Coverage of quantile confidence intervals


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.


References (1)

  1. Quantile confidence intervals for weighted samples (2020-12-08) 3 5 Mathematics Statistics Research
  1. Estimating quantile confidence intervals: Maritz-Jarrett vs. jackknife (2021-07-13) 3 Mathematics Statistics Research