high school Distribution of Sample Mean

Distribution of Sample Mean

If you keep drawing a sample from a population and taking its mean, those sample means form a distribution of their own. Remarkably, the mean of the sample means stays exactly the population mean μ, but their standard deviation shrinks to σ/√n, so the larger the sample, the more tightly the distribution gathers around the population mean. Since the width is inversely proportional to √n, making the sample four times bigger halves the spread, and once n is large enough the shape approaches a normal distribution. Grow the sample-size n slider and the distribution of sample means pulls in narrow around the population mean.

Bigger Samples Concentrate the Mean

Sample size n16

👀 See It

①The sample mean X̄ is itself a random variable that varies by sample

②As n grows, the distribution of X̄ narrows around the population mean

③The width is proportional to σ/√n — quadruple n and the width halves

④The population mean is μ. Central limit theorem: for large n, X̄ is nearly normal even if the population is not. The sketch is the bell only

Mean and Standard Deviation of X̄

Expectation and SD of the sample mean

E(X̄) = μ, σ(X̄) = σ/√n

Population mean μ, population SD σ, sample size n — the mean stays, the SD scales by 1/√n

Normal approximation

if n is large, X̄ ~ N(μ, σ²/n)

Even if the population is not normal, for large n the sample mean approaches a normal distribution

The Effect of √n

📏 Width Is Inversely Proportional to √n

①Since σ(X̄)=σ/√n, the SD is inversely proportional to √n

②Quadrupling n halves the SD

③To double the precision you must quadruple the sample

Compute It Directly

Example 1

From a population with mean 50 and SD 8, a sample of size 16 is drawn. Find the mean and SD of the sample mean.

Substitute into E(X̄)=μ, σ(X̄)=σ/√n.

E(X̄) = 50, σ(X̄) = 8/√16

Compute with √16=4.

σ(X̄) = 8/4 = 2

▸ mean 50, SD 2

The mean stays equal to the population mean; only the SD is divided by √n.

Example 2

From the same population, if the sample size grows to 64, what is the SD of the sample mean?

σ(X̄)=8/√64.

Compute with √64=8.

σ(X̄) = 8/8 = 1

▸ 1 (n ×4 → SD ×1/2)

Growing n from 16 to 64 (×4) halves the SD from 2 to 1.

Wrap-up

Key result

E(X̄)=μ, σ(X̄)=σ/√n, (large n) X̄ ~ N(μ, σ²/n)

The sample mean has mean μ and SD σ/√n — for large n it is normal

exam-style

From a population with SD 10, a sample of size 25 is drawn. What is the SD of the sample mean?

①0.4

②2

③5

④10

⑤50

▸ ② 2

Put σ=10, n=25 into σ(X̄)=σ/√n.

σ(X̄) = 10/√25

Compute with √25=5.

σ(X̄) = 10/5 = 2

🎯 Exam Points

①E(X̄)=μ (same as population mean)

②σ(X̄)=σ/√n (divide by √n)

③for large n, X̄~N(μ, σ²/n)

④double precision needs 4× the sample

⑤do not confuse the variance V(X̄)=σ²/n

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Statistical Estimation

Limit of a Sequence

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