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CIE A-Level Maths Study Notes

5.4.4 Distribution of the Sample Mean

This section will comprehensively explore how the sample mean behaves, especially when dealing with normal distributions or large sample sizes, and its implications in statistical analysis.

Introduction

The sample mean (Xˉ)(\bar{X}) is the average of a set of data from a population. It's key for understanding how averages of samples distribute, particularly with normal distributions or large samples.

Normal Distribution for Sample Mean

When samples come from a normally distributed population, Xˉ\bar{X} also has a normal distribution.

Key Points:

  • Mean: Same as the population mean (μ)(\mu).
  • Variance: Population variance divided by sample size (σ2/n)(\sigma^2/n).
  • Standard Deviation: σ/n\sigma/\sqrt{n}.

Central Limit Theorem (CLT)

CLT shows that Xˉ\bar{X} becomes normally distributed as sample size grows, regardless of the population's distribution.

Understanding CLT:

  • Large Samples: Typically, 30 or more.
  • Distribution: Xˉ\bar{X} becomes normal.
  • Mean and Variance: Mean is μ\mu, and variance is σ2/n\sigma^2/n.

Examples

Example 1: Normal Population

A population is normally distributed with a mean (μ)( \mu ) of 50 and a standard deviation (σ)( \sigma ) of 10. We are to find the distribution of the sample mean (Xˉ)( \bar{X} ) for a sample size (n)( n ) of 25.

Solution:

  • Population Parameters: μ=50\mu = 50, σ=10\sigma = 10.
  • Sample Size: n=25n = 25.
  • Calculating the Standard Deviation of ( \bar{X} ):
    • Formula: σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
    • Calculation: σXˉ=1025=105=2\sigma_{\bar{X}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2.
  • Result:
    • The sample mean Xˉ\bar{X} follows a normal distribution.
    • Mean of Xˉ=50 \bar{X} = 50 (same as the population mean).
    • Standard Deviation of Xˉ=2\bar{X} = 2.
Normal Population Graph

Example 2: Applying the CLT

Given a population with an unspecified distribution, a mean of 30, and a standard deviation of 6, we need to determine the characteristics of the sample mean's distribution for a sample size of 50.

Solution:

  • Population Parameters: μ=30\mu = 30, σ=6\sigma = 6.
  • Sample Size: n=50n = 50.
  • Applying the CLT: Given the large sample size, the CLT suggests that Xˉ\bar{X} will be approximately normally distributed.
  • Calculating the Standard Deviation of ( \bar{X} ):
    • Formula: σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
    • Calculation: σXˉ=6500.85\sigma_{\bar{X}} = \frac{6}{\sqrt{50}} \approx 0.85 (rounded off to two decimal places).
  • Result:
    • The sample mean Xˉ\bar{X} is approximately normally distributed.
    • Mean of Xˉ\bar{X} =30= 30 (same as the population mean).
    • Standard Deviation of Xˉ0.85\bar{X} ≈ 0.85.
Applying the CLT Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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