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Statistics

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The variance is a numerical index describing the dispersion of a set of scores around the mean of the distribution. The variance is calculated as the average of the squared deviations from the mean.

There are two versions of variance; the population variance, usually denoted by , and the sample variance, usually denoted by s2.

Formula for population variance:

Formula for Sample variance:

These formulas differ basically in their denominators. Also, their numerical values are practically the same when n, the number of observations, is large.


Steps to find the variance:
Step 1. Work out the mean (the simple average of the numbers).
Step 2. For each number subtract the mean and then square the result (the squared difference).
Step 3. Work out the average of those squared differences.


A couple has six children whose ages are 6, 8, 10, 12, 14 and 16. Find the variance in ages.

Step 1. Work out the population mean.

Step 2 and 3. For each number subtract the mean and then square the result. Work out the average of those squared differences.


There's a more efficient way to calculate the standard deviation for a group of numbers, shown in the following equation:

That is:




A couple has six children whose ages are 6, 8, 10, 12, 14 and 16. Find the variance in ages.

The sample mean is:

and

Then


The variance of a frequency distribution is given by:


The following table gives the frequency distribution of the number of computers sold during the past 30 weeks at a computer store.

Computers sold
Frequency (f)
[0-4)
2
[4-8)
3
[8-12)
4
[12-16)
2
[16-20)
1

Calculate the variance.

Solution:

Class interval
x
f
F
[0-4)
2
2
2
[4-8)
6
3
5
[8-12)
10
4
9
[12-16)
14
2
11
[16-20)
18
1
12

The mean number of computers sold per week was 9 computers.

Class interval
x
f
[0-4)
2
2
98
[4-8)
6
3
27
[8-12)
10
4
4
[12-16)
14
2
50
[16-20)
18
1
81

Properties:

  • Suppose there are two samples x1,...,xn and y1,...,yn where yi=xi+c, i=1,2,...,n and c>0. If the respective sample variances of the two samples are denoted by and then (Proof).
  • Suppose there are two samples x1,...,xn and y1,...,yn where yi=cxi, i=1,2,...,n and c>0. If the respective sample variances of the two samples are denoted by and then (Proof).