Normal Distribution

Size 5, Normal Distribution

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.668	0.682	-0.936	0.725	0.266	5
0.147	0.710	0.217	0.536	0.180	5
0.731	1.015	0.428	1.637	0.357	5
0.185	0.486	0.005	0.436	1.125	5
-0.698	0.473	-0.861	0.564	0.607	5
0.392	0.998	0.369	0.595	0.051	5
0.593	0.606	0.615	0.477	-0.049	5
-0.139	1.689	-0.137	1.231	-0.670	5
0.109	0.940	0.045	0.180	0.041	5
0.817	1.053	0.742	1.264	-0.279	5

Each row summarizes one independently drawn sample of size 5 from the normal distribution. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.805	0.000	NaN	5
2	-0.677	0.181	NaN	5
3	-0.592	0.195	-0.516	5
4	-0.427	0.366	0.576	5
5	-0.163	0.672	0.847	5
6	-0.053	0.658	0.342	5
7	-0.060	0.601	0.399	5
8	-0.090	0.563	0.573	5
9	-0.091	0.527	0.612	5
10	-0.039	0.524	0.346	5
20	0.022	0.537	0.564	5
30	0.036	0.488	0.389	5
40	0.065	0.453	0.331	5
50	0.094	0.451	0.231	5
60	0.090	0.449	0.138	5
70	0.078	0.458	0.110	5
80	0.037	0.451	0.228	5
90	0.049	0.447	0.189	5
100	0.026	0.444	0.211	5
200	0.019	0.432	0.084	5
300	0.008	0.437	-0.020	5
400	0.014	0.437	-0.019	5
500	0.000	0.440	-0.039	5
600	-0.020	0.445	-0.029	5
700	-0.033	0.445	-0.031	5
800	-0.026	0.444	-0.035	5
900	-0.022	0.449	-0.024	5
1,000	-0.023	0.451	-0.030	5

Statistics are computed cumulatively over the first 10 samples (reporting every 1), then every 10 through 100, and every 100 through 1,000. The standard error is the sample standard deviation of the sample means.

Size 25, Normal Distribution

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
0.055	1.102	0.140	1.280	-0.396	25
-0.231	0.966	-0.130	1.159	-0.138	25
0.049	1.411	0.149	1.898	0.018	25
0.108	0.855	0.045	1.230	-0.225	25
-0.177	0.946	0.019	1.141	0.023	25
0.133	1.122	0.137	1.211	-0.020	25
-0.204	0.832	-0.248	1.061	0.067	25
-0.035	1.149	0.259	1.581	-0.267	25
-0.158	1.164	-0.145	0.996	-0.284	25
-0.308	1.084	-0.454	1.177	0.400	25

Each row summarizes one independently drawn sample of size 25 from the normal distribution. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.423	0.000	NaN	25
2	-0.072	0.497	NaN	25
3	-0.043	0.355	-0.392	25
4	-0.022	0.293	-0.642	25
5	-0.028	0.254	-0.585	25
6	-0.058	0.238	-0.210	25
7	0.004	0.272	-0.149	25
8	0.005	0.252	-0.179	25
9	-0.010	0.240	0.009	25
10	-0.023	0.230	0.165	25
20	0.061	0.202	-0.416	25
30	-0.006	0.211	-0.139	25
40	0.012	0.205	-0.075	25
50	0.013	0.195	-0.013	25
60	0.005	0.196	-0.021	25
70	0.013	0.196	-0.100	25
80	-0.001	0.194	-0.063	25
90	-0.003	0.189	0.039	25
100	-0.012	0.197	-0.091	25
200	0.004	0.184	0.017	25
300	-0.000	0.180	0.096	25
400	-0.013	0.186	-0.070	25
500	-0.007	0.191	-0.029	25
600	-0.007	0.193	0.037	25
700	-0.010	0.196	0.043	25
800	-0.009	0.198	0.063	25
900	-0.009	0.198	0.053	25
1,000	-0.008	0.197	0.051	25

Size 100, Normal Distribution

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.111	1.136	-0.106	1.585	0.066	100
0.056	1.122	0.090	1.484	0.213	100
0.008	0.935	0.093	1.310	0.030	100
-0.074	0.945	-0.197	1.438	0.107	100
0.059	0.880	0.109	1.050	0.101	100
-0.100	1.025	-0.092	1.315	-0.013	100
-0.008	0.996	0.050	1.328	-0.282	100
-0.061	1.069	-0.005	1.259	-0.377	100
-0.141	1.060	-0.069	1.371	0.060	100
0.115	0.938	0.077	1.403	-0.101	100

Each row summarizes one independently drawn sample of size 100 from the normal distribution. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.009	0.000	NaN	100
2	-0.036	0.037	NaN	100
3	0.001	0.069	0.361	100
4	0.043	0.101	0.297	100
5	0.050	0.089	0.042	100
6	0.067	0.090	-0.280	100
7	0.052	0.092	0.049	100
8	0.062	0.090	-0.212	100
9	0.058	0.085	-0.093	100
10	0.036	0.108	-0.470	100
20	-0.004	0.094	0.188	100
30	-0.004	0.089	0.247	100
40	0.001	0.102	0.286	100
50	0.011	0.103	0.227	100
60	0.005	0.101	0.164	100
70	0.006	0.100	0.202	100
80	0.005	0.097	0.235	100
90	0.010	0.096	0.197	100
100	0.007	0.095	0.212	100
200	0.002	0.095	0.087	100
300	-0.005	0.100	0.086	100
400	-0.008	0.097	0.129	100
500	-0.007	0.098	0.168	100
600	-0.007	0.098	0.205	100
700	-0.007	0.097	0.185	100
800	-0.006	0.098	0.162	100
900	-0.006	0.099	0.150	100
1,000	-0.006	0.099	0.132	100

Size 1000, Normal Distribution

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.004	1.027	0.013	1.370	0.057	1,000
-0.036	1.017	-0.045	1.406	-0.038	1,000
-0.035	0.992	-0.026	1.352	-0.068	1,000
-0.031	1.027	-0.038	1.346	-0.025	1,000
-0.037	0.998	-0.054	1.326	-0.026	1,000
-0.015	1.000	0.005	1.357	-0.110	1,000
-0.104	0.988	-0.104	1.333	0.047	1,000
0.072	0.984	0.032	1.362	0.233	1,000
0.013	0.978	-0.002	1.314	-0.049	1,000
0.025	1.014	0.018	1.332	-0.072	1,000

Each row summarizes one independently drawn sample of size 1,000 from the normal distribution. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.029	0.000	NaN	1,000
2	-0.019	0.014	NaN	1,000
3	-0.001	0.032	0.560	1,000
4	0.004	0.028	-0.142	1,000
5	0.003	0.025	0.061	1,000
6	-0.001	0.024	0.400	1,000
7	-0.003	0.022	0.623	1,000
8	-0.004	0.021	0.816	1,000
9	-0.008	0.022	0.733	1,000
10	-0.006	0.022	0.493	1,000
20	0.007	0.034	0.083	1,000
30	0.006	0.033	0.191	1,000
40	0.004	0.034	0.512	1,000
50	0.002	0.034	0.177	1,000
60	0.001	0.034	0.069	1,000
70	-0.001	0.035	0.062	1,000
80	0.000	0.034	-0.004	1,000
90	-0.001	0.033	0.014	1,000
100	-0.002	0.034	0.133	1,000
200	-0.000	0.032	0.001	1,000
300	-0.001	0.033	0.013	1,000
400	-0.001	0.033	0.077	1,000
500	-0.000	0.033	0.081	1,000
600	-0.000	0.033	0.123	1,000
700	-0.000	0.033	0.094	1,000
800	-0.001	0.033	0.084	1,000
900	-0.001	0.032	0.047	1,000
1,000	-0.001	0.032	0.049	1,000

Positively Skewed

Size 5, Positively Skewed

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.429	0.572	-0.569	0.837	0.170	5
-0.021	0.574	-0.147	0.746	-0.019	5
-0.126	0.595	-0.169	0.387	0.739	5
0.062	0.620	0.281	0.962	-0.394	5
-0.352	0.473	-0.344	0.482	0.029	5
-0.566	0.510	-0.750	0.629	0.780	5
-0.432	0.696	-0.577	1.277	0.153	5
0.796	2.164	-0.515	2.919	0.669	5
-0.444	0.503	-0.384	0.446	0.618	5
0.022	1.009	-0.387	0.984	0.591	5

Each row summarizes one independently drawn sample of size 5 from the positively skewed. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.641	0.000	NaN	5
2	-0.600	0.058	NaN	5
3	-0.486	0.202	0.771	5
4	-0.298	0.412	0.763	5
5	-0.348	0.374	1.096	5
6	-0.275	0.379	0.488	5
7	-0.167	0.449	0.305	5
8	-0.146	0.420	0.160	5
9	-0.136	0.394	0.087	5
10	-0.135	0.371	0.081	5
20	-0.192	0.341	0.277	5
30	-0.130	0.360	0.059	5
40	-0.141	0.319	0.130	5
50	-0.147	0.305	0.140	5
60	-0.168	0.319	0.058	5
70	-0.184	0.323	0.097	5
80	-0.205	0.319	0.204	5
90	-0.185	0.331	0.335	5
100	-0.196	0.322	0.390	5
200	-0.199	0.360	0.739	5
300	-0.216	0.334	0.669	5
400	-0.223	0.331	0.698	5
500	-0.215	0.338	0.686	5
600	-0.216	0.336	0.668	5
700	-0.215	0.333	0.706	5
800	-0.211	0.333	0.700	5
900	-0.206	0.337	0.717	5
1,000	-0.205	0.340	0.846	5

Size 25, Positively Skewed

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.234	0.707	-0.535	0.852	1.102	25
-0.258	0.621	-0.489	0.854	0.668	25
-0.069	1.040	-0.368	0.489	2.137	25
-0.227	0.754	-0.513	1.327	0.934	25
0.040	0.972	-0.230	0.742	1.901	25
-0.382	0.633	-0.554	0.731	1.717	25
0.095	1.195	-0.301	0.993	2.004	25
0.000	0.988	-0.082	1.072	1.751	25
-0.314	0.772	-0.508	0.450	2.765	25
-0.292	0.761	-0.380	0.856	0.951	25

Each row summarizes one independently drawn sample of size 25 from the positively skewed. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.293	0.000	NaN	25
2	-0.160	0.188	NaN	25
3	-0.186	0.141	0.789	25
4	-0.157	0.128	-0.030	25
5	-0.226	0.189	-0.412	25
6	-0.239	0.172	-0.181	25
7	-0.203	0.184	-0.305	25
8	-0.196	0.172	-0.449	25
9	-0.186	0.163	-0.622	25
10	-0.177	0.156	-0.781	25
20	-0.182	0.154	-0.044	25
30	-0.213	0.157	0.191	25
40	-0.208	0.162	0.476	25
50	-0.204	0.164	0.350	25
60	-0.191	0.159	0.212	25
70	-0.193	0.156	0.255	25
80	-0.183	0.170	0.564	25
90	-0.175	0.169	0.536	25
100	-0.180	0.167	0.543	25
200	-0.195	0.162	0.670	25
300	-0.192	0.161	0.539	25
400	-0.200	0.160	0.567	25
500	-0.196	0.159	0.538	25
600	-0.197	0.158	0.503	25
700	-0.200	0.155	0.492	25
800	-0.197	0.154	0.478	25
900	-0.198	0.156	0.472	25
1,000	-0.198	0.155	0.458	25

Size 100, Positively Skewed

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.299	0.679	-0.423	0.772	1.411	100
-0.106	0.705	-0.303	0.718	1.241	100
-0.258	0.677	-0.441	0.735	1.695	100
-0.269	0.634	-0.473	0.673	1.291	100
-0.206	0.813	-0.461	0.839	1.783	100
-0.222	0.602	-0.359	0.688	1.179	100
-0.257	0.573	-0.331	0.642	1.225	100
-0.359	0.678	-0.524	0.751	1.398	100
-0.313	0.761	-0.499	0.691	2.560	100
-0.221	0.614	-0.361	0.560	1.502	100

Each row summarizes one independently drawn sample of size 100 from the positively skewed. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.271	0.000	NaN	100
2	-0.222	0.069	NaN	100
3	-0.201	0.060	-0.895	100
4	-0.223	0.066	-0.024	100
5	-0.182	0.108	0.600	100
6	-0.197	0.104	0.872	100
7	-0.192	0.096	0.701	100
8	-0.172	0.104	0.352	100
9	-0.174	0.098	0.426	100
10	-0.180	0.094	0.593	100
20	-0.197	0.088	0.762	100
30	-0.208	0.085	0.629	100
40	-0.205	0.087	0.382	100
50	-0.209	0.083	0.434	100
60	-0.213	0.079	0.503	100
70	-0.213	0.079	0.490	100
80	-0.205	0.079	0.390	100
90	-0.204	0.080	0.544	100
100	-0.203	0.079	0.474	100
200	-0.202	0.081	0.199	100
300	-0.202	0.081	0.115	100
400	-0.203	0.080	0.140	100
500	-0.205	0.078	0.124	100
600	-0.204	0.078	0.162	100
700	-0.205	0.079	0.189	100
800	-0.206	0.078	0.146	100
900	-0.206	0.079	0.114	100
1,000	-0.207	0.078	0.096	100

Size 1000, Positively Skewed

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Size
-0.215	0.711	-0.384	0.788	1.858	1,000
-0.262	0.741	-0.433	0.773	2.292	1,000
-0.198	0.770	-0.394	0.832	1.764	1,000
-0.170	0.845	-0.413	0.872	1.953	1,000
-0.189	0.771	-0.373	0.853	1.667	1,000
-0.238	0.741	-0.418	0.820	2.077	1,000
-0.188	0.792	-0.375	0.841	1.937	1,000
-0.198	0.799	-0.397	0.883	1.974	1,000
-0.174	0.832	-0.399	0.852	2.146	1,000
-0.200	0.801	-0.393	0.819	1.969	1,000

Each row summarizes one independently drawn sample of size 1,000 from the positively skewed. Values are reported to three decimal places.

The Central Limit Theorem in Action
Number of Samples	Mean of Means	Standard Error	Skewness	Sample Size
1	-0.200	0.000	NaN	1,000
2	-0.211	0.017	NaN	1,000
3	-0.211	0.012	-0.093	1,000
4	-0.211	0.010	-0.320	1,000
5	-0.208	0.010	-0.454	1,000
6	-0.209	0.010	-0.072	1,000
7	-0.207	0.011	-0.107	1,000
8	-0.205	0.011	-0.347	1,000
9	-0.207	0.011	-0.097	1,000
10	-0.206	0.011	-0.291	1,000
20	-0.209	0.018	-0.603	1,000
30	-0.202	0.023	-0.109	1,000
40	-0.206	0.025	0.185	1,000
50	-0.206	0.024	0.268	1,000
60	-0.205	0.024	0.344	1,000
70	-0.205	0.025	0.261	1,000
80	-0.205	0.024	0.275	1,000
90	-0.205	0.024	0.300	1,000
100	-0.204	0.024	0.277	1,000
200	-0.205	0.023	0.127	1,000
300	-0.205	0.024	0.263	1,000
400	-0.205	0.023	0.162	1,000
500	-0.204	0.024	0.030	1,000
600	-0.204	0.024	0.138	1,000
700	-0.204	0.024	0.119	1,000
800	-0.204	0.025	0.049	1,000
900	-0.204	0.024	0.022	1,000
1,000	-0.204	0.025	0.004	1,000

Methodology for Generating Values

This static HTML was generated by simulating samples from two populations:

Normal Distribution: mean μ = 0, standard deviation σ = 1.
Positively Skewed: lognormal with parameters (0, 0.6) transformed as exp(N(0,0.6)) - 1.4, matching the teaching app.

Per-sample summary table shows, for each of 10 independent samples of size n: mean, sample standard deviation (denominator n−1), median, interquartile range (IQR), and skewness.

CLT table accumulates sample means and reports at these checkpoints: after each of the first 10 samples, then every 10 samples through 100, and every 100 samples through 1,000.

Skewness uses the bias-corrected estimator employed in the original app: if m is the sample mean, s the sample standard deviation, and z_i=(x_i−m)/s, then skewness is √(n(n-1))/(n-2) * (1/n) * Σ z_i³ for n ≥ 3.

Median and IQR use linear interpolation (Hyndman–Fan Type 7) to compute quantiles.

All numeric values are formatted to three decimal places; integer counts use thousands separators.

Equivalent Text: Sampling & the Central Limit Theorem

Introduction