Independent, equal-chance draws with replacement from a known fixed population. Records can repeat. Current source values: 2, 4, 6, 8; population mean = 5. This is an ideal model illustrated by reproducible samples.

Sample size

Number of observations inside each mean. Extend or shorten the same prepared samples. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.

Sample number

Highlight another of the same 200 sample means, without changing the batch. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.

Selected sample 1 of 200. Each sample contains 1 observation from Bounded values. The repetition count 200 is different from sample size n = 1.

200 sample means

A sampling distribution describes a statistic over possible repeated samples. These bars count a finite batch of 200 sample means, not individual observations or an exact probability law. Orange dashed line: population mean 5. Indigo solid line: selected mean 6.00000.

Observed means range from 2.00000 to 8.00000. Each bin has width 0.25 and includes its lower endpoint, excluding its upper endpoint; the last includes 10. Count axis remains 0–200 as size changes.

Population SD

Spread of individual source values.

2.2361

σ = √mean[(value − 5)²]

Theoretical SE

SD of the model’s sample-mean distribution.

2.2361

σ / √n = 2.2361 / √1

Batch SD

Descriptive spread of these 200 means.

2.1821

√[Σ(mean − batch average)² / 200]

Sample 1: total 6 / size 1 = mean 6.00000. Signed gap from population mean 5 = 1.00000; absolute error = 1.00000. Theoretical SE = 2.236068.

Batch average of 200 means = 4.720000. Batch SD = 2.182109, computed around this batch average with denominator 200; it need not equal theoretical SE. Recent individual observations in the selected sample: 6.

Exact means of the 200 samples

Each sample has 1 observations. Selecting a sample changes its marker only.
Sample	Total	Mean
1 · selected	6	6.00000
2	2	2.00000
3	4	4.00000
4	8	8.00000
5	8	8.00000
6	6	6.00000
7	8	8.00000
8	2	2.00000
9	4	4.00000
10	4	4.00000
11	4	4.00000
12	6	6.00000
13	6	6.00000
14	2	2.00000
15	8	8.00000
16	4	4.00000
17	4	4.00000
18	8	8.00000
19	8	8.00000
20	4	4.00000
21	8	8.00000
22	2	2.00000
23	8	8.00000
24	2	2.00000
25	2	2.00000
26	4	4.00000
27	8	8.00000
28	6	6.00000
29	8	8.00000
30	4	4.00000
31	8	8.00000
32	2	2.00000
33	6	6.00000
34	4	4.00000
35	4	4.00000
36	6	6.00000
37	6	6.00000
38	4	4.00000
39	2	2.00000
40	2	2.00000
41	2	2.00000
42	4	4.00000
43	8	8.00000
44	6	6.00000
45	4	4.00000
46	8	8.00000
47	6	6.00000
48	4	4.00000
49	4	4.00000
50	2	2.00000
51	2	2.00000
52	6	6.00000
53	8	8.00000
54	4	4.00000
55	6	6.00000
56	8	8.00000
57	6	6.00000
58	4	4.00000
59	2	2.00000
60	6	6.00000
61	6	6.00000
62	6	6.00000
63	2	2.00000
64	2	2.00000
65	2	2.00000
66	4	4.00000
67	4	4.00000
68	2	2.00000
69	6	6.00000
70	2	2.00000
71	2	2.00000
72	4	4.00000
73	6	6.00000
74	2	2.00000
75	8	8.00000
76	4	4.00000
77	8	8.00000
78	4	4.00000
79	6	6.00000
80	6	6.00000
81	8	8.00000
82	8	8.00000
83	4	4.00000
84	6	6.00000
85	6	6.00000
86	8	8.00000
87	2	2.00000
88	4	4.00000
89	4	4.00000
90	4	4.00000
91	4	4.00000
92	2	2.00000
93	6	6.00000
94	6	6.00000
95	6	6.00000
96	2	2.00000
97	4	4.00000
98	2	2.00000
99	6	6.00000
100	4	4.00000
101	4	4.00000
102	2	2.00000
103	2	2.00000
104	6	6.00000
105	2	2.00000
106	8	8.00000
107	6	6.00000
108	2	2.00000
109	8	8.00000
110	6	6.00000
111	6	6.00000
112	4	4.00000
113	4	4.00000
114	4	4.00000
115	2	2.00000
116	8	8.00000
117	2	2.00000
118	2	2.00000
119	2	2.00000
120	8	8.00000
121	8	8.00000
122	8	8.00000
123	6	6.00000
124	4	4.00000
125	2	2.00000
126	2	2.00000
127	4	4.00000
128	2	2.00000
129	2	2.00000
130	2	2.00000
131	8	8.00000
132	4	4.00000
133	4	4.00000
134	2	2.00000
135	6	6.00000
136	2	2.00000
137	4	4.00000
138	6	6.00000
139	8	8.00000
140	8	8.00000
141	6	6.00000
142	6	6.00000
143	6	6.00000
144	8	8.00000
145	8	8.00000
146	4	4.00000
147	2	2.00000
148	2	2.00000
149	2	2.00000
150	6	6.00000
151	2	2.00000
152	2	2.00000
153	6	6.00000
154	4	4.00000
155	6	6.00000
156	4	4.00000
157	8	8.00000
158	4	4.00000
159	4	4.00000
160	2	2.00000
161	6	6.00000
162	6	6.00000
163	2	2.00000
164	2	2.00000
165	2	2.00000
166	6	6.00000
167	4	4.00000
168	6	6.00000
169	4	4.00000
170	6	6.00000
171	2	2.00000
172	4	4.00000
173	8	8.00000
174	8	8.00000
175	4	4.00000
176	2	2.00000
177	4	4.00000
178	2	2.00000
179	6	6.00000
180	4	4.00000
181	2	2.00000
182	8	8.00000
183	4	4.00000
184	4	4.00000
185	2	2.00000
186	4	4.00000
187	6	6.00000
188	2	2.00000
189	2	2.00000
190	8	8.00000
191	6	6.00000
192	8	8.00000
193	8	8.00000
194	4	4.00000
195	4	4.00000
196	6	6.00000
197	2	2.00000
198	8	8.00000
199	8	8.00000
200	4	4.00000

Exact histogram counts

40 bins of width 0.25. Counts sum to 200; empty bins included.
Mean interval	Count
[0, 0.25)	0
[0.25, 0.5)	0
[0.5, 0.75)	0
[0.75, 1)	0
[1, 1.25)	0
[1.25, 1.5)	0
[1.5, 1.75)	0
[1.75, 2)	0
[2, 2.25)	56
[2.25, 2.5)	0
[2.5, 2.75)	0
[2.75, 3)	0
[3, 3.25)	0
[3.25, 3.5)	0
[3.5, 3.75)	0
[3.75, 4)	0
[4, 4.25)	56
[4.25, 4.5)	0
[4.5, 4.75)	0
[4.75, 5)	0
[5, 5.25)	0
[5.25, 5.5)	0
[5.5, 5.75)	0
[5.75, 6)	0
[6, 6.25)	48
[6.25, 6.5)	0
[6.5, 6.75)	0
[6.75, 7)	0
[7, 7.25)	0
[7.25, 7.5)	0
[7.5, 7.75)	0
[7.75, 8)	0
[8, 8.25)	40
[8.25, 8.5)	0
[8.5, 8.75)	0
[8.75, 9)	0
[9, 9.25)	0
[9.25, 9.5)	0
[9.5, 9.75)	0
[9.75, 10]	0

Assumptions and reproducible sampling

For independent draws with replacement from a fixed finite-variance source, Var(sample mean) = population variance / n and SE = σ / √n. These known-source identities require no normal shape. We use population SD, not an estimated sample SD; dependent or without-replacement designs require other formulas.

Seed 1309 creates 200 nonoverlapping blocks of 400 uniform draw positions using a 32-bit generator. Size reveals each block’s prefix; source changes map the same positions to new values. Sample number highlights one existing mean without resampling. Finite pseudorandom illustrations do not prove independence or convergence.

Theoretical SE describes all possible means under the model. The histogram and descriptive Batch SD summarize only 200 realized means; denominator 200 treats that batch as the complete displayed list. Neither quantity bounds one sample’s error or guarantees that every larger prefix improves. More repeated samples refine a distribution illustration; they do not change n or theoretical SE. Normal approximation, estimated SE, confidence intervals, dependence and bias correction are outside this lesson.

Berkeley SticiGui · Standard error and the square-root rule

Open AI Grounds on a desktop

Sampling Distributions & Standard Error

200 sample means

Population SD

Theoretical SE

Batch SD