Sampling variation and distributions of sample means | Year 12

01 / LEARN

The idea, step by step

Imagine repeating the sampling process

A sampling distribution describes the values a statistic would take across repeated samples collected in the same way. It is not the distribution of individual people or objects in one sample. Different samples produce different means, even when the population has not changed.

For independent observations with finite population variance, the sample mean has mean mu and standard deviation sigma divided by the square root of n. This smaller spread is called the standard error of the mean.

E (\bar{X}) = μ, SD (\bar{X}) = \frac{σ}{\sqrt{n}}

Know why normality is being used

A normal population gives an exactly normal sample mean. Otherwise the central limit theorem supports an approximately normal mean for sufficiently large independent random samples. Strong skew or extreme outliers can require larger samples; no single sample-size rule makes all situations safe.

Increasing sample size reduces random variation but does not cure a biased sampling method. Before calculating a probability, state the sampling assumptions, identify the correct standard error, and say whether the normal result is exact or approximate.

A common trap

Believing that a large biased sample becomes representative.

Instead: A larger sample reduces random sampling variation, not a systematic selection bias.

02 / WORKED EXAMPLES

See the thinking

Example 1

A probability for a sample mean

A population has mean 100 and SD 20. For 25 independent observations from a normal population, find P(mean>108).

Standard error $= 20 / \sqrt{25} = 4$ .
The standardised cut-off is $(108 - 100) / 4 = 2$ .
The upper-tail probability is 0.0228; normality is exact here.

0.0228.

Example 2

What a bigger sample changes

A population has SD 18. Compare the standard errors for independent random samples of 36 and 144. Explain what changes.

For n=36, SE $= 18 / 6 = 3$ .
For n=144, SE $= 18 / 12 = 1.5$ .
Quadrupling sample size halves the standard error; it does not change the population SD.

3 and 1.5; the sample means become less variable.

Check your understanding

What sample size is needed to halve the standard error, all else unchanged?

Show answer

Four times the sample size.

Standard error is proportional to $1 / \sqrt{n}$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A population has SD 15. Find the standard error for n=25 independent observations.
2 marks
A population has mean 42. What is the mean of the sampling distribution of a random sample mean?
2 marks
Is the mean of 4 normal observations exactly normal or only approximately normal?
2 marks
Why may a sample of 30 be insufficient for a very skewed population?
2 marks
A sample mean has SE 2 at n=100. What is its SE at n=25 with the same population SD?
2 marks
A website surveys only volunteers. Would increasing its sample from 500 to 5000 remove self-selection bias?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A population has SD 15. Find the standard error for n=25 independent observations.
2 marks
$15 / \sqrt{25} = 3$ .
3.
A population has mean 42. What is the mean of the sampling distribution of a random sample mean?
2 marks
Under independent identically distributed sampling, the sample mean is unbiased for the population mean.
42.
Is the mean of 4 normal observations exactly normal or only approximately normal?
2 marks
Independent normal linear combinations are normal.
Exactly normal, assuming independence.
Why may a sample of 30 be insufficient for a very skewed population?
2 marks
Convergence depends on shape and tails, not only a fixed n threshold.
The approximation can still be poor.
A sample mean has SE 2 at n=100. What is its SE at n=25 with the same population SD?
2 marks
Reducing n by a factor of four doubles SE.
4.
A website surveys only volunteers. Would increasing its sample from 500 to 5000 remove self-selection bias?
2 marks
The same selection mechanism can remain biased even with a much larger sample.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods: continuous random variables, normal distributions and sampling distributions of sample means. Original lessons and practice.

Official curriculum source ↗

Back to Continuous probability and normal distributions