Galton Board: Sampling & the Sample Mean
Watch balls build a Binomial histogram, see the sample mean approach μ, and compare that with the sampling distribution of x̄.
Board setup
Peg rows (n)12
P(bounce right) = p0.50
Drop balls
Each ball's slot = number of right bounces — a draw from Binomial(n, p).
Population vs sample
Balls dropped0
| − μ|—
Law of large numbers: as more balls drop, the sample mean approaches μ (population mean).
Sampling distribution of mean
Sample size N
# of sample means0
mean of sample means—
Each point is the mean of N independent balls. Larger N → tighter cluster around μ. (
Why this matters (M1)
- One ball ≈ one observation from the population distribution (here Binomial).
- The histogram of many balls ≈ the population distribution.
- The histogram of many sample means ≈ the sampling distribution of the sample mean.
- Point estimate: use the sample mean to estimate μ — it gets closer as the sample grows.
Galton board
Indigo bars = observed counts · lilac = expected under Binomial(n, p)
Sample mean → population mean
Green path = running sample mean; dashed indigo = μ. Watch the green path settle onto μ.
Sampling distribution of the sample mean (N = 10)
Collect sample means from the sidebar to build this histogram.
Compare: ball histogram (population shape) vs this chart (how sample means spread). Try larger N — the sampling distribution gets narrower.