AI Grounds
Open AI Grounds on a desktop
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AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Compare sample means with a normal reference.
Independent draws from one fixed source with finite positive variance. Draw U uniformly from 0 to 1; value = 10 U⁴. Many values are low and fewer are high. This bounded source stays right-skewed.
Known population mean = 2; population SD = 2.6667. Sampling changes neither this source law nor its moments.
Observations inside each of the 1000 averages. Extend or shorten the same sample prefixes. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Compare −r ≤ Z < r. Include the lower edge and exclude the upper edge, matching the bins. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
1000 prepared samples, each with 1 observation from Skewed source. Interval: −1 ≤ Z < 1.
Z = (sample mean − population mean) / (population SD / √n). Theoretical Z mean is 0 and SD is 1 at every n; these moments do not guarantee a normal shape. The fixed standardized axis lets us compare shape, rather than simply shrinking raw averages.
Indigo bars: observed counts. Orange points and line: standard-normal expected counts in the same half-unit bins, rather than a density curve. The shaded interval is [−1, 1).
Finite histogram: 1000 means inside the plotted window, 0 below −6 and 0 above 6. All bins plus those tails total 1000; outside values are counted, not clamped into an edge bin.
Theoretical spread of raw sample means.
2.6667
σ / √n = 2.6667 / √1
Fraction of this finite batch in the interval.
84.1%
841 / 1000 with −1 ≤ Z < 1
Approximate sampling-law comparison.
68.27%
Φ(1) − Φ(−1)
841 of 1000 Z values satisfy −1 ≤ Z < 1, giving observed share 84.1%. Standard-normal reference = 68.27%. A finite simulated share is not an exact source probability or a guaranteed match.
| Z interval | Observed count | Normal expected count |
|---|---|---|
| [-6, -5.5) | 0 | 0.000 |
| [-5.5, -5) | 0 | 0.000 |
| [-5, -4.5) | 0 | 0.003 |
| [-4.5, -4) | 0 | 0.028 |
| [-4, -3.5) | 0 | 0.201 |
| [-3.5, -3) | 0 | 1.117 |
| [-3, -2.5) | 0 | 4.860 |
| [-2.5, -2) | 0 | 16.540 |
| [-2, -1.5) | 0 | 44.057 |
| [-1.5, -1) | 0 | 91.848 |
| [-1, -0.5) | 502 | 149.882 |
| [-0.5, 0) | 166 | 191.462 |
| [0, 0.5) | 102 | 191.462 |
| [0.5, 1) | 71 | 149.882 |
| [1, 1.5) | 52 | 91.848 |
| [1.5, 2) | 53 | 44.057 |
| [2, 2.5) | 31 | 16.540 |
| [2.5, 3) | 23 | 4.860 |
| [3, 3.5) | 0 | 1.117 |
| [3.5, 4) | 0 | 0.201 |
| [4, 4.5) | 0 | 0.028 |
| [4.5, 5) | 0 | 0.003 |
| [5, 5.5) | 0 | 0.000 |
| [5.5, 6] | 0 | 0.000 |
Sample 1 has total 3.053723 / size 1 = mean 3.053723; standardized Z = 0.395146. Its first 1 observations: 3.0537. This existing first sample is illustrative; no special outcome was selected. Display values are rounded; classification uses full precision.
X = 10 U⁴; E(X) = 10 / 5 = 2; Var(X) = 100 / 9 − 4 = 64 / 9. These are declared ideal source moments, not estimated from the displayed batch. Independent draws from a fixed finite-positive-variance source give mean μ and SE σ / √n for the sample mean. Standardization gives theoretical mean 0 and variance 1; normality is an additional asymptotic approximation.
A seed-1309 32-bit generator creates 1000 nonoverlapping blocks of 400 uniform positions. Each source transforms those positions through its declared rule. Size reveals the first n draws in every block; finite pseudorandom examples do not prove independence or convergence. Discrete outcomes can recur; the continuous source also retains the same law on each draw.
The CLT concerns standardized averages as n grows under its assumptions. Approximation quality depends on source and desired accuracy; no universal size-20 rule applies. Finite means may remain discrete and the observed batch may be jagged. Normal CDF values use the Abramowitz–Stegun 26.2.17 numerical approximation. The orange line joins expected bin counts, rather than plotting density. Counts outside −6 to 6 remain explicit. The selected interval includes its lower boundary and excludes its upper boundary; discrete endpoint mass can matter.
Individual observations do not become normal, and a finite histogram neither proves the theorem nor guarantees exact interval shares, monotone improvement or a bound on every sample error. Dependence, infinite-variance sources, confidence intervals, continuity corrections and rigorous approximation-error bounds are outside this lesson.
Berkeley SticiGui · Normal approximation and source dependence