Variance Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
A data set has n=5n=5 values with mean ฮผ=10\mu = 10 and โˆ‘xi2=530\sum x_i^2 = 530. Find the variance using the computational formula ฯƒ2=โˆ‘xi2nโˆ’ฮผ2\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2.

Solution

  1. 1
    Apply the computational formula: ฯƒ2=โˆ‘xi2nโˆ’ฮผ2\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2
  2. 2
    Substitute values: ฯƒ2=5305โˆ’102=106โˆ’100=6\sigma^2 = \frac{530}{5} - 10^2 = 106 - 100 = 6

Answer

ฯƒ2=6\sigma^2 = 6
The computational formula ฯƒ2=โˆ‘xi2nโˆ’ฮผ2\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 is algebraically equivalent to the definition but avoids computing each deviation separately, useful when only summary statistics are available.

About Variance

The variance is the average of the squared deviations from the mean: ฯƒ2=1nโˆ‘(xiโˆ’xห‰)2\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2. It is the square of the standard deviation.

Learn more about Variance โ†’

More Variance Examples

Example 1 medium

Calculate the population variance for the data set: [formula].

Example 2 hard

Two investments have the same mean return of 8%. Investment A returns: [formula]. Investment B retur

Example 3 easy

Calculate the population variance for: [formula].

โ† All Variance Examples Variance Concept