## Variance

The variance is defined as the average of the squared differences of the mean.

Population Variance \(= \sigma^2 = \cfrac{\sum(x – \mu)^2}{N}\)

Where:

\(\sigma^2 =\) population variance;

\(x =\) each value in the population;

\(\mu =\) the mean of the values; and

\(N =\) the number of values (the population).

Sample Variance \(= s^2 = \cfrac{\sum(x – \bar{x})^2}{N – 1}\)

Where:

\(s^2 =\) sample variance;

\(x =\) each value in the sample;

\(\bar{x} =\) the mean of the values; and

\(N =\) the number of values (the sample size).

### Examples:

a) Five friends measured the heights of their dogs (in millimetres). The heights at the shoulders were: 600mm, 470mm, 170mm, 430mm, and 300mm. Find the variance.

b) If the five dogs measured above are just a sample of a bigger population of dogs, find the variance.

**Solution:**

a) **Step 1:** Find the mean \((\mu)\).

\(\mu = \cfrac{600 + 470 + 170 + 430 + 300}{5} = \cfrac{1970}{5} = 394\).

**Step 2:** Subtract the mean from each score \((x – \mu)\).

\(600 – 394 = 206\)

\(470 – 394 = 76\)

\(170 – 394 = -224\)

\(430 – 394 = 36\)

\(300 – 394 = -94\)

**Step 3:** Square each deviation \(((x – \mu)^2)\).

\((206)^2 = 42,436\)

\((76)^2 = 5,776\)

\((-224)^2 = 50,176\)

\((36)^2 = 1,296\)

\((-94)^2 = 8,836\)

**Step 4:** Add the squared deviations \((\sum(x – \mu)^2)\).

\(42,436 + 5,776 + 50,176 + 1,296 + 8,836 = 108,520\)

**Step 5:** Divide the sum by the number of dogs \(\left(\cfrac{\sum(x – \mu)^2}{N}\right)\).

\(\cfrac{108,520}{5} = 21,704\)

b) **Step 1:** Find the mean \((\bar{x})\).

\(\mu = \cfrac{600 + 470 + 170 + 430 + 300}{5} = \cfrac{1970}{5} = 394\).

**Step 2:** Subtract the mean from each score \((x – \bar{x})\).

\(600 – 394 = 206\)

\(470 – 394 = 76\)

\(170 – 394 = -224\)

\(430 – 394 = 36\)

\(300 – 394 = -94\)

**Step 3:** Square each deviation \(((x – \bar{x})^2)\).

\((206)^2 = 42,436\)

\((76)^2 = 5,776\)

\((-224)^2 = 50,176\)

\((36)^2 = 1,296\)

\((-94)^2 = 8,836\)

**Step 4:** Add the squared deviations \((\sum(x – \bar{x})^2)\).

\(42,436 + 5,776 + 50,176 + 1,296 + 8,836 = 108,520\)

**Step 5:** Divide the sum by one less than the number of dogs \(\left(\cfrac{\sum(x – \bar{x})^2}{N – 1}\right)\).

\(\cfrac{108,520}{5 – 1} = \cfrac{108,520}{4} = 27,130\)