Calculating The Standard Deviation

In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though practically less robust than the expected deviation or average absolute deviation.
(Source : Wikipedia)

In this article we shall discuss about calculating Standard Deviation. Also we shall solve problems based on calculating standard deviation.

Calculating the Standard Deviation:

Formula for calculating Standard Deviation Questions :

It is nothing but standard deviation questions are calculated by taking square root for Variance.

Formula for finding mean,

barx   = (sum (X ) )/ n

Formula to solve standard deviation questions,

S = sqrt((sum(x - barx)^2) / (n -1))

Step 1: Calculate the average for given n numbers using the formula this is called mean of given numbers

Step 2: Find distance between each given numbers in the Data set from the calculated average value. This is called "deviation" from the mean value.

Step 3: Take the Square of each deviation value found From mean. This is squared deviation from mean.

Step 4: Calculate the sum  for all the Squared standard deviations .

Step 5: Now apply the Standard Deviation formula and find standard deviation formula. It will be the square root of variance.


Examples for Calculating the Standard deviation :

Examples for Calculating the Standard deviation are given below:

Problem 1:


The given data set are 18, 17, 14, 11 and 12. Calculating the Standard Deviation

Solution:

Mean: Calculate the average for the given values. To find the mean.
x  = (18 + 17 + 14 + 11 + 12)/ 5
= 72 / 5
= 14.4
Standard Deviation,

S = sqrt (( ( 18 - 14.4 )^2 + ( 17 - 14.4 ) ^2 + ( 14 - 14.4 ) ^2 + ( 11 - 14.4 ) ^2 + ( 12 - 14.4 ) ^2 )/ (5 - 1) )

=  sqrt(37.2/ 4 )

= sqrt( 9.3 )

S =  3.04959014
ANSWER:
Standard Deviation  S = 1.58113883


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Problem 2:

Find the Standard deviation of given Data  9, 10, 11, 12, 13, 14 and 15.
Solution:
barx = ( 9 + 10 + 11 + 12 + 13 + 14 + 15) / 7
barx = 84 / 7
barx = 12


X


X-barX


(X-barX)^2

9


9 - 12 = -3


9

10


10 - 12 = -2


4

11


11 - 12 = -1


1

12


12 - 12 =  0


0

13


13 - 12 = 1


1

14


14 -12 = 2


4

15


15 - 12 = 3


9


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Standard Deviation:

S =   sqrt((sum(x - barx)^2) / (n -1))

S =  sqrt( ( 9+4+1+0+1+4+9 ) / 6)
S = sqrt ( 28 / 6)
S =  sqrt(4.66667 )
S = 2.16024767