Variance Math Example 1

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

Example 1

medium
Calculate the population variance for the data set: {2,4,4,4,5,5,7,9}\{2, 4, 4, 4, 5, 5, 7, 9\}.

Solution

  1. 1
    Find the mean: ฮผ=2+4+4+4+5+5+7+98=408=5\mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5
  2. 2
    Calculate squared deviations: (2โˆ’5)2=9,ย (4โˆ’5)2=1,ย (4โˆ’5)2=1,ย (4โˆ’5)2=1,ย (5โˆ’5)2=0,ย (5โˆ’5)2=0,ย (7โˆ’5)2=4,ย (9โˆ’5)2=16(2-5)^2=9,\ (4-5)^2=1,\ (4-5)^2=1,\ (4-5)^2=1,\ (5-5)^2=0,\ (5-5)^2=0,\ (7-5)^2=4,\ (9-5)^2=16
  3. 3
    Sum squared deviations: 9+1+1+1+0+0+4+16=329+1+1+1+0+0+4+16 = 32
  4. 4
    Divide by nn: ฯƒ2=328=4\sigma^2 = \frac{32}{8} = 4

Answer

ฯƒ2=4\sigma^2 = 4
Variance measures how spread out values are from the mean. By squaring the deviations we ensure positive contributions and penalize larger deviations more heavily. Standard deviation would be 4=2\sqrt{4}=2.

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 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].

Example 4 medium

A data set has [formula] values with mean [formula] and [formula]. Find the variance using the compu

โ† All Variance Examples Variance Concept