Variance

Variance and Standard Deviation

In probability and statistics, the variance and standard deviation are the normal estimations of the squared difference of any variable from its mean value. In other terms, it calculates how far the random numbers are spread out from their mean value. In stats, the value of the square of the standard deviation is equal to the variance. The standard deviation is another central tool and the variance symbol denoted by σ2 or s2 or by var ( n ), where var stands for the variance. In this article, you will be learning about the definition of variance in statistics, the formula of variance. The properties of variance, and you’ll be able to understand the topic better with some solved problems. So, let us get started!

Variance Definition in Statistics

In statistics, the difference between the mean and its data points is known as the variance. In simpler terms, it is the measure of how far a set of data points are far from their mean value.  When you have to solve questions, you need to  use the following variance formula:

Var ( n ) = E [  (n - μ)2

If the formula has to be described in words, it goes by - the variance is the expectation of the deviation from its mean value that is squared. 

Here,  μ is  equal to the value of E ( n ) and the equation stated above can also be expressed as:

Var (n) = E  [(n - μ)2

Var (n) = E  [n2 - 2n E (n) + (E (n))2 ]

Var (n) = E (n2) - 2 E(n) E(n) + (E(n))2

Var (n) = E(x2) - (E(n))2

Since we have covered the formula of the variance, let us try and understand the relationship between standard deviation and the variance. 

Variance Formula in Statistics

Previously, we studied that the variance is the square of standard deviation. 

This can be expressed as Variance = σ2 = standard deviation2

Some of the other formulas that correspond with it are: 

  • Population deviation σ = \[\sqrt {\frac{{\sum {{{\left( {n - \mu } \right)}^2}} }}{N}} \]and

  • Standard deviation S =  \[\sqrt {\frac{{\sum {\left( {n - \underline n } \right)} }}{{N - 1}}} \]

Here,

X - Number of observations

μ - cumulative mean of values

N - Total values

Variance Properties

The variance of random variable n has the following properties.

  1. Var ( N + k ) = var ( N ), here k is constant value.

  2. Var ( kN ) = k2. Var ( N ), here k is constant value.

  3. Var ( rN + s) = r2.Var( N ), here r and s can be constants.

  4. If N1 , N2, N3, N4, . . . . . . . . . . . . . . Nn are n number of independent and random values or variables.

Variance Example

Question 1: Find the variance of the number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Solution:

Find the mean of the 10 values: Mean = ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ) / 10

 Mean = ( 55 ) / 10

Mean =  5.5

Value N

N - \[\overline N \]

  ( N -\[\overline N \])2

1

-4.5

20.25

2

-3.5

12.25

3

-2. 5

6.25

4

-1.5 

2.25

5

-0.5

0.25

6

+0.5 

0.25

7

+1.5

2.25

8

+2.5

6.25

9

+3.5

12.25

19

+4.5

20.25

Total

82.50


Now, we need to find the population variance

σ2 =  \[\sqrt {\frac{{\sum {\left( {n - \underline n } \right)} }}{{N - 1}}} \]

     = \[\frac{{82.5}}{{10}}\]

Question 2: Find the variance of the number 4, 2, 8, 6, 12, 17, 14, 20.

Solution:

Find the mean of the 10 values: Mean = ( 4+ 2 + 8 + 6 + 12 + 17 + 14 + 20 ) / 8

Mean = ( 83 ) / 8

Mean =  10.375

Value N

N - \[\overline N \]

  ( N -\[\overline N \])2

4

-6.375

40.640625

2

-8.375

70.140625

8

-2.375

5.640625

6

-4.375

19.140625

12

1.625

2.640625

17

6.625

43.890625

14

3.625

13.140625

20

9.625

19.25

Total

0

214.484375


Now, we need to find the population variance

σ2 =  \[\sqrt {\frac{{\sum {\left( {n - \underline n } \right)} }}{{N - 1}}} \]

     = \[\frac{{214.484375}}{8}\]

     = 26.8105469

Question 3: Find the variance of the number 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

Solution:

Find the mean of the 10 values: Mean = ( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ) / 10

Mean = ( 155 ) / 10

Mean =  15.5

Value N

N - \[\overline N \]

  ( N -\[\overline N \])2

1

-4.5

20.25

2

-3.5

12.25

3

-2. 5

6.25

4

-1.5 

2.25

5

-0.5

0.25

6

+0.5 

0.25

7

+1.5

2.25

8

+2.5

6.25

9

+3.5

12.25

19

+4.5

20.25

Total

82.50


Now, we need to find the population variance

σ2 =  \[\sqrt {\frac{{\sum {\left( {n - \underline n } \right)} }}{{N - 1}}} \]

     = \[\frac{{82.5}}{{10}}\]

     = 8.25

FAQ (Frequently Asked Questions)

1. What is the Variance?

In statistics, the difference between the mean and its data points is known as the variance. In simpler terms, it is the measure of how far a set of data points are far from their mean value.  When you have to solve questions, you need to  use the following equation:

Var ( n ) = E [  (n - μ)2

If the formula has to be described in words, it goes by - the variance is the expectation of the deviation from its mean value that is squared. 

2. What are the Properties of Variance?

-Var ( N + k ) = var ( N ), here k is constant value.

-Var ( kN ) = k2. Var ( N ), here k is constant value.

-Var ( rN + s) = r2.Var( N ), here r and s can be constants.

-If N1 , N2, N3, N4, . . . . . . . . . . . . . . Nn are n number of independent and random values or variables.