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Grade 11-12 (age 16-18)

high school Distribution of Sample Mean

Draw a sample of size n from a population and take its mean, repeat this many times, and the sample means themselves form a distribution. The mean of the sample means equals the population mean μ, but their standard deviation shrinks to σ/√n, so the larger the sample, the closer the sample means gather around the population mean. Because the width is inversely proportional to √n, making the sample 4 times larger halves the spread, and for large enough n the distribution comes close to normal whatever the shape of the population. Increase the sample size slider n and, compared with the faint dashed population curve, the blue distribution of the sample mean squeezes in around μ, while the line under the figure says what multiple of σ its standard deviation is.

Larger Samples Pull the Mean In

16
👀 Reading the figure
①The sample mean X̄ is a random variable too, since its value changes from sample to sample
②The faint dashed curve is the population distribution, and the blue curve is the distribution of the mean X̄ of a sample of size n
③As n grows, the distribution of X̄ gets narrower and taller around μ, and about 68% of X̄ falls in the shaded band μ ± σ/√n
④The figure draws a normal population, but by the central limit theorem X̄ gets close to normal for large n even when the population is not normal

Mean and Standard Deviation of the Sample Mean

Expected value and standard deviation of the sample mean
E(X̄) = μ, σ(X̄) = σ/√n
Population mean μ, population standard deviation σ, sample size n. The mean stays the same and the standard deviation becomes 1/√n times as large
Normal approximation
if n is large, X̄ ~ N(μ, σ²/n)
Even if the population is not normal, X̄ gets close to a normal distribution when n is large

The Effect of √n

📏 The width is inversely proportional to √n
①Since σ(X̄) = σ/√n, the standard deviation is inversely proportional to √n
②Multiplying n by 4 multiplies the standard deviation by 1/2
③Halving the spread takes a sample 4 times larger, and cutting it to a tenth takes one 100 times larger

Worked Examples

Example 1
A sample of size 16 is drawn from a population with mean 50 and standard deviation 8. Find the mean and the standard deviation of the sample mean.
1
Substitute into E(X̄) = μ and σ(X̄) = σ/√n.
E(X̄) = 50, σ(X̄) = 8/√16
2
Divide by √16 = 4.
σ(X̄) = 8/4 = 2
▸ mean 50, standard deviation 2
The mean stays equal to the population mean; only the standard deviation is divided by √n.
Example 2
For the same population, what is the standard deviation of the sample mean when the sample size is raised to 64?
1
σ(X̄) = 8/√64.
2
Divide by √64 = 8.
σ(X̄) = 8/8 = 1
▸ 1 (n is 4 times larger, so the standard deviation is 1/2 as large)
Raising n 4 times, from 16 to 64, halves the standard deviation from 2 to 1.

Summary

Key results
E(X̄)=μ, σ(X̄)=σ/√n, (large n) X̄ ~ N(μ, σ²/n)
The sample mean has mean μ and standard deviation σ/√n, and is close to normal for large n
2021 CSAT Math type, adapted
A sample of size 25 is drawn from a population with standard deviation 10. What is the standard deviation of the sample mean?
①0.4
②2
③5
④10
⑤50
▸ ② 2
1
Put σ = 10 and n = 25 into σ(X̄) = σ/√n.
σ(X̄) = 10/√25
2
Divide by √25 = 5.
σ(X̄) = 10/5 = 2
🎯 Exam Points
①E(X̄) = μ, equal to the population mean
②σ(X̄) = σ/√n, divided by √n
③For large n, X̄ is close to N(μ, σ²/n)
④To halve the spread, make the sample 4 times larger
⑤Do not confuse the variance V(X̄) = σ²/n with the standard deviation σ/√n
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Limit of a Sequence

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