Variance

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 $$\left(\cfrac{\sum(x – \bar{x})^2}{N – 1}\right)$$.
$$\cfrac{108,520}{5 – 1} = \cfrac{108,520}{4} = 27,130$$