Size 5

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Min	Max	\|x̄ − μ\|	Size
22.600	6.504	22.000	9.000	0.103	15	31	1.400	5
24.800	5.070	26.000	5.000	0.195	19	32	0.800	5
23.400	5.899	23.000	8.000	-0.249	15	29	0.600	5
21.200	4.550	23.000	6.000	-0.282	15	26	2.800	5
22.400	5.030	22.000	9.000	0.048	17	28	1.600	5
25.200	5.357	27.000	5.000	-0.104	18	32	1.200	5
25.400	6.731	27.000	7.000	-0.492	15	32	1.400	5
23.000	6.819	25.000	1.000	-0.984	11	28	1.000	5
23.600	4.278	23.000	8.000	0.067	19	28	0.400	5
22.800	7.662	25.000	6.000	-0.760	10	29	1.200	5

Each row summarizes one independently drawn sample of size 5 from the SWLS population (scores 5–35). The |x̄ − μ| column shows how far the sample mean is from the population mean (μ = 24). Values are reported to three decimals.

Representativeness & CLT (Cumulative Sample Means)
Number of Samples	Mean of Means	Standard Error	Skewness of Means	Best \|x̄ − μ\|	Worst \|x̄ − μ\|	Sample Size
1	29.200	0.000	NaN	5.200	5.200	5
2	27.600	2.263	NaN	2.000	5.200	5
3	24.933	4.888	-0.935	2.000	5.200	5
4	25.050	3.998	-0.934	1.400	5.200	5
5	25.320	3.515	-1.221	1.400	5.200	5
6	25.467	3.164	-1.407	1.400	5.200	5
7	25.943	3.151	-1.472	1.400	5.200	5
8	25.250	3.514	-0.813	1.400	5.200	5
9	24.578	3.857	-0.459	1.400	5.200	5
10	24.600	3.637	-0.496	0.800	5.200	5
20	24.620	3.175	-0.735	0.200	6.000	5
30	24.233	3.179	-0.520	0.000	6.800	5
40	24.285	3.011	-0.510	0.000	6.800	5
50	24.080	2.918	-0.424	0.000	6.800	5
60	24.223	2.788	-0.504	0.000	6.800	5
70	24.140	2.692	-0.419	0.000	6.800	5
80	23.983	2.710	-0.346	0.000	6.800	5
90	23.922	2.926	-0.290	0.000	6.800	5
100	23.816	2.907	-0.239	0.000	6.800	5
200	23.724	2.845	-0.244	0.000	9.600	5
300	23.709	2.731	-0.218	0.000	9.600	5
400	23.771	2.681	-0.192	0.000	9.600	5
500	23.874	2.625	-0.226	0.000	9.600	5
600	23.886	2.645	-0.220	0.000	9.600	5
700	23.919	2.689	-0.225	0.000	9.600	5
800	23.909	2.718	-0.235	0.000	9.600	5
900	23.904	2.707	-0.192	0.000	9.600	5
1,000	23.892	2.709	-0.194	0.000	9.600	5

Cumulative results after each of the first 10 samples, then every 10 through 100, and every 100 through 1,000. “Best |x̄ − μ|” is the smallest absolute deviation from the population mean (μ = 24) observed so far; “Worst |x̄ − μ|” is the largest.

Size 25

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Min	Max	\|x̄ − μ\|	Size
23.320	6.945	24.000	6.000	-0.636	7	34	0.680	25
23.600	6.570	25.000	10.000	-0.482	7	33	0.400	25
23.400	5.401	23.000	7.000	0.113	15	33	0.600	25
23.200	6.519	25.000	10.000	-0.102	11	35	0.800	25
22.400	6.564	23.000	6.000	-0.593	8	34	1.600	25
23.640	5.866	23.000	6.000	-0.065	11	33	0.360	25
22.920	5.729	23.000	6.000	-0.431	9	33	1.080	25
23.720	6.181	24.000	8.000	0.146	13	34	0.280	25
25.080	5.758	25.000	7.000	-0.456	13	35	1.080	25
24.440	5.731	25.000	6.000	-0.626	10	35	0.440	25

Each row summarizes one independently drawn sample of size 25 from the SWLS population (scores 5–35). The |x̄ − μ| column shows how far the sample mean is from the population mean (μ = 24). Values are reported to three decimals.

Representativeness & CLT (Cumulative Sample Means)
Number of Samples	Mean of Means	Standard Error	Skewness of Means	Best \|x̄ − μ\|	Worst \|x̄ − μ\|	Sample Size
1	23.480	0.000	NaN	0.520	0.520	25
2	24.260	1.103	NaN	0.520	1.040	25
3	24.293	0.782	-0.381	0.360	1.040	25
4	23.900	1.013	-0.089	0.360	1.280	25
5	24.168	1.063	-0.507	0.360	1.280	25
6	23.813	1.288	-0.282	0.360	1.960	25
7	23.851	1.180	-0.418	0.080	1.960	25
8	24.105	1.307	-0.326	0.080	1.960	25
9	24.000	1.262	-0.051	0.080	1.960	25
10	23.944	1.203	0.110	0.080	1.960	25
20	23.830	1.358	0.471	0.080	2.440	25
30	23.996	1.218	0.281	0.040	2.440	25
40	24.045	1.183	0.270	0.000	2.440	25
50	24.142	1.263	0.288	0.000	3.240	25
60	24.031	1.214	0.405	0.000	3.240	25
70	24.056	1.212	0.384	0.000	3.240	25
80	24.091	1.190	0.325	0.000	3.240	25
90	24.040	1.157	0.392	0.000	3.240	25
100	24.041	1.158	0.427	0.000	3.240	25
200	24.123	1.128	-0.045	0.000	3.360	25
300	24.093	1.134	-0.103	0.000	3.360	25
400	24.025	1.139	-0.118	0.000	3.520	25
500	23.969	1.147	-0.115	0.000	4.000	25
600	23.956	1.158	-0.103	0.000	4.000	25
700	23.965	1.162	-0.064	0.000	4.000	25
800	23.975	1.181	-0.053	0.000	4.000	25
900	23.979	1.176	-0.093	0.000	4.000	25
1,000	23.994	1.185	-0.084	0.000	4.000	25

Size 100

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Min	Max	\|x̄ − μ\|	Size
23.850	6.051	24.000	8.500	-0.205	10	34	0.150	100
23.970	5.702	24.000	8.250	-0.031	10	35	0.030	100
24.490	5.580	24.000	9.000	0.056	12	35	0.490	100
23.300	5.766	24.000	6.250	-0.451	7	35	0.700	100
24.100	5.519	24.000	8.000	-0.104	11	34	0.100	100
23.690	5.844	24.000	9.000	-0.230	10	35	0.310	100
23.210	6.456	23.000	8.000	-0.483	6	34	0.790	100
23.420	6.611	23.000	11.250	-0.150	6	34	0.580	100
24.500	5.344	25.000	7.000	-0.391	12	34	0.500	100
24.650	6.267	25.000	8.250	-0.230	9	35	0.650	100

Each row summarizes one independently drawn sample of size 100 from the SWLS population (scores 5–35). The |x̄ − μ| column shows how far the sample mean is from the population mean (μ = 24). Values are reported to three decimals.

Representativeness & CLT (Cumulative Sample Means)
Number of Samples	Mean of Means	Standard Error	Skewness of Means	Best \|x̄ − μ\|	Worst \|x̄ − μ\|	Sample Size
1	24.710	0.000	NaN	0.710	0.710	100
2	24.110	0.849	NaN	0.490	0.710	100
3	23.847	0.754	1.609	0.490	0.710	100
4	24.007	0.694	0.022	0.490	0.710	100
5	23.732	0.861	-0.031	0.490	1.370	100
6	23.705	0.773	0.123	0.430	1.370	100
7	23.721	0.707	0.029	0.180	1.370	100
8	23.840	0.735	-0.222	0.180	1.370	100
9	23.933	0.743	-0.437	0.180	1.370	100
10	23.933	0.700	-0.449	0.070	1.370	100
20	23.996	0.607	-0.240	0.060	1.370	100
30	24.141	0.657	0.283	0.040	1.860	100
40	24.140	0.608	0.224	0.020	1.860	100
50	24.105	0.606	0.008	0.020	1.860	100
60	24.089	0.610	0.096	0.020	1.860	100
70	24.075	0.587	0.113	0.020	1.860	100
80	24.040	0.584	0.122	0.020	1.860	100
90	24.031	0.576	0.096	0.020	1.860	100
100	24.046	0.571	0.081	0.020	1.860	100
200	24.034	0.576	0.161	0.010	1.860	100
300	24.032	0.591	0.121	0.010	1.860	100
400	24.027	0.607	-0.021	0.010	1.880	100
500	24.010	0.601	-0.037	0.000	1.880	100
600	24.022	0.591	-0.039	0.000	1.880	100
700	24.023	0.591	-0.000	0.000	1.880	100
800	24.029	0.606	-0.039	0.000	1.880	100
900	24.024	0.615	-0.018	0.000	1.880	100
1,000	24.023	0.609	-0.019	0.000	1.880	100

Size 1000

A Sampling of Ten Samples
Mean	Stdev	Median	IQR	Skewness	Min	Max	\|x̄ − μ\|	Size
23.937	6.094	24.000	9.000	-0.371	5	35	0.063	1,000
23.989	5.824	24.000	8.000	-0.311	5	35	0.011	1,000
24.193	5.840	24.000	9.000	-0.353	6	35	0.193	1,000
24.058	6.102	24.000	9.000	-0.388	6	35	0.058	1,000
23.907	5.940	24.000	8.000	-0.407	6	35	0.093	1,000
24.082	5.974	25.000	9.000	-0.436	6	35	0.082	1,000
23.959	6.237	25.000	9.000	-0.426	5	35	0.041	1,000
24.173	6.068	24.000	9.000	-0.314	6	35	0.173	1,000
23.883	6.121	24.000	9.000	-0.312	5	35	0.117	1,000
24.277	6.005	25.000	8.000	-0.406	5	35	0.277	1,000

Each row summarizes one independently drawn sample of size 1,000 from the SWLS population (scores 5–35). The |x̄ − μ| column shows how far the sample mean is from the population mean (μ = 24). Values are reported to three decimals.

Representativeness & CLT (Cumulative Sample Means)
Number of Samples	Mean of Means	Standard Error	Skewness of Means	Best \|x̄ − μ\|	Worst \|x̄ − μ\|	Sample Size
1	23.906	0.000	NaN	0.094	0.094	1,000
2	23.852	0.076	NaN	0.094	0.202	1,000
3	23.857	0.055	-0.795	0.094	0.202	1,000
4	23.930	0.154	1.494	0.094	0.202	1,000
5	23.928	0.133	1.527	0.082	0.202	1,000
6	23.884	0.160	0.590	0.082	0.333	1,000
7	23.903	0.154	0.140	0.017	0.333	1,000
8	23.913	0.145	-0.102	0.017	0.333	1,000
9	23.927	0.142	-0.361	0.017	0.333	1,000
10	23.925	0.134	-0.304	0.017	0.333	1,000
20	23.954	0.150	0.088	0.017	0.333	1,000
30	23.980	0.164	-0.141	0.017	0.333	1,000
40	23.997	0.166	0.046	0.013	0.382	1,000
50	23.997	0.173	0.063	0.013	0.382	1,000
60	24.002	0.175	-0.026	0.013	0.382	1,000
70	23.991	0.173	0.052	0.013	0.382	1,000
80	24.001	0.173	0.061	0.013	0.382	1,000
90	24.007	0.168	0.001	0.011	0.382	1,000
100	24.000	0.170	0.039	0.011	0.382	1,000
200	24.011	0.178	-0.064	0.002	0.455	1,000
300	24.007	0.178	0.048	0.001	0.476	1,000
400	24.001	0.181	0.123	0.000	0.586	1,000
500	24.004	0.183	0.003	0.000	0.586	1,000
600	23.998	0.186	-0.006	0.000	0.586	1,000
700	23.995	0.189	0.051	0.000	0.644	1,000
800	24.001	0.189	0.022	0.000	0.644	1,000
900	24.003	0.187	0.026	0.000	0.644	1,000
1,000	24.001	0.186	0.048	0.000	0.644	1,000

Methodology for Generating Values

Population. SWLS scores are integers from 5 to 35. We simulate a truncated normal so that the truncated population has mean μ = 24 and SD = 6. We solve for parameters (μ_sim, σ_sim) of an underlying normal and truncate to [5,35].

Sampling. Each sample of size n is drawn independently with the same truncation and integer rounding used in the app.

Statistics. Per-sample: Stdev is the sample SD (n−1); skewness is m₃/n divided by SD³ (uncorrected). For the cumulative table (sample means), skewness is the bias‑corrected estimator. Median and IQR use linear interpolation (Hyndman–Fan Type 7) when needed. Standard Error is the sample SD of sample means.

Equivalent Text: A Deep Look into Sample Representativeness (Random Selection)

Introduction

Size 5

Size 25

Size 100

Size 1000

Methodology for Generating Values