AI Grounds
Open AI Grounds on a desktop
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Compare estimates from repeated samples.
The ideal model selects each population record with equal probability on every independent draw, with replacement: records can repeat. These fixed toy sources use reproducible simulated samples. Population details start hidden.
Show more or fewer observations from the same prepared sample, without resampling it. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Select another prepared sample from the same source at the same size. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Showing sample 1 of 20, with 4 observations from Bounded values. Population records and mean are hidden until you reveal them.
Each row is one sample mean, not an individual observation. A double ring marks the selected sample. The population reference stays hidden until revealed.
Twenty observed means range from 2.5000 to 7.5000. This is a finite illustrative batch. The horizontal scale uses this source’s full value range 2–8; changing Sample size does not shrink the axis or the source.
Statistic from the selected observations.
3.0000
12 / 4
Fixed source parameter, initially unknown.
Hidden
Reveal population to compare
Selected estimate’s distance from truth.
Hidden
|sample mean − population mean|
Sample 1: total 12 / size 4 = mean 3.0000. Population mean and error remain hidden.
Recent individual observations: 6, 2, 2, 2. This selected sample’s individual values range from 2 to 6. These are observations, not averages.
| Sample | Total | Mean |
|---|---|---|
| 1 · selected | 12 | 3.0000 |
| 2 | 14 | 3.5000 |
| 3 | 20 | 5.0000 |
| 4 | 18 | 4.5000 |
| 5 | 18 | 4.5000 |
| 6 | 16 | 4.0000 |
| 7 | 20 | 5.0000 |
| 8 | 24 | 6.0000 |
| 9 | 22 | 5.5000 |
| 10 | 12 | 3.0000 |
| 11 | 24 | 6.0000 |
| 12 | 24 | 6.0000 |
| 13 | 26 | 6.5000 |
| 14 | 30 | 7.5000 |
| 15 | 14 | 3.5000 |
| 16 | 12 | 3.0000 |
| 17 | 18 | 4.5000 |
| 18 | 10 | 2.5000 |
| 19 | 24 | 6.0000 |
| 20 | 20 | 5.0000 |
A population parameter describes the source; a sample statistic estimates it. The ideal independent, fixed-source, equal-record model uses replacement, so repeated values are expected and sample size can exceed the number of source records. This is not sampling without replacement or a census. All supplied sources are bounded and have finite means.
A seed-1309 deterministic 32-bit generator creates 20 nonoverlapping blocks of 1000 positions each. Sample number selects a block; Sample size reveals its prefix. Next sample chooses the next block, stopping at 20; the exact editor can revisit any prepared block. Changing source remaps those same positions. Finite pseudorandom illustrations are not proof of independence, randomness, monotone improvement or exact finite estimates.
Larger independent samples from a fixed bounded source tend to reduce mean-estimate variability. A particular sample or larger prefix can still be worse, and the original observations do not contract around their mean. Biased sampling, formal sampling distributions, standard error and confidence intervals require later lessons.
Berkeley SticiGui · Samples, statistics and populations