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

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

**Nature**: As $\bar{X}$ 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 $(\mu)$. 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: $\frac{\sigma^2}{n}$, where $\sigma^2$ is population variance and $(n)$ is sample size.

## Examples

### Example 1: Student Scores

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

**Solution:**

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

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

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

**Formula:**$\text{Var}(\bar{X}) = \frac{\sigma^2}{n}$.**Calculation:**$\text{Var}(\bar{X}) = \frac{25}{40} = 0.625$.

**Conclusion**: The variance of the sample mean is 0.625, suggesting that $\bar{X}$ 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.

#### Solution:

**1. Distribution of Sample Mean**:

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

**2. Expected Value and Variance**:

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

**Interpretation**:

**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.

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

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.