# Variance

How to find the variance of a data: formula, examples, and their solutions.

## Formula

The variance V(*X*) means

how far the values are from the mean *x*.

Mean (Average)

The distance between a value and the mean is

(*x* - *x*).

But it leaves a sign: [+] or [-].

To undo the effect of the signs,

we square (*x* - *x*): (*x* - *x*)^{2}.

If you add up these (*x* - *x*)^{2} for each case,

you get the variance V(*X*).

Sigma notation

If the values are far from the mean,

the variance gets bigger.

If the values are close to the mean

the variance gets smaller.

## Example 1: Variance of 60, 70, 80, 90, 100

First, find the mean of the data.

The values are 60, 70, 80, 90, 100.

There are 5 values.

So the mean *x* is

(60 + 70 + 80 + 90 + 100)/5.

Mean (Average)

60 + 100 = 160

70 + 90 = 160

160 + 160 = 320

320 + 80 = 400

400/5 = 80

So *x* = 80.

Draw a three column table like this.

Write the titles*x*_{i}, *x*_{i} - *x*, and (*x*_{i} - *x*)^{2}.

Write the given values in the *x*_{i} column.

*x* = 80

So write the value of *x*_{i} - *x*, *x*_{i} - 80,

in the next column.

60 - 80 = -20

70 - 80 = -10

80 - 80 = 0

90 - 80 = 10

100 - 80 = 20

Write the squares of the second column

in the (*x*_{i} - *x*)^{2} column.

(-20)^{2} = 400

(-10)^{2} = 100

0^{2} = 0

10^{2} = 100

20^{2} = 400

Then write the sum of the third column

in the next row:

400 + 100 + 0 + 100 + 400 = 1000.

This 1000 is the sum of (*x*_{i} - *x*)^{2}.

The sum of (*x*_{i} - *x*)^{2} is 1000.*n* = 5

So V(*X*) = 1000/5.

1000/5 = 200

So V(*X*) = 200.

## Example 2: Variance of 70, 75, 80, 85, 90

First, find the mean of the data.

The values are 70, 75, 80, 85, 90.

There are 5 values.

So the mean *x* is

(70 + 75 + 80 + 85 + 90)/5.

Mean (Average)

70 + 90 = 160

75 + 85 = 160

160 + 160 = 320

320 + 80 = 400

400/5 = 80

So *x* = 80.

Draw a three column table like this.

Write the titles*x*_{i}, *x*_{i} - *x*, and (*x*_{i} - *x*)^{2}.

Write the given values in the *x*_{i} column.

*x* = 80

So write the value of *x*_{i} - *x*, *x*_{i} - 80,

in the next column.

70 - 80 = -10

75 - 85 = -5

80 - 80 = 0

85 - 80 = 5

90 - 80 = 10

Write the squares of the second column

in the (*x*_{i} - *x*)^{2} column.

(-10)^{2} = 100

(-5)^{2} = 25

0^{2} = 0

5^{2} = 25

10^{2} = 100

Then write the sum of the third column

in the next row:

100 + 25 + 0 + 25 + 100 = 250.

This 250 is the sum of (*x*_{i} - *x*)^{2}.

The sum of (*x*_{i} - *x*)^{2} is 250.*n* = 5

So V(*X*) = 250/5.

250/5 = 50

So V(*X*) = 50.

The data in last example is [60, 70, 80, 90, 100].

The data in this example is [70, 75, 80, 85, 90].

Its obvious that

the values of the data [60, 70, 80, 90, 100]

are further from the mean 80

than the values of the data [70, 75, 80, 85, 90].

This is why

the variance of [60, 70, 80, 90, 100], V(*X*) = 200,

is greater than

the variance of [70, 75, 80, 85, 90], V(*X*) = 50.