# Coverage of quantile confidence intervals

by 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)


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   500    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   500    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   500    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   500    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   500    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.