Standard Deviation

Standard Deviation is defined as "the measure of the variability". If the spread is more then the standard deviation will also be more. Standard deviation can also be stated as the "square root of the variance". Standard deviation has its application in computing the the annual returns of the banks

Arithmetic Mean:

A mathematical representation of the series of the number which is totalled as the sum of all the number in series and it is divided by the total number of values in the set.

Variability :

When the  standard deviation is raised to the power of two , the value which we get is the variance i.e Variance is equal to the square of the standard deviation. It is the measure of the degree of the spread among certain set of the values. Variance is also stated in the other way as follows"Variance is the measure of tendancy of individual values that vary from the arithmetic mean value

Formula to determine the Standard deviation :

[Formula of Standard deviation]

Where  ,     Σ = sum of

xi = Individual scores

m = arithmetic mean of all scores

n = Size of the number of scores

Variance :

Variance = σ2

Example

Example 1: To find the standard deviation of  2 , 4 , 6 , 8 , 10

Step-1 :Calculate the mean and the deviation

xi       m        xi-m         (xi-m)2

2        6        -4              16

4         6        -2              4

6        6          0              0

8        6          2              4

10       6         4             16

Step-2 :Find the sum of  (x-m)2

16 + 4 + 0 + 4 + 16 = 40

Step-3 :n = 5 , the total number of values , find n-1

5-1 = 4

Step-4 :Now find standard deviation

√40 / √4 = 3.162

3.162 is the standard deviation

To find the variance just square the standard deviation value

Step-5 : Variance = (3.162)2 = 9.998244

Example 2

Example 2 :To find the standard deviation of 3 , 6 , 9 , 12 , 15

Step-1 : Calculate the mean deviation

xi          m          xi-m           (xi-m)2

3           9           -6               36

6           9           -3                 9

9           9            0                 0

12          9            3                 9

15          9           6                36

Step-2 :

calculate the sum of (x-m)2

36+9+0+9+36 = 90

Step-3 :

n=5 ,the total number of values , find n-1

5-1 = 4

Step-4 :

Now let us find the standard deviation

√90/√4 = 4.743

The standard deviation is 4.743

Step-5 :

Variance = (4.743)2

= 22.496