Variance with Examples in Statistics

 

Certainly! In statistics, variance is a measure of how spread out the values of a random variable are around the mean or expected value. It quantifies the dispersion or variability of a random variable from its mean, providing important insights into the consistency or unpredictability of outcomes. Let's explore the concept of variance in detail, along with examples and numerical illustrations.

Definition and Notation:

For a random variable $X$ with mean $\mu = E(X)$ , the variance $\text{Var}(X)$ is defined as:

$\text{Var}(X) = E[(X - \mu)^2]$

Alternatively, it can be expressed as:

$\text{Var}(X) = E(X^2) - [E(X)]^2$

Where:

• $E(X)$ is the expected value or mean of $X$ .
• $E(X^2)$ is the expected value of $X^2$ .

Interpretation:

The variance measures how much the values of $X$ deviate from the mean $E(X)$ . A higher variance indicates that the values are more spread out from the mean, while a lower variance indicates the values are closer to the mean, suggesting greater consistency.

Examples:

Example 1: Coin Toss

Consider a fair coin toss where $X$ is a random variable representing the number of heads in a single toss.

• Random Variable $X$ : Possible outcomes are $\{0, 1\}$ .
• Probability Mass Function (PMF): $P(X = 0) = P(X = 1) = 0.5$ .

To find $\text{Var}(X)$ :

1. Calculate $E(X)$ : $E(X) = 0 \cdot 0.5 + 1 \cdot 0.5 = 0.5$

2. Calculate $E(X^2)$ : $E(X^2) = 0^2 \cdot 0.5 + 1^2 \cdot 0.5 = 0 + 0.5 = 0.5$

3. Find $\text{Var}(X)$ : $\text{Var}(X) = E(X^2) - [E(X)]^2 = 0.5 - (0.5)^2 = 0.5 - 0.25 = 0.25$

So, the variance of $X$ , the number of heads in a fair coin toss, is $0.25$ .

Example 2: Normal Distribution

Let $X$ follow a normal distribution $N(\mu, \sigma^2)$ , where $\mu$ is the mean and $\sigma^2$ is the variance.

• Random Variable $X$ : $X \sim N(\mu, \sigma^2)$ .
• Probability Density Function (PDF): $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right)$ .

To find $\text{Var}(X)$ :

1. Mean $\mu$ : $E(X) = \mu$ .

2. Calculate $E(X^2)$ : $E(X^2) = \int_{-\infty}^{\infty} x^2 \cdot f(x) \, dx = \mu^2 + \sigma^2$

3. Find $\text{Var}(X)$ : $\text{Var}(X) = E(X^2) - [E(X)]^2 = \mu^2 + \sigma^2 - \mu^2 = \sigma^2$

Thus, the variance of $X$ in a normal distribution $N(\mu, \sigma^2)$ is $\sigma^2$ .

Importance of Variance:

• Measurement of Spread: Variance quantifies the spread or dispersion of data points around the mean.
• Decision Making: It helps in assessing risk, uncertainty, and variability in various fields such as finance, engineering, and natural sciences.
• Comparison: Variance allows comparison of the variability of different datasets or processes.

Conclusion:

Variance is a crucial statistical measure that provides insights into the dispersion of random variables around their mean values. Whether in simple discrete scenarios like coin tosses or continuous distributions like the normal distribution, understanding variance enhances our ability to analyze and interpret data, making it a fundamental concept in statistical analysis and inference.