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Maths Formulas

Variance Formula

Variance is a measure of how data points differ from the mean value. According to the simple terms, it is a measure of how far a set of data i.e. numbers are spread out from their mean i.e. average value. Variance is a statistical measure that tells us how measured data vary from the average value of the set of data. In other words, a variance is the mean of the squares of the deviations from the arithmetic mean of a data set. In this article, we will discuss the variance formula.

Variance Formula

What is a Variance?

Variance is used in how far a set of numbers are spread out. This is different from finding the average, or the mean, of numbers. For example, take a look at the following numbers: 12, 8, 10, 10, 8, 12. If we add these numbers together and divide by the total numbers in the data set, which in this case is 6, you will get an average of 10. Notice that these numbers are all pretty close to the number 10.

Now take a look at this set of data: 28, 4, 6, 4, 2, 16. We can notice that there is a greater difference between the numbers in the second set of data versus the first set of data. Although both sets of data have an average of ten. We show these differences in data by using variance.

There are two types of variance, one is population and the other is a sample. Population is having all members of a specified group. If we were to collect data on just the members of our household, then everyone living in would be considered the population. A sample is a part of the population used to describe the whole group.

If we collect data on the members of our household and only collect data about two members out of the five members. Then this will be considered a sample. Other examples of population and samples would be the total members of a school versus only the members of a class in the school or a random selection of 50 members of a school, which will also be a sample.

Variance Formula

For the purpose of solving questions, it is,

\( Var(X)=E[(X-\mu)^2] \)

Var(X) will represent the variance.

This means that variance is the expectation of the deviation of a given random set of data from its mean value and then squared.

Here, X is the data,

µ is the mean value equal to E(X), so the above equation may also be expressed as,

\( Var(X)=E[X-E(X)^2]\)

\( Or Var(X) = E(X^2)- E(X)^2\)

Solved Examples

Example: Find the variance of the numbers 3, 8, 6, 10, 12, 9, 11, 10, 12, 7.

Solution:

Step 1: First compute the mean of the 10 values given.

\( \bar X = \frac{3+8+6+10+12+9+11+10+12+7}{10} \)

= \( \frac{88}{10} \)

= 8.8

Step 2: Make a table as following with three columns, one for the X values, the second for the deviations and the third for squared deviations.

Value (X) \( X – \bar X \) \( (X-\bar X)^2\)
3 -5.8 33.64
8 -0.8 0.64
6 -2.8 7.84
10 1.2 1.44
12 3.2 10.24
9 0.2 0.04
11 2.2 4.84
10 1.2 1.44
12 3.2 10.24
7 -1.8 3.24
Total 0 73.6

Step 3:

As the data is not given as sample data, thus we use the formula for population variance.

= \( \frac{73.6} {10} \)

= 7.36

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