Variance

Variance

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

Formula

The variance means how far the values are from the mean.

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

The following data show 5 test scores of a student. 60, 70, 80, 90, 100. Find the variance of the data.

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
xi, xi - x, and (xi - x)2.

Write the given values in the xi column.

x = 80

So write the value of xi - x, xi - 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 (xi - x)2 column.

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

The sum of (xi - 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

The following data show 5 test scores of a student. 70, 75, 80, 85, 90. Find the variance of the data.

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
xi, xi - x, and (xi - x)2.

Write the given values in the xi column.

x = 80

So write the value of xi - x, xi - 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 (xi - x)2 column.

(-10)2 = 100
(-5)2 = 25
02 = 0
52 = 25
102 = 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 (xi - x)2.

The sum of (xi - 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.