Edexcel IAL - Statistics 3- 3.2 Distribution of the Sample Mean- Study notes  - New syllabus

The Distribution of the Sample Mean

When a random sample is taken from a population, the sample mean \( \mathrm{\bar{X}} \) is itself a random variable. Understanding its distribution is essential for making inferences about the population mean.

Definition of the Sample Mean

For a random sample of size \( \mathrm{n} \) with observations \( \mathrm{X_1, X_2, \dots, X_n} \), the sample mean is defined as 

\( \mathrm{\bar{X} = \dfrac{1}{n}\sum_{i=1}^{n} X_i} \)

Since the sample observations are random variables, \( \mathrm{\bar{X}} \) is also a random variable.

Mean of the Sample Mean

The expected value of the sample mean is equal to the population mean:

\( \mathrm{E(\bar{X}) = \mu} \)

This means that \( \mathrm{\bar{X}} \) is an unbiased estimator of \( \mathrm{\mu} \).

Variance of the Sample Mean

The variance of the sample mean is given by

\( \mathrm{Var(\bar{X}) = \dfrac{\sigma^2}{n}} \)

As the sample size increases, the variance decreases, meaning the sample mean becomes more precise.

Distribution When the Population is Normal

If the population distribution is normal,

\( \mathrm{X \sim N(\mu,\,\sigma^2)} \)

then the distribution of the sample mean is also normal:

\( \mathrm{\bar{X} \sim N\!\left(\mu,\,\dfrac{\sigma^2}{n}\right)} \)

This result may be used directly. No proof is required.

Interpretation

• The sample mean has the same mean as the population
• The spread of the sample mean decreases as \( \mathrm{n} \) increases
• Larger samples give more reliable estimates of the population mean

Key Formula Summary

\( \mathrm{E(\bar{X}) = \mu} \)

\( \mathrm{Var(\bar{X}) = \sigma^2/n} \)

If \( \mathrm{X \sim N(\mu, \sigma^2)} \), then \( \mathrm{\bar{X} \sim N(\mu, \sigma^2/n)} \)