Topic 4 of 7

Sampling distributions and the central limit theorem

Step 17 · Statistics and probability topics

See that a statistic like a sample mean changes from sample to sample, measure that variation with the standard error, and use the central limit theorem to describe it.

Sample statistics vary

A parameter, such as the population mean μ, is a fixed number we usually do not know. A statistic, such as the sample mean x̄, is worked out from a sample, and it changes from one random sample to the next. The pattern of those values is the sampling distribution.

Example: Many samples

A population has mean 50. Five random samples give means 48.2, 51.5, 49.7, 50.9 and 50.1. What do they show?

Each sample mean misses 50 by a little, sometimes above and sometimes below.
They cluster around 50.

AnswerThe sample mean varies, but centers on the population mean

Example: Random sampling matters

A survey of gym members finds people exercise 5 hours a week. Can this estimate the whole town?

Gym members are not a random sample of the town.
They probably exercise more than most.

AnswerNo: the sample is biased

The standard error

The standard deviation of a sampling distribution is called the standard error. For a sample mean it is σ ÷ √n, so a sample four times as large has half the standard error.

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \qquad \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}

Example: A sample mean

Adult weights have σ = 15 kg. What is the standard error of the mean weight of 25 people?

σ ÷ √n is 15 ÷ 5.

Answer3 kg

Example: A sample proportion

40% of voters favor a plan. What is the standard error of the sample proportion p̂ for a sample of 600?

$\sqrt{\frac{0.4 \times 0.6}{600}} = \sqrt{0.0004}$ .

Answer0.02, or 2 percentage points

The central limit theorem

For large enough random samples, the sampling distribution of x̄ is approximately normal, whatever the shape of the population. This is the central limit theorem, and it is why normal tables work for averages.

Means of 1,000 samples of size 40 from a skewed population with mean 50 form a bell shape around 50.

Example: Using the theorem

Waiting times are skewed, with mean 8 minutes and SD 6 minutes. For a random sample of 36 waits, what is the probability that the mean is over 9 minutes?

The standard error is 6 ÷ 6 = 1.
z = (9 − 8) ÷ 1, which is 1.
About 16% of a normal distribution lies above z = 1.

AnswerAbout 0.16

Try it yourself

Work each one on paper first, then open the answer to check.

Is the mean of a sample a parameter or a statistic?
Show answer
A statistic
σ = 20 and n = 100. What is the standard error of x̄?
Show answer
2
How much larger must a sample be to halve the standard error?
Show answer
4 times as large
p = 0.5 and n = 100. What is the standard error of p̂?
Show answer
0.05
A population has mean 70. What is the mean of the sampling distribution of x̄?
Show answer
70
Scores have μ = 100 and σ = 15. For samples of 25, what share of sample means fall between 94 and 106?
Show answer
About 95%