 0.974 0.974 0.968 0.972 0.967 0.968 0.966 0.966 0.966 0.967 0.963 0.960 0.966 0.965 0.958 0.966 0.957 0.963 0.958 0.964 0.959 0.957 0.960 0.961 0.960 0.958 0.958 0.961 0.955 
214  0.787 0.863 0.933 0.956 0.968 0.968 0.964 0.969 0.971 0.971 0.973 0.976 0.974 0.975 0.979 0.975 0.979 0.978 0.977 0.977 0.978 0.977 0.979 0.978 0.980 0.979 0.979 0.980 0.980 0.979 0.977 0.980 0.979 0.981 0.980 0.977 0.980 0.977 0.977 0.977 0.973 0.977 0.976 0.976 0.976 0.977 0.977 0.974 0.976 
215  0.726 0.859 0.932 0.963 0.972 0.971 0.964 0.970 0.978 0.979 0.972 0.979 0.980 0.977 0.980 0.984 0.982 0.980 0.982 0.985 0.983 0.982 0.981 0.985 0.982 0.983 0.984 0.981 0.983 0.983 0.981 0.979 0.982 0.982 0.982 0.980 0.981 0.982 0.980 0.980 0.978 0.980 0.979 0.981 0.980 0.976 0.978 0.977 0.978 
216  0.537 0.694 0.792 0.862 0.914 0.938 0.805 0.852 0.897 0.922 0.815 0.854 0.897 0.920 0.833 0.864 0.899 0.919 0.849 0.878 0.900 0.916 0.860 0.885 0.908 0.882 0.869 0.894 0.916 0.860 0.879 0.901 0.919 0.866 0.890 0.912 0.909 0.888 0.900 0.916 0.886 0.894 0.907 0.924 0.890 0.899 0.913 0.919 0.899 
300  0.903 0.915 0.931 0.940 0.944 0.959 0.958 0.953 0.952 0.952 0.953 0.948 0.944 0.947 0.944 0.946 0.946 0.943 0.946 0.949 0.943 0.942 0.942 0.943 0.948 0.946 0.947 0.941 0.941 0.938 0.940 0.937 0.938 0.936 0.940 0.942 0.940 0.939 0.945 0.938 0.936 0.937 0.938 0.939 0.937 0.940 0.939 0.935 0.936 
301  0.861 0.902 0.903 0.910 0.927 0.938 0.943 0.935 0.925 0.923 0.929 0.924 0.916 0.919 0.923 0.921 0.913 0.912 0.914 0.917 0.911 0.911 0.913 0.910 0.907 0.904 0.912 0.913 0.908 0.912 0.904 0.906 0.908 0.904 0.900 0.907 0.906 0.908 0.910 0.912 0.903 0.908 0.908 0.904 0.906 0.908 0.908 0.908 0.913 
302  0.811 0.891 0.890 0.895 0.923 0.933 0.931 0.902 0.903 0.920 0.918 0.896 0.901 0.913 0.908 0.897 0.894 0.904 0.909 0.892 0.898 0.906 0.899 0.888 0.901 0.904 0.895 0.889 0.901 0.899 0.895 0.896 0.903 0.901 0.888 0.902 0.904 0.899 0.901 0.901 0.905 0.899 0.897 0.905 0.901 0.893 0.901 0.911 0.908 
302  0.858 0.900 0.904 0.910 0.926 0.941 0.945 0.933 0.920 0.924 0.929 0.924 0.923 0.919 0.925 0.921 0.914 0.910 0.914 0.915 0.909 0.911 0.910 0.912 0.912 0.903 0.909 0.910 0.906 0.910 0.904 0.906 0.908 0.910 0.909 0.906 0.912 0.907 0.904 0.905 0.911 0.915 0.906 0.906 0.904 0.908 0.910 0.909 0.907 
304  0.851 0.872 0.919 0.942 0.956 0.959 0.958 0.963 0.968 0.970 0.970 0.972 0.967 0.970 0.976 0.974 0.979 0.979 0.980 0.979 0.980 0.984 0.980 0.984 0.982 0.984 0.988 0.987 0.986 0.986 0.990 0.988 0.989 0.990 0.990 0.989 0.991 0.993 0.991 0.994 0.989 0.992 0.993 0.993 0.993 0.992 0.992 0.994 0.994 
305  0.901 0.944 0.945 0.935 0.941 0.931 0.936 0.933 0.928 0.932 0.933 0.927 0.927 0.925 0.923 0.925 0.925 0.924 0.921 0.920 0.924 0.924 0.923 0.923 0.920 0.921 0.925 0.923 0.929 0.922 0.923 0.923 0.919 0.924 0.921 0.925 0.923 0.930 0.927 0.925 0.925 0.932 0.919 0.927 0.925 0.930 0.926 0.929 0.928 
306  0.873 0.875 0.876 0.885 0.906 0.926 0.925 0.912 0.896 0.905 0.913 0.912 0.899 0.902 0.907 0.907 0.897 0.902 0.904 0.905 0.899 0.901 0.901 0.900 0.903 0.900 0.902 0.908 0.898 0.902 0.904 0.902 0.897 0.899 0.902 0.901 0.900 0.901 0.900 0.904 0.903 0.903 0.901 0.899 0.906 0.903 0.903 0.904 0.900 
307  0.791 0.839 0.886 0.905 0.920 0.941 0.940 0.932 0.920 0.929 0.935 0.932 0.927 0.929 0.930 0.931 0.927 0.928 0.929 0.931 0.930 0.927 0.935 0.930 0.929 0.927 0.933 0.931 0.927 0.929 0.931 0.929 0.929 0.925 0.930 0.929 0.925 0.927 0.929 0.929 0.924 0.925 0.929 0.926 0.928 0.926 0.928 0.923 0.924 
400  0.911 0.915 0.934 0.942 0.955 0.952 0.957 0.957 0.956 0.953 0.955 0.953 0.953 0.948 0.949 0.952 0.951 0.952 0.949 0.950 0.946 0.947 0.950 0.947 0.944 0.946 0.947 0.951 0.948 0.949 0.949 0.949 0.949 0.949 0.950 0.946 0.948 0.947 0.948 0.946 0.951 0.947 0.947 0.949 0.947 0.951 0.947 0.944 0.948 
401  0.883 0.918 0.929 0.942 0.949 0.946 0.951 0.945 0.950 0.949 0.944 0.943 0.940 0.944 0.945 0.945 0.942 0.944 0.942 0.943 0.941 0.943 0.943 0.940 0.945 0.944 0.942 0.944 0.944 0.945 0.946 0.944 0.944 0.950 0.947 0.945 0.947 0.943 0.947 0.948 0.946 0.945 0.946 0.946 0.948 0.949 0.946 0.941 0.944 
402  0.852 0.918 0.920 0.927 0.945 0.935 0.936 0.945 0.940 0.935 0.938 0.930 0.936 0.932 0.935 0.931 0.935 0.935 0.930 0.935 0.934 0.939 0.940 0.934 0.939 0.936 0.937 0.935 0.942 0.935 0.937 0.940 0.935 0.943 0.942 0.937 0.942 0.946 0.942 0.947 0.944 0.946 0.943 0.946 0.945 0.947 0.948 0.947 0.947 
403  0.886 0.917 0.929 0.944 0.948 0.944 0.949 0.949 0.950 0.949 0.943 0.944 0.945 0.939 0.944 0.943 0.936 0.944 0.941 0.944 0.941 0.941 0.944 0.940 0.946 0.943 0.944 0.943 0.945 0.943 0.945 0.946 0.945 0.946 0.944 0.947 0.947 0.943 0.947 0.944 0.943 0.947 0.943 0.947 0.948 0.942 0.951 0.943 0.945

The coverage situation got worse. Now we need at least 4-5 elements in the sample to get an accurate prediction in simple cases. For some distributions (e.g., 305, 306), we are not able to get reliable predictions even on large sample sizes (while we got a good coverage percentage for the median).

Now it’s time for the $90^\textrm{th}$ percentile:

                              Quantile = 0.90, ConfidenceLevel = 95%
Dist    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50
0    0.744 0.699 0.723 0.758 0.798 0.821 0.850 0.867 0.877 0.893 0.904 0.911 0.922 0.931 0.930 0.939 0.940 0.941 0.945 0.945 0.947 0.949 0.948 0.951 0.953 0.950 0.947 0.952 0.952 0.949 0.953 0.953 0.952 0.954 0.949 0.951 0.953 0.950 0.948 0.948 0.950 0.954 0.950 0.948 0.949 0.954 0.949 0.949 0.949 
1    0.753 0.690 0.722 0.763 0.790 0.828 0.844 0.870 0.886 0.897 0.910 0.919 0.924 0.921 0.933 0.936 0.938 0.938 0.944 0.944 0.949 0.945 0.950 0.952 0.950 0.950 0.956 0.949 0.954 0.952 0.950 0.951 0.954 0.948 0.948 0.950 0.946 0.949 0.946 0.948 0.952 0.948 0.949 0.946 0.949 0.952 0.946 0.951 0.953 
2    0.749 0.700 0.721 0.769 0.796 0.829 0.850 0.871 0.887 0.899 0.910 0.918 0.928 0.936 0.937 0.948 0.948 0.950 0.953 0.955 0.956 0.952 0.961 0.961 0.964 0.963 0.963 0.962 0.963 0.963 0.964 0.964 0.966 0.969 0.965 0.963 0.969 0.964 0.966 0.968 0.968 0.968 0.967 0.966 0.972 0.970 0.967 0.970 0.971 
100  0.740 0.643 0.652 0.685 0.726 0.760 0.787 0.822 0.836 0.847 0.865 0.873 0.886 0.896 0.905 0.912 0.918 0.927 0.927 0.935 0.934 0.944 0.940 0.947 0.944 0.948 0.946 0.951 0.950 0.952 0.951 0.955 0.955 0.952 0.952 0.956 0.954 0.958 0.953 0.954 0.949 0.954 0.951 0.952 0.954 0.957 0.957 0.952 0.953 
101  0.740 0.630 0.649 0.679 0.725 0.750 0.780 0.804 0.831 0.852 0.862 0.875 0.888 0.898 0.909 0.921 0.917 0.919 0.931 0.936 0.939 0.937 0.941 0.945 0.945 0.952 0.946 0.954 0.952 0.949 0.951 0.952 0.952 0.957 0.951 0.949 0.951 0.957 0.951 0.952 0.954 0.951 0.954 0.954 0.953 0.953 0.951 0.952 0.954 
102  0.752 0.631 0.647 0.693 0.715 0.751 0.788 0.809 0.827 0.854 0.859 0.877 0.889 0.897 0.906 0.908 0.920 0.921 0.929 0.930 0.938 0.942 0.940 0.946 0.946 0.950 0.948 0.948 0.949 0.952 0.953 0.953 0.953 0.954 0.954 0.954 0.950 0.951 0.958 0.955 0.954 0.952 0.949 0.954 0.953 0.952 0.953 0.955 0.952 
103  0.729 0.617 0.629 0.662 0.698 0.737 0.767 0.798 0.823 0.842 0.855 0.871 0.887 0.893 0.907 0.913 0.919 0.927 0.933 0.934 0.941 0.942 0.946 0.949 0.952 0.952 0.958 0.961 0.962 0.962 0.961 0.961 0.962 0.963 0.964 0.966 0.966 0.965 0.966 0.967 0.968 0.965 0.967 0.965 0.969 0.969 0.966 0.968 0.966 
104  0.740 0.643 0.654 0.686 0.727 0.755 0.785 0.805 0.823 0.847 0.865 0.878 0.882 0.903 0.905 0.904 0.916 0.925 0.927 0.928 0.937 0.939 0.942 0.943 0.944 0.948 0.946 0.953 0.951 0.953 0.951 0.947 0.950 0.954 0.954 0.948 0.954 0.952 0.953 0.952 0.951 0.951 0.951 0.953 0.955 0.952 0.952 0.954 0.955 
111  0.744 0.634 0.648 0.693 0.724 0.768 0.787 0.808 0.825 0.847 0.866 0.875 0.896 0.896 0.907 0.914 0.924 0.923 0.929 0.930 0.939 0.936 0.943 0.944 0.948 0.945 0.952 0.950 0.954 0.954 0.955 0.953 0.955 0.957 0.954 0.953 0.955 0.954 0.954 0.955 0.956 0.956 0.951 0.950 0.950 0.951 0.952 0.953 0.954 
130  0.508 0.462 0.511 0.567 0.622 0.666 0.702 0.738 0.770 0.797 0.826 0.842 0.842 0.865 0.884 0.889 0.901 0.905 0.912 0.922 0.926 0.929 0.931 0.939 0.936 0.943 0.942 0.946 0.950 0.949 0.959 0.950 0.952 0.956 0.958 0.955 0.959 0.958 0.958 0.959 0.957 0.959 0.960 0.961 0.959 0.959 0.961 0.960 0.960 
131  0.563 0.500 0.549 0.608 0.634 0.684 0.733 0.761 0.786 0.810 0.834 0.845 0.865 0.874 0.888 0.890 0.903 0.914 0.917 0.919 0.929 0.931 0.938 0.938 0.941 0.943 0.944 0.946 0.950 0.949 0.955 0.953 0.950 0.956 0.957 0.957 0.952 0.957 0.956 0.959 0.955 0.955 0.952 0.954 0.954 0.958 0.956 0.959 0.955 
140  0.742 0.624 0.641 0.691 0.712 0.749 0.787 0.802 0.830 0.854 0.868 0.878 0.890 0.902 0.911 0.915 0.920 0.932 0.935 0.936 0.944 0.943 0.944 0.951 0.950 0.953 0.958 0.956 0.960 0.960 0.956 0.962 0.958 0.959 0.962 0.963 0.964 0.964 0.961 0.962 0.963 0.964 0.962 0.964 0.962 0.965 0.966 0.962 0.959 
141  0.743 0.625 0.638 0.691 0.719 0.756 0.780 0.813 0.833 0.856 0.867 0.880 0.891 0.905 0.915 0.919 0.925 0.925 0.931 0.936 0.943 0.949 0.949 0.950 0.956 0.952 0.958 0.954 0.956 0.959 0.957 0.962 0.960 0.960 0.962 0.960 0.965 0.964 0.964 0.962 0.964 0.966 0.963 0.966 0.963 0.967 0.968 0.967 0.966 
142  0.733 0.624 0.647 0.682 0.718 0.758 0.779 0.806 0.839 0.845 0.867 0.885 0.896 0.899 0.917 0.922 0.918 0.933 0.937 0.939 0.941 0.946 0.951 0.949 0.956 0.954 0.954 0.951 0.960 0.959 0.956 0.961 0.963 0.957 0.964 0.965 0.965 0.963 0.963 0.963 0.965 0.965 0.965 0.962 0.963 0.964 0.965 0.965 0.966 
143  0.732 0.622 0.637 0.672 0.719 0.751 0.782 0.802 0.825 0.855 0.864 0.882 0.892 0.905 0.914 0.917 0.924 0.924 0.933 0.934 0.945 0.945 0.953 0.953 0.955 0.956 0.956 0.961 0.961 0.962 0.964 0.967 0.965 0.962 0.965 0.966 0.966 0.969 0.967 0.970 0.967 0.965 0.970 0.969 0.967 0.964 0.972 0.969 0.972 
144  0.721 0.601 0.620 0.656 0.695 0.738 0.766 0.790 0.808 0.837 0.858 0.878 0.886 0.895 0.907 0.915 0.922 0.925 0.938 0.937 0.943 0.945 0.951 0.952 0.950 0.956 0.959 0.961 0.960 0.964 0.962 0.963 0.962 0.968 0.967 0.968 0.967 0.966 0.969 0.969 0.971 0.972 0.972 0.976 0.972 0.973 0.973 0.974 0.976 
145  0.727 0.621 0.640 0.687 0.710 0.755 0.779 0.796 0.819 0.838 0.866 0.883 0.886 0.900 0.911 0.915 0.922 0.928 0.935 0.939 0.947 0.950 0.947 0.951 0.950 0.956 0.955 0.960 0.962 0.961 0.965 0.965 0.965 0.962 0.966 0.967 0.966 0.967 0.967 0.968 0.965 0.968 0.969 0.970 0.969 0.968 0.974 0.970 0.973 
146  0.736 0.616 0.633 0.668 0.696 0.745 0.769 0.796 0.831 0.842 0.861 0.873 0.887 0.897 0.906 0.917 0.924 0.929 0.934 0.934 0.941 0.947 0.950 0.952 0.955 0.953 0.956 0.957 0.959 0.959 0.963 0.967 0.962 0.965 0.968 0.968 0.965 0.966 0.964 0.967 0.970 0.971 0.969 0.970 0.969 0.972 0.971 0.972 0.972 
147  0.729 0.611 0.623 0.662 0.701 0.732 0.773 0.794 0.816 0.836 0.856 0.874 0.888 0.894 0.906 0.914 0.917 0.923 0.930 0.938 0.944 0.948 0.950 0.950 0.956 0.957 0.958 0.962 0.954 0.963 0.962 0.965 0.967 0.965 0.967 0.965 0.968 0.966 0.968 0.967 0.971 0.969 0.968 0.970 0.970 0.970 0.972 0.972 0.972 
148  0.729 0.598 0.620 0.666 0.709 0.735 0.765 0.795 0.815 0.838 0.865 0.877 0.889 0.894 0.907 0.914 0.926 0.930 0.929 0.935 0.942 0.946 0.947 0.955 0.956 0.955 0.959 0.961 0.960 0.965 0.964 0.965 0.968 0.964 0.965 0.970 0.967 0.966 0.966 0.970 0.969 0.970 0.969 0.967 0.970 0.974 0.967 0.970 0.971 
149  0.733 0.614 0.627 0.668 0.710 0.741 0.765 0.788 0.811 0.839 0.863 0.875 0.883 0.897 0.906 0.910 0.917 0.925 0.931 0.940 0.942 0.942 0.948 0.947 0.953 0.952 0.960 0.960 0.959 0.959 0.964 0.963 0.967 0.968 0.964 0.967 0.968 0.968 0.972 0.972 0.969 0.967 0.970 0.967 0.973 0.969 0.971 0.971 0.972 
150  0.725 0.610 0.632 0.652 0.700 0.729 0.767 0.793 0.818 0.838 0.863 0.876 0.884 0.897 0.905 0.914 0.914 0.926 0.929 0.943 0.939 0.946 0.953 0.954 0.952 0.955 0.959 0.962 0.958 0.964 0.962 0.966 0.967 0.965 0.967 0.969 0.969 0.967 0.968 0.970 0.970 0.970 0.973 0.971 0.968 0.972 0.975 0.971 0.970 
151  0.726 0.610 0.616 0.650 0.706 0.738 0.756 0.792 0.815 0.839 0.861 0.870 0.877 0.900 0.907 0.914 0.921 0.925 0.930 0.938 0.943 0.947 0.950 0.954 0.954 0.953 0.960 0.956 0.962 0.963 0.960 0.966 0.964 0.967 0.967 0.966 0.970 0.969 0.971 0.969 0.970 0.969 0.974 0.970 0.972 0.970 0.975 0.971 0.975 
152  0.729 0.599 0.624 0.657 0.698 0.731 0.757 0.781 0.816 0.836 0.864 0.872 0.883 0.904 0.905 0.907 0.921 0.926 0.931 0.941 0.942 0.943 0.945 0.954 0.954 0.955 0.957 0.960 0.959 0.961 0.966 0.963 0.966 0.964 0.970 0.967 0.969 0.970 0.969 0.968 0.972 0.970 0.974 0.973 0.971 0.974 0.973 0.973 0.973 
153  0.743 0.631 0.646 0.681 0.725 0.764 0.784 0.810 0.829 0.853 0.864 0.880 0.891 0.900 0.913 0.915 0.921 0.929 0.930 0.932 0.937 0.941 0.945 0.945 0.944 0.951 0.951 0.954 0.955 0.956 0.956 0.953 0.954 0.956 0.958 0.958 0.956 0.959 0.957 0.958 0.958 0.960 0.958 0.959 0.955 0.959 0.959 0.961 0.962 
154  0.743 0.632 0.648 0.686 0.727 0.760 0.788 0.813 0.829 0.848 0.860 0.872 0.883 0.899 0.909 0.910 0.922 0.924 0.926 0.939 0.938 0.934 0.939 0.944 0.947 0.951 0.947 0.954 0.955 0.956 0.955 0.956 0.958 0.956 0.956 0.955 0.955 0.955 0.956 0.955 0.957 0.956 0.957 0.956 0.958 0.956 0.958 0.959 0.955 
155  0.744 0.641 0.645 0.686 0.723 0.750 0.773 0.807 0.826 0.841 0.865 0.877 0.885 0.898 0.907 0.913 0.920 0.922 0.928 0.935 0.939 0.940 0.944 0.943 0.948 0.952 0.946 0.951 0.952 0.958 0.953 0.947 0.952 0.958 0.955 0.958 0.953 0.952 0.955 0.955 0.955 0.955 0.949 0.957 0.952 0.955 0.952 0.953 0.955 
200  0.756 0.698 0.716 0.752 0.785 0.808 0.829 0.852 0.869 0.879 0.889 0.903 0.907 0.910 0.921 0.925 0.932 0.933 0.940 0.942 0.947 0.948 0.948 0.950 0.951 0.953 0.952 0.955 0.956 0.955 0.954 0.958 0.959 0.956 0.957 0.956 0.958 0.956 0.956 0.956 0.955 0.956 0.957 0.957 0.958 0.957 0.958 0.956 0.955 
201  0.760 0.724 0.745 0.776 0.806 0.827 0.843 0.863 0.878 0.890 0.895 0.905 0.916 0.924 0.929 0.926 0.931 0.935 0.943 0.943 0.946 0.947 0.950 0.954 0.954 0.955 0.955 0.955 0.959 0.959 0.954 0.957 0.956 0.957 0.957 0.959 0.960 0.960 0.953 0.958 0.956 0.955 0.956 0.958 0.952 0.957 0.959 0.959 0.957 
202  0.744 0.783 0.819 0.845 0.857 0.875 0.877 0.892 0.901 0.910 0.915 0.924 0.927 0.934 0.934 0.941 0.937 0.945 0.942 0.950 0.949 0.947 0.953 0.953 0.953 0.956 0.956 0.956 0.958 0.957 0.959 0.956 0.952 0.957 0.959 0.958 0.958 0.957 0.957 0.957 0.958 0.957 0.956 0.958 0.954 0.956 0.956 0.953 0.954 
203  0.714 0.835 0.873 0.902 0.913 0.918 0.925 0.931 0.932 0.932 0.938 0.945 0.941 0.942 0.948 0.951 0.948 0.952 0.954 0.953 0.953 0.954 0.956 0.955 0.960 0.957 0.961 0.957 0.960 0.959 0.957 0.959 0.957 0.959 0.954 0.959 0.957 0.957 0.953 0.953 0.959 0.957 0.957 0.956 0.955 0.958 0.951 0.957 0.954 
204  0.745 0.784 0.812 0.835 0.856 0.864 0.879 0.887 0.904 0.909 0.921 0.919 0.927 0.939 0.940 0.943 0.944 0.948 0.946 0.952 0.951 0.957 0.959 0.962 0.961 0.956 0.961 0.962 0.963 0.966 0.963 0.965 0.970 0.965 0.963 0.965 0.969 0.964 0.966 0.965 0.964 0.966 0.970 0.966 0.969 0.969 0.969 0.968 0.965 
205  0.756 0.785 0.809 0.828 0.848 0.865 0.875 0.884 0.895 0.906 0.918 0.920 0.932 0.934 0.937 0.941 0.950 0.950 0.950 0.955 0.956 0.958 0.962 0.962 0.964 0.962 0.965 0.966 0.969 0.965 0.970 0.967 0.970 0.972 0.967 0.967 0.969 0.968 0.970 0.969 0.971 0.972 0.973 0.972 0.974 0.969 0.973 0.972 0.972 
206  0.737 0.780 0.794 0.811 0.829 0.841 0.854 0.863 0.875 0.889 0.909 0.909 0.920 0.920 0.930 0.939 0.940 0.945 0.947 0.950 0.954 0.956 0.959 0.957 0.957 0.964 0.965 0.967 0.965 0.968 0.969 0.966 0.969 0.973 0.969 0.974 0.969 0.972 0.976 0.971 0.975 0.973 0.975 0.974 0.976 0.975 0.972 0.979 0.977 
210  0.597 0.639 0.704 0.769 0.817 0.859 0.889 0.908 0.915 0.935 0.936 0.949 0.950 0.958 0.963 0.960 0.963 0.970 0.968 0.972 0.970 0.973 0.972 0.973 0.972 0.972 0.977 0.973 0.972 0.977 0.975 0.974 0.976 0.974 0.975 0.973 0.970 0.970 0.972 0.971 0.974 0.973 0.973 0.972 0.971 0.972 0.972 0.971 0.969 
211  0.813 0.791 0.770 0.783 0.795 0.803 0.830 0.838 0.852 0.866 0.877 0.892 0.895 0.903 0.908 0.918 0.923 0.932 0.934 0.935 0.940 0.944 0.946 0.949 0.946 0.948 0.949 0.952 0.952 0.951 0.952 0.954 0.953 0.957 0.953 0.952 0.954 0.954 0.951 0.958 0.953 0.955 0.953 0.954 0.955 0.954 0.954 0.955 0.952 
212  0.600 0.638 0.712 0.780 0.829 0.863 0.888 0.904 0.922 0.936 0.945 0.952 0.954 0.963 0.967 0.967 0.969 0.972 0.972 0.977 0.977 0.977 0.980 0.978 0.980 0.982 0.979 0.982 0.983 0.983 0.983 0.982 0.982 0.980 0.981 0.981 0.983 0.979 0.979 0.980 0.980 0.982 0.981 0.981 0.980 0.978 0.981 0.983 0.979 
213  0.807 0.784 0.771 0.764 0.787 0.809 0.822 0.843 0.852 0.865 0.880 0.890 0.897 0.911 0.914 0.925 0.933 0.932 0.940 0.944 0.945 0.945 0.952 0.953 0.956 0.955 0.956 0.958 0.958 0.956 0.957 0.963 0.960 0.962 0.964 0.962 0.963 0.965 0.963 0.962 0.964 0.962 0.963 0.965 0.965 0.966 0.969 0.963 0.966 
214  0.704 0.804 0.844 0.871 0.881 0.896 0.901 0.913 0.918 0.927 0.929 0.928 0.930 0.933 0.942 0.945 0.944 0.947 0.952 0.952 0.956 0.954 0.954 0.958 0.958 0.959 0.961 0.956 0.959 0.959 0.965 0.963 0.960 0.961 0.960 0.961 0.959 0.964 0.960 0.960 0.958 0.958 0.956 0.958 0.959 0.959 0.958 0.960 0.958 
215  0.706 0.832 0.876 0.910 0.917 0.927 0.931 0.928 0.937 0.938 0.944 0.939 0.948 0.949 0.952 0.953 0.952 0.953 0.955 0.955 0.957 0.957 0.960 0.962 0.959 0.959 0.961 0.962 0.959 0.958 0.957 0.960 0.961 0.959 0.960 0.960 0.962 0.962 0.962 0.961 0.961 0.957 0.958 0.961 0.964 0.963 0.958 0.960 0.961 
216  0.531 0.691 0.804 0.864 0.915 0.939 0.955 0.970 0.978 0.985 0.987 0.990 0.991 0.993 0.994 0.993 0.993 0.993 0.992 0.991 0.992 0.991 0.991 0.991 0.988 0.991 0.991 0.991 0.991 0.993 0.990 0.989 0.988 0.987 0.987 0.986 0.987 0.987 0.986 0.984 0.986 0.984 0.983 0.985 0.983 0.981 0.983 0.982 0.980 
300  0.754 0.711 0.731 0.776 0.807 0.836 0.855 0.878 0.892 0.903 0.921 0.922 0.926 0.929 0.935 0.942 0.946 0.946 0.954 0.952 0.956 0.957 0.960 0.963 0.961 0.963 0.960 0.964 0.967 0.960 0.966 0.962 0.962 0.964 0.962 0.966 0.962 0.966 0.960 0.963 0.963 0.959 0.962 0.967 0.961 0.956 0.959 0.963 0.963 
301  0.772 0.731 0.781 0.828 0.878 0.907 0.928 0.942 0.947 0.959 0.963 0.963 0.969 0.974 0.968 0.971 0.973 0.973 0.973 0.973 0.975 0.977 0.978 0.976 0.976 0.976 0.976 0.975 0.975 0.976 0.978 0.973 0.974 0.974 0.972 0.976 0.975 0.971 0.972 0.969 0.972 0.970 0.970 0.968 0.969 0.968 0.968 0.970 0.967 
302  0.769 0.736 0.806 0.858 0.898 0.926 0.948 0.959 0.963 0.970 0.976 0.978 0.979 0.978 0.985 0.982 0.984 0.981 0.983 0.982 0.982 0.982 0.984 0.983 0.986 0.981 0.983 0.980 0.982 0.981 0.980 0.980 0.980 0.980 0.978 0.978 0.976 0.979 0.975 0.979 0.977 0.975 0.976 0.975 0.973 0.972 0.972 0.975 0.969 
302  0.771 0.727 0.791 0.840 0.873 0.902 0.926 0.940 0.945 0.954 0.960 0.964 0.966 0.967 0.970 0.970 0.974 0.974 0.970 0.973 0.972 0.977 0.976 0.977 0.975 0.978 0.976 0.976 0.977 0.973 0.973 0.975 0.972 0.976 0.975 0.971 0.977 0.971 0.970 0.972 0.970 0.971 0.968 0.968 0.972 0.968 0.967 0.969 0.964 
304  0.642 0.561 0.586 0.632 0.687 0.727 0.776 0.804 0.829 0.858 0.878 0.898 0.907 0.919 0.929 0.934 0.936 0.941 0.947 0.945 0.947 0.946 0.950 0.949 0.948 0.944 0.947 0.950 0.951 0.955 0.957 0.962 0.954 0.961 0.959 0.954 0.958 0.959 0.960 0.964 0.961 0.961 0.960 0.963 0.961 0.963 0.957 0.957 0.960 
305  0.710 0.613 0.672 0.744 0.789 0.836 0.873 0.895 0.917 0.931 0.944 0.948 0.955 0.959 0.964 0.966 0.970 0.969 0.971 0.976 0.974 0.977 0.977 0.976 0.979 0.976 0.978 0.977 0.979 0.979 0.978 0.979 0.981 0.980 0.980 0.981 0.978 0.979 0.980 0.981 0.977 0.978 0.978 0.978 0.978 0.979 0.977 0.977 0.978 
306  0.734 0.701 0.765 0.819 0.864 0.886 0.919 0.932 0.943 0.949 0.955 0.957 0.961 0.967 0.967 0.972 0.969 0.971 0.971 0.974 0.972 0.972 0.977 0.975 0.975 0.974 0.976 0.975 0.975 0.975 0.978 0.975 0.974 0.973 0.973 0.974 0.974 0.972 0.975 0.974 0.972 0.972 0.973 0.969 0.969 0.970 0.971 0.967 0.968 
307  0.694 0.738 0.794 0.850 0.888 0.913 0.934 0.951 0.955 0.963 0.967 0.973 0.972 0.975 0.976 0.976 0.980 0.977 0.980 0.977 0.978 0.981 0.978 0.980 0.980 0.977 0.978 0.980 0.979 0.979 0.981 0.976 0.978 0.977 0.977 0.974 0.976 0.973 0.973 0.976 0.974 0.973 0.974 0.970 0.969 0.969 0.970 0.970 0.968 
400  0.758 0.689 0.732 0.765 0.795 0.829 0.851 0.872 0.890 0.905 0.912 0.924 0.932 0.939 0.942 0.944 0.949 0.953 0.952 0.952 0.957 0.953 0.958 0.960 0.962 0.962 0.962 0.965 0.964 0.960 0.965 0.965 0.967 0.961 0.965 0.961 0.964 0.961 0.965 0.963 0.961 0.964 0.963 0.963 0.964 0.961 0.960 0.964 0.962 
401  0.755 0.706 0.732 0.780 0.819 0.856 0.881 0.900 0.915 0.926 0.941 0.945 0.950 0.956 0.964 0.963 0.965 0.964 0.973 0.973 0.973 0.971 0.971 0.975 0.975 0.973 0.973 0.975 0.975 0.977 0.976 0.975 0.976 0.975 0.976 0.976 0.977 0.973 0.977 0.978 0.977 0.978 0.976 0.975 0.976 0.976 0.975 0.972 0.976 
402  0.765 0.695 0.733 0.780 0.824 0.861 0.890 0.917 0.931 0.947 0.959 0.968 0.970 0.974 0.978 0.978 0.980 0.983 0.978 0.979 0.982 0.982 0.983 0.982 0.983 0.983 0.986 0.987 0.984 0.985 0.985 0.985 0.984 0.985 0.984 0.985 0.986 0.982 0.985 0.985 0.984 0.982 0.985 0.984 0.985 0.984 0.982 0.985 0.983 
403  0.760 0.696 0.732 0.778 0.816 0.851 0.877 0.899 0.921 0.933 0.938 0.945 0.951 0.958 0.961 0.967 0.964 0.966 0.968 0.969 0.972 0.971 0.974 0.971 0.975 0.975 0.976 0.974 0.979 0.974 0.977 0.976 0.974 0.975 0.978 0.974 0.976 0.974 0.973 0.977 0.978 0.977 0.976 0.973 0.974 0.975 0.975 0.973 0.975

The coverage picture significantly changed! Now we need at least 30 sample elements to get an accurate prediction in most cases.

Let’s compare it with confidence intervals around the $95^\textrm{th}$ percentile:

                              Quantile = 0.95, ConfidenceLevel = 95%
Dist    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50
0    0.623 0.539 0.560 0.601 0.641 0.667 0.714 0.722 0.746 0.773 0.788 0.797 0.818 0.830 0.848 0.856 0.866 0.874 0.882 0.890 0.895 0.900 0.905 0.907 0.912 0.914 0.917 0.923 0.924 0.925 0.930 0.932 0.936 0.938 0.938 0.939 0.943 0.941 0.945 0.943 0.948 0.946 0.946 0.947 0.948 0.947 0.947 0.950 0.954 
1    0.630 0.531 0.552 0.596 0.627 0.664 0.699 0.731 0.751 0.781 0.787 0.801 0.821 0.827 0.847 0.856 0.861 0.880 0.874 0.884 0.894 0.895 0.900 0.903 0.914 0.912 0.919 0.926 0.926 0.925 0.929 0.931 0.940 0.940 0.936 0.937 0.942 0.940 0.950 0.943 0.944 0.950 0.946 0.949 0.948 0.949 0.949 0.951 0.953 
2    0.624 0.513 0.529 0.565 0.585 0.606 0.627 0.660 0.721 0.774 0.792 0.795 0.816 0.832 0.844 0.853 0.863 0.869 0.882 0.890 0.892 0.900 0.912 0.914 0.928 0.924 0.929 0.931 0.935 0.938 0.943 0.946 0.947 0.942 0.946 0.951 0.953 0.955 0.959 0.959 0.959 0.961 0.964 0.963 0.964 0.967 0.966 0.969 0.968 
100  0.564 0.397 0.407 0.443 0.479 0.523 0.549 0.589 0.615 0.645 0.665 0.678 0.710 0.729 0.739 0.767 0.771 0.789 0.810 0.813 0.828 0.831 0.843 0.840 0.855 0.862 0.867 0.885 0.882 0.887 0.883 0.899 0.905 0.906 0.909 0.910 0.914 0.917 0.922 0.924 0.925 0.924 0.926 0.932 0.931 0.934 0.937 0.939 0.938 
101  0.556 0.401 0.405 0.449 0.473 0.515 0.563 0.575 0.614 0.631 0.660 0.686 0.707 0.732 0.743 0.764 0.776 0.794 0.797 0.816 0.825 0.830 0.843 0.849 0.857 0.869 0.871 0.876 0.875 0.882 0.898 0.898 0.893 0.905 0.910 0.915 0.912 0.918 0.919 0.921 0.923 0.929 0.929 0.930 0.933 0.933 0.937 0.936 0.941 
102  0.558 0.403 0.409 0.446 0.487 0.523 0.541 0.588 0.610 0.636 0.667 0.688 0.716 0.731 0.743 0.761 0.776 0.790 0.803 0.813 0.821 0.828 0.839 0.847 0.858 0.865 0.871 0.875 0.878 0.893 0.892 0.894 0.903 0.903 0.901 0.901 0.913 0.918 0.918 0.924 0.924 0.924 0.931 0.932 0.931 0.933 0.932 0.939 0.937 
103  0.517 0.339 0.338 0.370 0.398 0.447 0.483 0.519 0.545 0.584 0.602 0.633 0.659 0.671 0.699 0.719 0.729 0.754 0.765 0.782 0.791 0.801 0.819 0.820 0.831 0.841 0.844 0.856 0.862 0.867 0.867 0.884 0.881 0.888 0.894 0.898 0.896 0.905 0.909 0.911 0.916 0.919 0.919 0.918 0.927 0.931 0.924 0.934 0.938 
104  0.557 0.395 0.412 0.444 0.477 0.521 0.554 0.588 0.614 0.640 0.665 0.692 0.702 0.724 0.733 0.755 0.775 0.781 0.804 0.815 0.826 0.827 0.845 0.846 0.857 0.858 0.874 0.879 0.880 0.882 0.892 0.891 0.903 0.908 0.906 0.915 0.911 0.914 0.918 0.926 0.928 0.925 0.926 0.926 0.933 0.939 0.934 0.938 0.936 
111  0.559 0.394 0.411 0.446 0.484 0.524 0.551 0.583 0.615 0.652 0.666 0.691 0.704 0.731 0.738 0.755 0.773 0.797 0.796 0.814 0.818 0.831 0.840 0.850 0.854 0.863 0.871 0.874 0.881 0.886 0.897 0.896 0.900 0.902 0.909 0.905 0.914 0.918 0.919 0.922 0.926 0.929 0.929 0.932 0.936 0.931 0.931 0.940 0.940 
130  0.288 0.237 0.269 0.315 0.352 0.391 0.435 0.466 0.516 0.555 0.569 0.598 0.629 0.660 0.675 0.694 0.712 0.727 0.749 0.771 0.777 0.788 0.805 0.816 0.816 0.832 0.836 0.844 0.864 0.865 0.865 0.875 0.882 0.887 0.895 0.894 0.897 0.900 0.907 0.910 0.920 0.914 0.916 0.926 0.924 0.926 0.927 0.930 0.934 
131  0.345 0.279 0.311 0.354 0.393 0.432 0.479 0.513 0.535 0.572 0.608 0.627 0.662 0.674 0.693 0.709 0.726 0.755 0.771 0.783 0.797 0.806 0.813 0.823 0.837 0.842 0.844 0.861 0.867 0.867 0.879 0.891 0.880 0.893 0.893 0.902 0.906 0.907 0.911 0.911 0.915 0.920 0.920 0.928 0.930 0.928 0.930 0.934 0.931 
140  0.550 0.363 0.370 0.393 0.423 0.464 0.494 0.540 0.589 0.627 0.654 0.677 0.700 0.729 0.747 0.756 0.776 0.790 0.796 0.816 0.817 0.833 0.838 0.856 0.861 0.871 0.872 0.879 0.884 0.894 0.895 0.898 0.903 0.910 0.908 0.923 0.920 0.923 0.926 0.931 0.934 0.930 0.932 0.939 0.941 0.941 0.943 0.944 0.948 
141  0.558 0.357 0.365 0.385 0.430 0.458 0.485 0.508 0.576 0.643 0.662 0.677 0.701 0.719 0.743 0.755 0.768 0.784 0.802 0.811 0.822 0.836 0.839 0.857 0.871 0.869 0.881 0.889 0.893 0.892 0.898 0.902 0.907 0.910 0.916 0.920 0.923 0.926 0.929 0.929 0.930 0.935 0.938 0.935 0.939 0.941 0.944 0.943 0.950 
142  0.542 0.361 0.355 0.382 0.413 0.445 0.473 0.497 0.534 0.632 0.664 0.692 0.711 0.723 0.750 0.762 0.770 0.784 0.795 0.811 0.814 0.830 0.847 0.859 0.862 0.871 0.884 0.883 0.889 0.894 0.900 0.906 0.912 0.914 0.915 0.918 0.923 0.927 0.925 0.932 0.933 0.935 0.938 0.942 0.940 0.944 0.944 0.945 0.944 
143  0.532 0.329 0.331 0.353 0.377 0.411 0.443 0.497 0.555 0.610 0.635 0.655 0.685 0.708 0.721 0.738 0.754 0.759 0.777 0.795 0.805 0.820 0.830 0.844 0.851 0.862 0.870 0.879 0.882 0.884 0.895 0.892 0.898 0.907 0.911 0.913 0.914 0.923 0.925 0.924 0.932 0.933 0.926 0.937 0.936 0.938 0.947 0.943 0.947 
144  0.485 0.255 0.237 0.249 0.271 0.306 0.346 0.430 0.510 0.544 0.573 0.590 0.611 0.622 0.646 0.656 0.675 0.690 0.700 0.719 0.735 0.757 0.780 0.789 0.798 0.811 0.816 0.823 0.829 0.835 0.843 0.845 0.852 0.858 0.868 0.872 0.877 0.886 0.885 0.889 0.893 0.893 0.899 0.903 0.905 0.910 0.914 0.916 0.915 
145  0.521 0.311 0.309 0.325 0.349 0.363 0.420 0.494 0.559 0.600 0.624 0.656 0.676 0.689 0.700 0.714 0.720 0.733 0.743 0.748 0.757 0.778 0.797 0.822 0.841 0.848 0.854 0.859 0.864 0.875 0.869 0.879 0.880 0.874 0.877 0.883 0.886 0.894 0.904 0.910 0.917 0.923 0.926 0.924 0.920 0.929 0.929 0.928 0.936 
146  0.507 0.313 0.311 0.333 0.362 0.415 0.448 0.506 0.541 0.575 0.588 0.629 0.642 0.663 0.691 0.702 0.716 0.721 0.743 0.762 0.780 0.789 0.802 0.818 0.817 0.833 0.840 0.848 0.853 0.856 0.873 0.867 0.876 0.878 0.888 0.892 0.890 0.896 0.906 0.903 0.906 0.913 0.916 0.917 0.916 0.922 0.927 0.924 0.931 
147  0.506 0.308 0.293 0.319 0.351 0.398 0.451 0.491 0.526 0.564 0.590 0.615 0.632 0.649 0.675 0.690 0.703 0.723 0.736 0.754 0.769 0.789 0.796 0.815 0.820 0.827 0.833 0.843 0.843 0.853 0.859 0.866 0.875 0.875 0.881 0.890 0.889 0.896 0.892 0.908 0.904 0.905 0.912 0.917 0.921 0.922 0.918 0.921 0.931 
148  0.505 0.295 0.287 0.304 0.331 0.387 0.436 0.480 0.527 0.568 0.585 0.617 0.630 0.651 0.668 0.682 0.697 0.709 0.732 0.743 0.758 0.781 0.786 0.799 0.809 0.824 0.831 0.836 0.845 0.859 0.858 0.860 0.864 0.874 0.877 0.888 0.891 0.894 0.898 0.904 0.902 0.906 0.913 0.915 0.923 0.915 0.919 0.923 0.928 
149  0.499 0.301 0.278 0.296 0.330 0.369 0.420 0.466 0.525 0.563 0.582 0.605 0.627 0.643 0.664 0.676 0.695 0.712 0.722 0.747 0.762 0.775 0.789 0.810 0.810 0.830 0.829 0.841 0.849 0.843 0.859 0.857 0.867 0.872 0.877 0.889 0.888 0.889 0.899 0.897 0.908 0.909 0.909 0.911 0.915 0.913 0.917 0.926 0.920 
150  0.494 0.280 0.269 0.285 0.324 0.351 0.404 0.471 0.530 0.549 0.587 0.596 0.624 0.637 0.660 0.671 0.695 0.704 0.716 0.741 0.759 0.771 0.792 0.798 0.810 0.815 0.828 0.838 0.846 0.850 0.848 0.861 0.864 0.871 0.876 0.880 0.885 0.887 0.893 0.899 0.896 0.906 0.910 0.913 0.914 0.914 0.918 0.926 0.917 
151  0.496 0.288 0.267 0.275 0.317 0.341 0.409 0.474 0.512 0.555 0.577 0.596 0.626 0.633 0.660 0.670 0.683 0.699 0.708 0.735 0.749 0.764 0.797 0.795 0.814 0.818 0.818 0.830 0.843 0.843 0.849 0.852 0.855 0.870 0.874 0.881 0.889 0.894 0.889 0.898 0.899 0.907 0.907 0.910 0.911 0.916 0.918 0.917 0.921 
152  0.491 0.278 0.260 0.278 0.302 0.337 0.399 0.454 0.513 0.553 0.577 0.599 0.613 0.637 0.660 0.669 0.687 0.695 0.716 0.728 0.754 0.765 0.789 0.793 0.809 0.822 0.821 0.832 0.836 0.837 0.848 0.860 0.858 0.864 0.872 0.879 0.882 0.886 0.897 0.896 0.897 0.901 0.901 0.904 0.906 0.910 0.917 0.918 0.919 
153  0.564 0.388 0.378 0.413 0.454 0.493 0.520 0.552 0.596 0.631 0.668 0.685 0.709 0.726 0.739 0.764 0.782 0.786 0.801 0.817 0.826 0.834 0.841 0.855 0.863 0.869 0.879 0.884 0.886 0.892 0.899 0.901 0.901 0.911 0.915 0.916 0.916 0.920 0.925 0.925 0.932 0.929 0.931 0.933 0.934 0.941 0.938 0.940 0.940 
154  0.556 0.389 0.400 0.435 0.475 0.499 0.537 0.561 0.618 0.640 0.666 0.694 0.711 0.725 0.745 0.770 0.770 0.778 0.796 0.813 0.822 0.833 0.846 0.859 0.860 0.867 0.875 0.880 0.875 0.883 0.891 0.900 0.897 0.906 0.908 0.912 0.923 0.920 0.925 0.924 0.922 0.929 0.934 0.930 0.939 0.938 0.937 0.942 0.939 
155  0.562 0.398 0.404 0.447 0.478 0.512 0.549 0.575 0.606 0.628 0.666 0.679 0.708 0.722 0.748 0.763 0.778 0.793 0.799 0.815 0.826 0.829 0.847 0.851 0.864 0.866 0.873 0.875 0.886 0.890 0.899 0.897 0.905 0.907 0.905 0.914 0.917 0.919 0.922 0.923 0.926 0.929 0.925 0.930 0.931 0.932 0.938 0.936 0.941 
200  0.611 0.486 0.500 0.509 0.548 0.578 0.601 0.638 0.650 0.682 0.700 0.718 0.731 0.753 0.770 0.786 0.793 0.815 0.814 0.827 0.838 0.846 0.855 0.860 0.866 0.876 0.875 0.880 0.894 0.894 0.902 0.906 0.906 0.911 0.917 0.920 0.918 0.926 0.925 0.925 0.927 0.930 0.927 0.934 0.932 0.941 0.930 0.941 0.944 
201  0.637 0.516 0.517 0.540 0.562 0.595 0.613 0.640 0.671 0.691 0.706 0.718 0.745 0.755 0.773 0.789 0.803 0.814 0.814 0.828 0.845 0.851 0.853 0.864 0.869 0.876 0.883 0.886 0.889 0.890 0.901 0.907 0.907 0.912 0.911 0.922 0.919 0.923 0.924 0.928 0.927 0.934 0.934 0.935 0.938 0.941 0.944 0.942 0.938 
202  0.653 0.612 0.591 0.612 0.625 0.634 0.654 0.674 0.685 0.708 0.725 0.742 0.751 0.770 0.780 0.782 0.802 0.814 0.829 0.831 0.845 0.845 0.857 0.861 0.866 0.881 0.886 0.884 0.888 0.893 0.900 0.903 0.907 0.913 0.912 0.918 0.917 0.924 0.924 0.931 0.931 0.933 0.933 0.936 0.938 0.933 0.938 0.939 0.938 
203  0.666 0.709 0.709 0.715 0.715 0.723 0.725 0.727 0.728 0.740 0.741 0.761 0.768 0.776 0.790 0.803 0.809 0.821 0.821 0.836 0.836 0.845 0.858 0.859 0.867 0.874 0.879 0.887 0.899 0.888 0.900 0.905 0.913 0.912 0.913 0.915 0.918 0.925 0.923 0.924 0.929 0.928 0.932 0.935 0.936 0.934 0.940 0.935 0.939 
204  0.658 0.560 0.550 0.553 0.567 0.575 0.585 0.619 0.659 0.687 0.708 0.726 0.742 0.760 0.774 0.785 0.797 0.809 0.820 0.826 0.840 0.851 0.866 0.872 0.879 0.881 0.890 0.892 0.900 0.896 0.901 0.906 0.911 0.919 0.921 0.925 0.929 0.928 0.928 0.936 0.936 0.936 0.943 0.944 0.944 0.945 0.949 0.949 0.948 
205  0.656 0.529 0.497 0.497 0.502 0.514 0.525 0.558 0.617 0.676 0.686 0.706 0.721 0.734 0.748 0.757 0.774 0.790 0.803 0.807 0.818 0.839 0.847 0.861 0.867 0.867 0.880 0.881 0.890 0.890 0.895 0.903 0.906 0.908 0.917 0.916 0.923 0.927 0.936 0.928 0.934 0.944 0.938 0.941 0.941 0.942 0.950 0.944 0.944 
206  0.633 0.452 0.394 0.364 0.379 0.376 0.391 0.444 0.558 0.582 0.604 0.624 0.640 0.649 0.666 0.677 0.686 0.695 0.713 0.725 0.748 0.761 0.789 0.803 0.808 0.816 0.816 0.825 0.834 0.837 0.843 0.848 0.859 0.851 0.869 0.871 0.877 0.889 0.891 0.898 0.896 0.894 0.900 0.903 0.903 0.905 0.910 0.910 0.914 
210  0.499 0.527 0.580 0.627 0.671 0.701 0.741 0.757 0.775 0.799 0.811 0.822 0.833 0.830 0.856 0.855 0.861 0.872 0.874 0.878 0.886 0.890 0.896 0.902 0.901 0.905 0.910 0.917 0.910 0.917 0.924 0.925 0.926 0.932 0.930 0.934 0.934 0.932 0.929 0.941 0.941 0.939 0.944 0.943 0.945 0.943 0.948 0.945 0.952 
211  0.707 0.550 0.515 0.523 0.535 0.561 0.576 0.610 0.634 0.657 0.677 0.705 0.722 0.734 0.762 0.773 0.788 0.795 0.810 0.822 0.834 0.837 0.839 0.855 0.862 0.867 0.874 0.880 0.885 0.888 0.892 0.899 0.907 0.911 0.913 0.911 0.916 0.919 0.920 0.925 0.926 0.927 0.930 0.929 0.933 0.934 0.942 0.938 0.941 
212  0.500 0.501 0.550 0.586 0.617 0.655 0.671 0.704 0.746 0.772 0.797 0.812 0.818 0.831 0.837 0.847 0.855 0.867 0.870 0.880 0.883 0.889 0.895 0.901 0.910 0.910 0.912 0.917 0.922 0.928 0.928 0.932 0.929 0.930 0.939 0.939 0.939 0.938 0.944 0.945 0.947 0.952 0.951 0.952 0.955 0.953 0.954 0.959 0.956 
213  0.701 0.514 0.470 0.466 0.477 0.501 0.520 0.548 0.605 0.645 0.671 0.702 0.710 0.725 0.748 0.766 0.779 0.797 0.796 0.818 0.820 0.845 0.846 0.855 0.872 0.874 0.883 0.888 0.893 0.899 0.902 0.909 0.912 0.916 0.920 0.923 0.929 0.925 0.928 0.932 0.934 0.935 0.942 0.945 0.944 0.944 0.945 0.947 0.951 
214  0.655 0.638 0.631 0.631 0.648 0.647 0.663 0.676 0.692 0.719 0.736 0.739 0.761 0.765 0.787 0.798 0.805 0.821 0.819 0.829 0.839 0.847 0.864 0.864 0.877 0.875 0.882 0.893 0.893 0.899 0.901 0.903 0.910 0.914 0.917 0.918 0.922 0.926 0.926 0.933 0.934 0.934 0.935 0.939 0.934 0.942 0.939 0.940 0.949 
215  0.654 0.700 0.705 0.715 0.714 0.714 0.712 0.725 0.728 0.748 0.742 0.766 0.777 0.786 0.785 0.799 0.817 0.824 0.824 0.832 0.848 0.854 0.862 0.871 0.874 0.884 0.877 0.892 0.893 0.899 0.906 0.909 0.910 0.914 0.921 0.916 0.921 0.926 0.923 0.933 0.930 0.934 0.937 0.937 0.940 0.939 0.944 0.943 0.946 
216  0.522 0.685 0.792 0.855 0.900 0.928 0.945 0.951 0.959 0.966 0.967 0.969 0.962 0.969 0.964 0.966 0.962 0.961 0.961 0.958 0.957 0.960 0.957 0.956 0.957 0.952 0.955 0.954 0.955 0.957 0.954 0.953 0.955 0.956 0.952 0.953 0.956 0.956 0.949 0.954 0.950 0.951 0.951 0.954 0.954 0.956 0.953 0.956 0.954 
300  0.628 0.527 0.547 0.569 0.604 0.632 0.656 0.684 0.716 0.724 0.741 0.751 0.770 0.785 0.797 0.808 0.821 0.841 0.844 0.852 0.857 0.868 0.870 0.875 0.882 0.888 0.895 0.903 0.902 0.902 0.908 0.912 0.919 0.918 0.922 0.924 0.928 0.926 0.929 0.932 0.931 0.935 0.938 0.936 0.939 0.939 0.946 0.943 0.944 
301  0.667 0.622 0.675 0.708 0.733 0.759 0.780 0.799 0.799 0.816 0.825 0.836 0.840 0.841 0.851 0.859 0.856 0.873 0.874 0.875 0.883 0.887 0.890 0.896 0.898 0.897 0.911 0.908 0.910 0.910 0.920 0.913 0.923 0.927 0.926 0.928 0.929 0.929 0.933 0.937 0.935 0.938 0.941 0.944 0.942 0.945 0.943 0.945 0.944 
302  0.695 0.670 0.739 0.784 0.825 0.849 0.861 0.866 0.873 0.883 0.887 0.886 0.889 0.890 0.894 0.893 0.898 0.887 0.893 0.898 0.901 0.902 0.909 0.912 0.907 0.912 0.912 0.919 0.919 0.926 0.925 0.923 0.928 0.928 0.930 0.935 0.935 0.933 0.936 0.937 0.940 0.942 0.943 0.941 0.947 0.949 0.944 0.945 0.943 
302  0.673 0.631 0.669 0.721 0.744 0.769 0.787 0.799 0.815 0.817 0.826 0.836 0.845 0.848 0.855 0.863 0.861 0.867 0.878 0.884 0.882 0.890 0.891 0.896 0.899 0.902 0.907 0.906 0.918 0.915 0.919 0.921 0.917 0.925 0.928 0.930 0.932 0.938 0.934 0.936 0.934 0.940 0.940 0.942 0.944 0.949 0.944 0.950 0.949 
304  0.551 0.430 0.506 0.574 0.624 0.676 0.724 0.753 0.793 0.815 0.839 0.857 0.867 0.881 0.896 0.907 0.919 0.924 0.925 0.936 0.938 0.944 0.942 0.949 0.950 0.955 0.958 0.958 0.962 0.963 0.964 0.964 0.967 0.966 0.967 0.968 0.971 0.970 0.972 0.972 0.975 0.971 0.973 0.971 0.976 0.974 0.975 0.974 0.976 
305  0.606 0.530 0.595 0.648 0.696 0.739 0.763 0.790 0.820 0.834 0.849 0.863 0.871 0.875 0.886 0.900 0.902 0.905 0.909 0.916 0.919 0.920 0.924 0.927 0.929 0.930 0.935 0.936 0.936 0.943 0.936 0.941 0.945 0.944 0.941 0.948 0.948 0.948 0.949 0.953 0.953 0.951 0.954 0.953 0.957 0.956 0.955 0.954 0.960 
306  0.632 0.600 0.633 0.683 0.713 0.739 0.759 0.782 0.798 0.810 0.815 0.828 0.835 0.840 0.852 0.861 0.862 0.867 0.867 0.875 0.881 0.891 0.887 0.895 0.905 0.904 0.906 0.907 0.913 0.913 0.916 0.919 0.924 0.922 0.925 0.929 0.927 0.928 0.934 0.933 0.940 0.937 0.938 0.939 0.943 0.939 0.946 0.946 0.945 
307  0.630 0.643 0.703 0.749 0.782 0.807 0.814 0.829 0.837 0.840 0.852 0.853 0.861 0.866 0.866 0.870 0.874 0.880 0.883 0.880 0.887 0.892 0.890 0.898 0.903 0.906 0.904 0.911 0.912 0.919 0.919 0.927 0.922 0.922 0.925 0.932 0.935 0.933 0.937 0.934 0.939 0.937 0.940 0.943 0.947 0.942 0.946 0.946 0.946 
400  0.620 0.542 0.554 0.589 0.630 0.649 0.681 0.709 0.730 0.751 0.769 0.779 0.804 0.806 0.828 0.832 0.839 0.853 0.861 0.868 0.872 0.879 0.888 0.896 0.904 0.908 0.905 0.912 0.914 0.916 0.923 0.917 0.922 0.928 0.927 0.935 0.927 0.935 0.939 0.939 0.940 0.944 0.945 0.947 0.947 0.947 0.948 0.946 0.949 
401  0.634 0.576 0.618 0.659 0.703 0.741 0.761 0.791 0.806 0.824 0.830 0.850 0.854 0.868 0.881 0.880 0.883 0.898 0.903 0.906 0.910 0.910 0.913 0.918 0.921 0.920 0.926 0.934 0.930 0.932 0.927 0.939 0.938 0.943 0.941 0.941 0.947 0.944 0.946 0.948 0.950 0.946 0.951 0.948 0.952 0.954 0.957 0.955 0.956 
402  0.650 0.607 0.665 0.735 0.780 0.814 0.847 0.876 0.891 0.902 0.915 0.915 0.929 0.931 0.931 0.939 0.945 0.939 0.941 0.938 0.944 0.949 0.947 0.950 0.947 0.946 0.951 0.951 0.950 0.954 0.955 0.953 0.955 0.958 0.954 0.958 0.962 0.957 0.958 0.961 0.957 0.956 0.958 0.960 0.957 0.959 0.960 0.957 0.964 
403  0.641 0.568 0.613 0.668 0.712 0.736 0.773 0.791 0.801 0.821 0.833 0.854 0.866 0.863 0.881 0.882 0.889 0.894 0.899 0.907 0.911 0.916 0.923 0.919 0.925 0.922 0.929 0.927 0.930 0.935 0.938 0.940 0.941 0.943 0.945 0.944 0.942 0.945 0.948 0.950 0.949 0.952 0.953 0.950 0.955 0.950 0.955 0.953 0.959

It’s still possible to get an accurate prediction in some cases, but we need a huge sample and a specific distribution.

Finally, let’s look at the $99^\textrm{th}$ percentile:

                              Quantile = 0.99, ConfidenceLevel = 95%
Dist    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48   49   50
0    0.383 0.267 0.274 0.290 0.309 0.341 0.361 0.367 0.389 0.405 0.441 0.447 0.461 0.463 0.493 0.494 0.509 0.536 0.527 0.544 0.562 0.574 0.581 0.593 0.584 0.601 0.618 0.621 0.632 0.630 0.643 0.650 0.653 0.673 0.682 0.679 0.693 0.689 0.700 0.703 0.716 0.711 0.722 0.729 0.732 0.736 0.742 0.752 0.755 
1    0.387 0.268 0.273 0.297 0.316 0.332 0.355 0.378 0.395 0.420 0.422 0.433 0.459 0.471 0.489 0.498 0.513 0.530 0.536 0.552 0.562 0.565 0.576 0.587 0.594 0.609 0.614 0.625 0.634 0.642 0.633 0.651 0.662 0.665 0.678 0.687 0.689 0.691 0.702 0.711 0.709 0.719 0.719 0.725 0.729 0.735 0.738 0.752 0.754 
2    0.369 0.242 0.234 0.237 0.258 0.272 0.276 0.290 0.301 0.312 0.317 0.321 0.321 0.341 0.347 0.346 0.348 0.355 0.360 0.361 0.360 0.365 0.378 0.378 0.368 0.366 0.372 0.379 0.379 0.375 0.382 0.378 0.376 0.385 0.378 0.376 0.384 0.376 0.378 0.375 0.380 0.388 0.385 0.375 0.369 0.375 0.381 0.373 0.369 
100  0.173 0.085 0.087 0.094 0.112 0.125 0.141 0.157 0.168 0.185 0.196 0.203 0.220 0.232 0.240 0.253 0.265 0.286 0.289 0.302 0.308 0.330 0.345 0.353 0.357 0.365 0.382 0.382 0.394 0.414 0.421 0.424 0.429 0.441 0.454 0.455 0.461 0.471 0.485 0.489 0.499 0.512 0.519 0.526 0.538 0.534 0.542 0.542 0.554 
101  0.171 0.080 0.085 0.099 0.112 0.123 0.147 0.157 0.163 0.181 0.192 0.203 0.215 0.231 0.238 0.253 0.273 0.271 0.295 0.304 0.305 0.331 0.335 0.347 0.363 0.358 0.388 0.379 0.394 0.398 0.422 0.424 0.428 0.451 0.451 0.468 0.472 0.474 0.484 0.494 0.506 0.500 0.517 0.525 0.531 0.546 0.541 0.553 0.570 
102  0.170 0.084 0.089 0.099 0.111 0.122 0.137 0.147 0.165 0.178 0.193 0.206 0.217 0.225 0.258 0.251 0.273 0.278 0.288 0.305 0.308 0.325 0.337 0.349 0.351 0.374 0.385 0.402 0.399 0.404 0.421 0.418 0.432 0.443 0.453 0.462 0.472 0.482 0.489 0.494 0.499 0.507 0.515 0.525 0.533 0.530 0.542 0.550 0.561 
103  0.045 0.046 0.052 0.061 0.074 0.091 0.106 0.119 0.135 0.143 0.161 0.171 0.184 0.192 0.195 0.211 0.234 0.237 0.244 0.263 0.281 0.287 0.296 0.306 0.321 0.329 0.335 0.349 0.354 0.361 0.378 0.380 0.400 0.413 0.409 0.423 0.441 0.435 0.443 0.463 0.465 0.476 0.480 0.487 0.492 0.504 0.507 0.521 0.526 
104  0.165 0.090 0.088 0.094 0.114 0.117 0.137 0.154 0.163 0.184 0.198 0.210 0.212 0.233 0.256 0.255 0.272 0.282 0.286 0.307 0.314 0.326 0.333 0.347 0.361 0.367 0.387 0.392 0.395 0.413 0.414 0.427 0.438 0.436 0.456 0.458 0.469 0.464 0.486 0.491 0.496 0.510 0.512 0.518 0.537 0.531 0.543 0.547 0.561 
111  0.166 0.082 0.086 0.097 0.107 0.128 0.140 0.159 0.160 0.173 0.193 0.199 0.217 0.229 0.247 0.256 0.264 0.274 0.290 0.317 0.314 0.327 0.328 0.349 0.347 0.368 0.386 0.387 0.399 0.403 0.412 0.427 0.427 0.432 0.455 0.457 0.462 0.483 0.482 0.490 0.503 0.508 0.505 0.524 0.523 0.539 0.539 0.553 0.565 
130  0.048 0.033 0.037 0.046 0.054 0.066 0.071 0.080 0.090 0.097 0.107 0.119 0.135 0.145 0.151 0.158 0.175 0.183 0.195 0.198 0.210 0.219 0.230 0.242 0.249 0.259 0.268 0.274 0.291 0.285 0.300 0.318 0.323 0.327 0.340 0.353 0.354 0.374 0.374 0.387 0.392 0.407 0.413 0.418 0.430 0.447 0.445 0.460 0.459 
131  0.070 0.049 0.051 0.064 0.078 0.081 0.095 0.109 0.115 0.126 0.145 0.154 0.169 0.175 0.194 0.194 0.204 0.220 0.231 0.242 0.248 0.265 0.270 0.274 0.298 0.306 0.323 0.323 0.336 0.340 0.358 0.362 0.375 0.379 0.392 0.393 0.412 0.415 0.426 0.441 0.446 0.448 0.456 0.468 0.477 0.487 0.489 0.501 0.508 
140  0.081 0.028 0.027 0.030 0.033 0.036 0.042 0.050 0.051 0.056 0.066 0.066 0.074 0.078 0.079 0.089 0.089 0.099 0.099 0.100 0.110 0.117 0.114 0.124 0.122 0.126 0.133 0.136 0.137 0.144 0.156 0.158 0.152 0.160 0.172 0.173 0.185 0.175 0.177 0.191 0.193 0.204 0.202 0.222 0.224 0.231 0.238 0.252 0.261 
141  0.058 0.019 0.018 0.019 0.018 0.021 0.025 0.030 0.034 0.035 0.033 0.040 0.039 0.048 0.049 0.051 0.055 0.055 0.061 0.063 0.066 0.065 0.070 0.073 0.079 0.080 0.080 0.082 0.089 0.087 0.098 0.095 0.097 0.098 0.101 0.102 0.108 0.109 0.110 0.117 0.119 0.127 0.122 0.130 0.138 0.139 0.148 0.155 0.166 
142  0.030 0.006 0.004 0.007 0.007 0.008 0.010 0.010 0.011 0.011 0.012 0.015 0.017 0.018 0.018 0.016 0.020 0.021 0.024 0.020 0.023 0.024 0.025 0.027 0.030 0.028 0.030 0.034 0.031 0.030 0.035 0.036 0.035 0.039 0.035 0.039 0.038 0.039 0.038 0.043 0.044 0.043 0.044 0.050 0.048 0.049 0.052 0.056 0.059 
143  0.026 0.017 0.022 0.028 0.031 0.038 0.043 0.053 0.064 0.068 0.081 0.090 0.096 0.107 0.122 0.137 0.145 0.159 0.175 0.189 0.203 0.224 0.224 0.235 0.239 0.254 0.250 0.269 0.277 0.286 0.292 0.300 0.312 0.316 0.328 0.328 0.343 0.340 0.365 0.355 0.377 0.373 0.398 0.387 0.407 0.411 0.416 0.415 0.427 
144  0.040 0.049 0.064 0.073 0.091 0.097 0.119 0.131 0.141 0.155 0.175 0.182 0.208 0.216 0.228 0.239 0.258 0.266 0.277 0.292 0.305 0.320 0.331 0.347 0.353 0.372 0.382 0.390 0.402 0.412 0.420 0.431 0.435 0.456 0.471 0.471 0.474 0.490 0.496 0.510 0.521 0.533 0.539 0.541 0.550 0.562 0.565 0.573 0.587 
145  0.037 0.059 0.083 0.093 0.117 0.136 0.150 0.163 0.178 0.201 0.217 0.236 0.252 0.265 0.275 0.282 0.310 0.323 0.337 0.343 0.359 0.372 0.381 0.394 0.398 0.410 0.428 0.443 0.458 0.459 0.474 0.487 0.506 0.501 0.521 0.528 0.537 0.542 0.566 0.552 0.575 0.588 0.594 0.603 0.603 0.616 0.615 0.628 0.644 
146  0.042 0.042 0.056 0.069 0.083 0.095 0.110 0.128 0.137 0.148 0.165 0.173 0.193 0.193 0.216 0.220 0.230 0.244 0.257 0.276 0.281 0.302 0.312 0.329 0.340 0.344 0.354 0.363 0.372 0.383 0.393 0.404 0.419 0.420 0.432 0.441 0.444 0.457 0.472 0.474 0.484 0.490 0.501 0.509 0.515 0.530 0.535 0.537 0.541 
147  0.043 0.045 0.058 0.070 0.083 0.094 0.102 0.130 0.138 0.151 0.161 0.183 0.186 0.201 0.217 0.227 0.244 0.262 0.265 0.282 0.293 0.293 0.322 0.332 0.337 0.355 0.357 0.376 0.382 0.396 0.399 0.408 0.419 0.433 0.434 0.451 0.453 0.469 0.466 0.479 0.483 0.493 0.506 0.518 0.515 0.532 0.542 0.547 0.559 
148  0.043 0.043 0.058 0.068 0.084 0.099 0.108 0.124 0.139 0.150 0.164 0.171 0.191 0.206 0.216 0.233 0.247 0.250 0.273 0.286 0.288 0.298 0.311 0.333 0.343 0.342 0.357 0.369 0.387 0.398 0.405 0.418 0.421 0.439 0.442 0.451 0.461 0.465 0.475 0.490 0.497 0.505 0.509 0.518 0.528 0.537 0.539 0.559 0.552 
149  0.039 0.047 0.060 0.072 0.083 0.098 0.112 0.129 0.132 0.149 0.166 0.176 0.193 0.210 0.223 0.230 0.248 0.260 0.273 0.291 0.297 0.310 0.314 0.333 0.340 0.365 0.373 0.382 0.391 0.390 0.400 0.423 0.429 0.438 0.446 0.461 0.473 0.478 0.478 0.484 0.494 0.505 0.507 0.526 0.534 0.536 0.549 0.556 0.569 
150  0.037 0.048 0.061 0.072 0.085 0.101 0.109 0.120 0.137 0.150 0.161 0.181 0.197 0.209 0.223 0.236 0.251 0.258 0.270 0.283 0.297 0.306 0.324 0.334 0.356 0.363 0.369 0.376 0.387 0.392 0.407 0.424 0.432 0.439 0.448 0.456 0.465 0.482 0.493 0.503 0.498 0.511 0.533 0.522 0.535 0.541 0.544 0.558 0.553 
151  0.043 0.049 0.057 0.073 0.082 0.095 0.118 0.130 0.142 0.156 0.172 0.183 0.200 0.206 0.220 0.226 0.246 0.262 0.277 0.286 0.297 0.318 0.329 0.332 0.353 0.358 0.372 0.385 0.387 0.410 0.410 0.429 0.426 0.453 0.454 0.466 0.474 0.480 0.495 0.500 0.505 0.513 0.515 0.543 0.532 0.545 0.555 0.560 0.581 
152  0.043 0.043 0.056 0.071 0.089 0.097 0.112 0.122 0.136 0.162 0.161 0.182 0.198 0.205 0.217 0.232 0.245 0.263 0.277 0.290 0.302 0.306 0.322 0.339 0.351 0.363 0.364 0.383 0.396 0.406 0.409 0.416 0.426 0.444 0.458 0.465 0.463 0.480 0.491 0.495 0.499 0.521 0.528 0.533 0.550 0.552 0.555 0.563 0.570 
153  0.134 0.062 0.063 0.068 0.080 0.088 0.093 0.106 0.114 0.121 0.141 0.143 0.154 0.161 0.178 0.178 0.190 0.196 0.207 0.210 0.215 0.232 0.235 0.250 0.248 0.258 0.264 0.269 0.275 0.292 0.295 0.299 0.304 0.306 0.322 0.320 0.330 0.330 0.343 0.342 0.354 0.355 0.360 0.361 0.375 0.387 0.387 0.399 0.405 
154  0.154 0.078 0.080 0.089 0.100 0.107 0.124 0.136 0.145 0.157 0.174 0.179 0.188 0.206 0.220 0.238 0.238 0.252 0.257 0.261 0.271 0.296 0.295 0.310 0.318 0.319 0.332 0.347 0.354 0.359 0.367 0.379 0.383 0.393 0.388 0.407 0.421 0.412 0.422 0.437 0.442 0.448 0.452 0.460 0.466 0.478 0.480 0.486 0.496 
155  0.165 0.083 0.087 0.090 0.103 0.120 0.131 0.143 0.160 0.169 0.179 0.192 0.197 0.219 0.231 0.240 0.254 0.266 0.277 0.281 0.298 0.308 0.316 0.335 0.337 0.346 0.364 0.368 0.372 0.386 0.396 0.402 0.410 0.419 0.418 0.429 0.438 0.453 0.455 0.467 0.470 0.480 0.490 0.489 0.500 0.501 0.507 0.522 0.517 
200  0.253 0.111 0.110 0.118 0.131 0.143 0.164 0.178 0.182 0.201 0.206 0.222 0.236 0.243 0.260 0.277 0.285 0.295 0.315 0.311 0.330 0.334 0.352 0.359 0.378 0.379 0.383 0.396 0.406 0.423 0.426 0.434 0.443 0.444 0.460 0.462 0.485 0.489 0.495 0.505 0.516 0.526 0.522 0.525 0.537 0.547 0.550 0.557 0.568 
201  0.286 0.128 0.115 0.125 0.136 0.140 0.162 0.168 0.180 0.198 0.208 0.227 0.228 0.244 0.262 0.274 0.278 0.292 0.302 0.312 0.325 0.332 0.347 0.357 0.362 0.388 0.393 0.408 0.409 0.422 0.429 0.430 0.455 0.441 0.466 0.469 0.487 0.493 0.498 0.504 0.503 0.507 0.512 0.533 0.545 0.549 0.555 0.559 0.561 
202  0.383 0.174 0.142 0.148 0.150 0.151 0.176 0.182 0.195 0.199 0.216 0.227 0.235 0.242 0.259 0.266 0.286 0.298 0.313 0.314 0.326 0.338 0.354 0.370 0.372 0.379 0.397 0.399 0.407 0.422 0.434 0.434 0.446 0.449 0.470 0.476 0.477 0.487 0.492 0.498 0.510 0.513 0.525 0.531 0.542 0.542 0.553 0.561 0.564 
203  0.506 0.263 0.207 0.202 0.183 0.183 0.192 0.195 0.197 0.211 0.222 0.228 0.240 0.248 0.259 0.276 0.287 0.301 0.300 0.320 0.322 0.340 0.353 0.351 0.374 0.382 0.384 0.410 0.408 0.422 0.427 0.447 0.450 0.454 0.462 0.466 0.479 0.484 0.487 0.496 0.520 0.519 0.524 0.530 0.543 0.541 0.552 0.562 0.572 
204  0.264 0.065 0.046 0.047 0.047 0.047 0.052 0.056 0.056 0.060 0.064 0.073 0.077 0.078 0.080 0.082 0.091 0.096 0.099 0.099 0.105 0.112 0.113 0.107 0.117 0.116 0.125 0.127 0.131 0.132 0.145 0.139 0.146 0.148 0.149 0.159 0.160 0.163 0.171 0.166 0.177 0.183 0.181 0.190 0.196 0.196 0.207 0.217 0.220 
205  0.028 0.020 0.019 0.026 0.032 0.035 0.044 0.052 0.062 0.066 0.078 0.084 0.092 0.107 0.119 0.132 0.142 0.157 0.164 0.186 0.202 0.215 0.228 0.226 0.241 0.258 0.254 0.264 0.276 0.283 0.290 0.298 0.299 0.314 0.331 0.332 0.335 0.340 0.344 0.364 0.367 0.373 0.374 0.396 0.402 0.399 0.411 0.426 0.419 
206  0.042 0.043 0.060 0.074 0.084 0.099 0.108 0.125 0.144 0.160 0.170 0.184 0.203 0.213 0.227 0.248 0.256 0.265 0.273 0.303 0.299 0.317 0.337 0.335 0.359 0.367 0.376 0.382 0.396 0.414 0.429 0.426 0.437 0.450 0.456 0.464 0.472 0.490 0.495 0.504 0.509 0.529 0.533 0.538 0.558 0.566 0.563 0.567 0.579 
210  0.321 0.174 0.171 0.178 0.190 0.194 0.211 0.222 0.229 0.244 0.254 0.271 0.279 0.294 0.303 0.306 0.320 0.320 0.337 0.342 0.353 0.367 0.379 0.384 0.401 0.402 0.414 0.415 0.432 0.432 0.448 0.455 0.471 0.471 0.472 0.486 0.497 0.496 0.509 0.517 0.517 0.529 0.546 0.542 0.547 0.551 0.560 0.568 0.574 
211  0.358 0.123 0.112 0.120 0.123 0.136 0.152 0.164 0.173 0.185 0.203 0.212 0.224 0.238 0.252 0.263 0.278 0.284 0.298 0.315 0.321 0.337 0.334 0.345 0.365 0.370 0.385 0.388 0.397 0.402 0.420 0.434 0.445 0.445 0.457 0.468 0.475 0.485 0.486 0.501 0.492 0.510 0.508 0.528 0.536 0.535 0.547 0.549 0.569 
212  0.232 0.074 0.062 0.058 0.062 0.070 0.069 0.072 0.077 0.083 0.080 0.082 0.083 0.093 0.095 0.098 0.098 0.101 0.107 0.110 0.114 0.117 0.119 0.127 0.133 0.130 0.136 0.141 0.143 0.143 0.144 0.149 0.157 0.156 0.154 0.163 0.166 0.169 0.174 0.176 0.189 0.187 0.196 0.198 0.202 0.206 0.215 0.228 0.229 
213  0.236 0.045 0.034 0.033 0.035 0.041 0.047 0.047 0.052 0.055 0.059 0.067 0.069 0.074 0.079 0.086 0.090 0.090 0.094 0.094 0.099 0.108 0.111 0.108 0.115 0.120 0.121 0.126 0.128 0.130 0.135 0.140 0.137 0.148 0.147 0.153 0.151 0.156 0.163 0.165 0.174 0.177 0.186 0.180 0.189 0.191 0.201 0.200 0.217 
214  0.398 0.162 0.149 0.142 0.138 0.145 0.153 0.161 0.166 0.173 0.185 0.194 0.206 0.227 0.224 0.235 0.239 0.260 0.276 0.278 0.284 0.291 0.299 0.306 0.325 0.327 0.326 0.347 0.344 0.355 0.369 0.372 0.376 0.389 0.407 0.404 0.400 0.405 0.422 0.427 0.427 0.446 0.435 0.458 0.459 0.459 0.463 0.468 0.484 
215  0.498 0.240 0.190 0.181 0.178 0.167 0.171 0.177 0.177 0.192 0.195 0.203 0.218 0.212 0.229 0.239 0.243 0.256 0.272 0.277 0.280 0.294 0.305 0.306 0.315 0.328 0.336 0.345 0.356 0.366 0.360 0.376 0.389 0.390 0.395 0.408 0.411 0.413 0.427 0.429 0.432 0.435 0.438 0.451 0.453 0.464 0.474 0.472 0.477 
216  0.468 0.629 0.705 0.726 0.738 0.730 0.730 0.710 0.688 0.658 0.632 0.616 0.581 0.571 0.541 0.528 0.519 0.504 0.491 0.479 0.471 0.465 0.453 0.459 0.465 0.451 0.454 0.447 0.456 0.465 0.458 0.472 0.477 0.480 0.491 0.492 0.503 0.507 0.515 0.519 0.522 0.538 0.542 0.543 0.553 0.552 0.568 0.571 0.572 
300  0.314 0.157 0.142 0.150 0.149 0.176 0.184 0.198 0.209 0.216 0.236 0.239 0.258 0.258 0.274 0.291 0.291 0.318 0.327 0.328 0.346 0.348 0.357 0.376 0.387 0.400 0.395 0.411 0.422 0.426 0.436 0.447 0.456 0.460 0.477 0.476 0.488 0.484 0.497 0.507 0.514 0.529 0.539 0.542 0.550 0.552 0.563 0.566 0.572 
301  0.411 0.263 0.226 0.225 0.220 0.230 0.235 0.234 0.245 0.256 0.260 0.274 0.278 0.282 0.294 0.306 0.308 0.325 0.323 0.341 0.353 0.361 0.380 0.377 0.384 0.387 0.412 0.415 0.423 0.442 0.446 0.448 0.461 0.477 0.476 0.482 0.483 0.507 0.507 0.501 0.517 0.539 0.549 0.550 0.539 0.554 0.560 0.561 0.571 
302  0.467 0.378 0.347 0.328 0.317 0.315 0.312 0.303 0.306 0.308 0.312 0.302 0.315 0.321 0.321 0.316 0.330 0.332 0.337 0.358 0.357 0.360 0.377 0.385 0.384 0.396 0.402 0.420 0.422 0.430 0.440 0.442 0.449 0.466 0.472 0.490 0.496 0.489 0.505 0.517 0.518 0.528 0.527 0.540 0.542 0.542 0.564 0.564 0.569 
302  0.415 0.276 0.241 0.236 0.241 0.252 0.260 0.255 0.265 0.267 0.280 0.288 0.299 0.297 0.305 0.322 0.336 0.342 0.352 0.365 0.375 0.389 0.388 0.409 0.407 0.421 0.428 0.443 0.453 0.465 0.464 0.472 0.481 0.496 0.496 0.517 0.514 0.530 0.526 0.537 0.538 0.554 0.562 0.570 0.572 0.579 0.579 0.592 0.599 
304  0.268 0.238 0.243 0.260 0.272 0.295 0.304 0.331 0.337 0.355 0.361 0.365 0.385 0.409 0.415 0.432 0.420 0.427 0.441 0.437 0.460 0.469 0.459 0.474 0.469 0.480 0.491 0.482 0.501 0.511 0.523 0.515 0.533 0.535 0.527 0.545 0.551 0.554 0.564 0.554 0.568 0.579 0.576 0.577 0.597 0.589 0.604 0.594 0.617 
305  0.335 0.224 0.210 0.210 0.231 0.245 0.258 0.269 0.275 0.292 0.296 0.307 0.306 0.321 0.328 0.337 0.348 0.358 0.371 0.378 0.380 0.388 0.399 0.399 0.415 0.420 0.431 0.428 0.445 0.454 0.468 0.466 0.477 0.476 0.489 0.491 0.503 0.511 0.509 0.525 0.528 0.547 0.544 0.551 0.553 0.568 0.570 0.570 0.579 
306  0.378 0.216 0.203 0.203 0.204 0.214 0.222 0.230 0.243 0.244 0.253 0.268 0.277 0.283 0.294 0.298 0.308 0.322 0.323 0.342 0.350 0.369 0.368 0.381 0.385 0.389 0.415 0.411 0.427 0.431 0.445 0.444 0.458 0.467 0.480 0.489 0.487 0.494 0.502 0.521 0.525 0.523 0.543 0.533 0.542 0.548 0.560 0.568 0.573 
307  0.436 0.318 0.275 0.274 0.276 0.268 0.260 0.269 0.268 0.275 0.273 0.283 0.282 0.293 0.309 0.311 0.315 0.330 0.326 0.343 0.346 0.365 0.375 0.382 0.397 0.396 0.406 0.413 0.425 0.424 0.436 0.449 0.452 0.463 0.473 0.482 0.487 0.505 0.506 0.511 0.526 0.522 0.539 0.541 0.548 0.555 0.565 0.571 0.584 
400  0.335 0.180 0.163 0.166 0.185 0.191 0.210 0.227 0.233 0.245 0.250 0.251 0.273 0.293 0.303 0.309 0.322 0.336 0.335 0.348 0.361 0.377 0.373 0.384 0.391 0.402 0.411 0.428 0.441 0.447 0.449 0.454 0.466 0.474 0.481 0.492 0.495 0.504 0.514 0.510 0.526 0.532 0.557 0.549 0.549 0.566 0.575 0.579 0.581 
401  0.388 0.259 0.230 0.236 0.238 0.250 0.251 0.266 0.275 0.287 0.293 0.301 0.304 0.313 0.319 0.332 0.342 0.356 0.357 0.369 0.371 0.386 0.393 0.405 0.408 0.415 0.427 0.431 0.445 0.449 0.462 0.466 0.468 0.482 0.480 0.499 0.495 0.519 0.523 0.520 0.532 0.535 0.547 0.550 0.561 0.572 0.572 0.579 0.585 
402  0.433 0.379 0.368 0.363 0.372 0.374 0.379 0.375 0.378 0.386 0.379 0.382 0.376 0.381 0.383 0.392 0.390 0.394 0.403 0.400 0.408 0.402 0.422 0.415 0.436 0.438 0.443 0.447 0.455 0.464 0.466 0.478 0.475 0.487 0.490 0.504 0.505 0.518 0.521 0.521 0.539 0.543 0.533 0.557 0.558 0.567 0.576 0.578 0.581 
403  0.388 0.270 0.240 0.248 0.247 0.260 0.271 0.275 0.285 0.296 0.302 0.312 0.316 0.328 0.338 0.349 0.362 0.364 0.373 0.385 0.390 0.394 0.402 0.409 0.424 0.431 0.439 0.435 0.451 0.456 0.473 0.477 0.491 0.492 0.510 0.513 0.518 0.521 0.540 0.541 0.555 0.544 0.553 0.560 0.567 0.577 0.587 0.583 0.599

We don’t have any accurate predictions anymore! Even with 50-element samples, the coverage percentage is about 50% in most cases. For some distributions, the coverage percentage of the 95% confidence interval may be less than 10%!

Conclusion

As we can see, statistical predictions are not always accurate. Any statement about the statistical properties of some “generic” data may be invalid for your particular cases. What should we do about that?

Well, it’s always a good idea to verify all of the chosen statistical approaches on your data. If you can’t do it with real data, try to generate random data that is close to your expectations about reality. In the above simulation, I used synthetic latency distributions that are close to some distribution produced by performance measurements of real software. However, you shouldn’t use the results of this simulation as a reference table. The coverage percentage heavily depends on the distributions you use. Also, I applied the Maritz-Jarrett method to estimate confidence intervals, but there are other approaches (e.g., density estimations or bootstrap) that may give another coverage picture.

You shouldn’t blindly trust any theoretical assumptions. Instead of it, you should check how these assumptions work in the context of problems you are trying to solve.