Variance

How to find the variance of the given data: formula, 2 examples, and their solutions.

Formula

Formula

V(X) = [(x1 - x)2 + (x2 - x)2 + (x3 - x)2 + ...]/n

V(X): Variance
x1, x2, x3, ...: Values of the data
x: Mean
n: Number of the values

The variance V(X) means
how far the values are from the mean x.

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

Example

Solution

First find the mean x of the data.

(sum) = 60 + 70 + 80 + 90 + 100

60 + 100 = 160
70 + 90 = 160

160 + 160 = 320

320 + 80 = 400

There are 5 values.

So n = 5.

(sum) = 400
n = 5

Then x = 400/5.

400/5 = 80

So x = 80.

Make a 3 column table like this.

Name the titles
xi, xi - x, and (xi - x)2.

Write the values in the xi column:
60, 70, 80, 90, 100.

Make an empty row
at the bottom of the table.

Write the xi - x column.

x = 80
So write xi - 80.

60 - 80 = -20
70 - 80 = -10
80 - 80 = 0
90 - 80 = 10
100 - 80 = 20

Write the (xi - x)2 column.

Square the xi - x column.

(-20)2 = 400
(-10)2 = 100
02 = 0
102 = 100
202 = 400

Add up the (xi - x)2 column.

400 + 100 + 0 + 100 + 400
= 500 + 500
= 1000

The sum of (xi - x)2 is 1000.

n = 5

So the variance is
V(X) = 1000/5.

1000/5 = 200

So 200 is the answer.

Example 2

Example

Solution

To find the variance from a frequency table,
make a 6 column table like this.

Name the titles
xi, fi, xifi, xi - x, (xi - x)2, and , (xi - x)2fi.

Write the scores in the xi column:
0, 1, 2, 3, 4, 5.

Write the frequencies in the fi column:
1, 1, 4, 7, 5, 2.

Make an empty row
at the bottom of the table.

Add up the fi column.

1 + 1 + 4 + 7 + 5 + 2
= 2 + 11 + 7
= 9 + 11
= 20

The sum of the frequencies is n.

So n = 20.

Write the xifi column.

Multiply xi and fi.

0⋅1 = 0
1⋅1 = 1
2⋅4 = 8
3⋅7 = 21
4⋅5 = 20
5⋅2 = 10

Add up the xifi column.

0 + 1 + 8 + 21 + 20 + 10
= 9 + 21 + 30
= 30 + 30
= 60

The sum of xifi is the sum of the values.

So (sum) = 60.

(sum) = 60
n = 20

Then the mean is
x = 60/20.

60/20 = 3

So x = 3.

Write the xi - x column.

x = 3

So write xi - 3.

0 - 3 = -3
1 - 3 = -2
2 - 3 = -1
3 - 3 = 0
4 - 3 = 1
5 - 3 = 2

Write the (xi - x)2 column.

Square the xi - x column.

(-3)2 = 9
(-2)2 = 4
(-1)2 = 1
02 = 0
12 = 1
22 = 4

Write the (xi - x)2fi column.

Multiply fi and (xi - x)2.

1⋅9 = 9
1⋅4 = 4
4⋅1 = 4
7⋅0 = 0
5⋅1 = 5
2⋅4 = 8

Add up the (xi - x)fi column.

9 + 4 + 4 + 0 + 5 + 8
= 13 + 4 + 13
= 17 + 13
= 30

The sum of (xi - x)fi is 30.

n = 20

So the variance is
V(X) = 30/20.

30/20 = 3/2

So 3/2 is the answer.