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CIE A-Level Maths Study Notes

5.4.3 Sample Mean as a Random Variable

A deep understanding of the sample mean as a random variable is essential. This section expands on the properties of the sample mean, its significance in statistical analysis, and how these properties are applied in problem-solving scenarios.

Introduction to Sample Mean

The sample mean, Xˉ\bar{X}, is crucial in statistics for indicating a sample's average value, selected from a larger population.

Definition: Calculated as Xˉ=i=1nXin\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}, where XiX_i are sample observations and nn is the sample size.

Nature: As Xˉ\bar{X} varies with each sample, it's considered a random variable, essential for estimating the population's central tendency.

Properties of Sample Mean

1. Expected Value (Mean): The average of sample means equals the population mean (μ)(\mu). It shows the sample mean is a good guess for the population mean.

2. Variance: How spread out the sample means are. Less spread with bigger samples. Formula: σ2n\frac{\sigma^2}{n}, where σ2\sigma^2 is population variance and (n)(n) is sample size.


Example 1: Student Scores

A sample of 40 students' scores, with a population variance (σ2)(\sigma^2) of 25. Calculate the expected value and variance of sample mean.


1. Expected Value of Sample Mean (E(Xˉ))(E(\bar{X})):

  • E(Xˉ)E(\bar{X}) equals the population mean (μ)(\mu).
  • If sample mean (Xˉ)(\bar{X}) is 68, then E(Xˉ)=68E(\bar{X}) = 68.

2. Variance of Sample Mean (Var(Xˉ))(\text{Var}(\bar{X})):

  • Formula: Var(Xˉ)=σ2n\text{Var}(\bar{X}) = \frac{\sigma^2}{n}.
  • Calculation: Var(Xˉ)=2540=0.625\text{Var}(\bar{X}) = \frac{25}{40} = 0.625.

Conclusion: The variance of the sample mean is 0.625, suggesting that Xˉ\bar{X} is a reliable estimator of the population mean.

Example 2: Central Limit Theorem (CLT)

For a sample of 50 from a non-normal distribution with mean 75 and variance 36, apply the Central Limit Theorem to determine the expected value and variance of the sample mean.


1. Distribution of Sample Mean:

  • With n=50n = 50 (adequate for CLT), the sample mean (Xˉ)(\bar{X})approximates a normal distribution.

2. Expected Value and Variance:

  • Expected Value (E(Xˉ))(E(\bar{X})): According to CLT, E(Xˉ)=μ=75E(\bar{X}) = \mu = 75.
  • Variance (Var(Xˉ))(\text{Var}(\bar{X})): Var(Xˉ)=σ2n=3650=0.72.\text{Var}(\bar{X}) = \frac{\sigma^2}{n} = \frac{36}{50} = 0.72.


  • Sample Mean's Distribution: Norma with mean = 75.
  • Variance = 0.72: This shows the spread of sample means around 75.
  • Implication: Normal distribution analysis is applicable to (\bar{X}), despite the original population's non-normal distribution.
Central Limit Theorem Graph

Demonstrating the effect of the Central Limit Theorem, with a sample mean distribution becoming normal.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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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