How to calculate Standard deviation

Introduction to Standard deviation  : In 1983 karl peasron  denoted  standard deviation by  letter s (small sigma) of the Greek alphabet  as a measure of dispersion . It is defined as the positive square root of the arithmetic mean of the squares of the deviations of the given observations from their arithmetic mean. Thus, if X1, X2, ….., Xn  is a set of observations .

Formula for Standard Deviation is given by s = sqrt(1/n summation (X – X bar)^2) where X bar = 1/n summation X, is the arithmetic mean of the given values .Here are steps for  Calculating Standard Deviation first we will compute the arithmetic mean X bar by the formula  X bar = 1/n summation .

Then we will Compute the deviation (X – X bar) of each observation from arithmetic mean i.e., obtain X1 – X bar, X2 – X bar, ….., Xn – X bar. then we will Square each of the deviations obtained   such as  compute (X1 – X bar)^2, (X2 – X bar)^2, (Xn – X bar)^2.Find the sum of the squared deviations in given by Summation (X – X bar)^2 = (X1 – X bar)^2 + (X2 – X bar)^2 + ….+ (Xn – X bar)^2.

Divide this sum  by n to obtain 1/n summation (X – X bar)^2.

now we will take the positive square root of the value obtained The resulting value gives the standard deviation of the distribution. In case of frequency distribution, the standard deviation equation is given by
s = sqrt (1/N summation f(X – X bar)^2),
where X is the value of the variable or the mid value of the class ( in case of grouped or continuous frequency distribution);
f is the corresponding  frequency of the value X ;
N = summation f, is the total frequency and  X bar = 1/N summation fX.
Is the arithmetic mean of the distribution.

Steps for computation of standard deviation  in case of frequency distribution Compute X bar by the formula given above or the usual step deviation formula.
Compute deviations (X – X bar) from the mean for each value of the variable X.
Obtain the squares of the deviations obtained in
step 2 i.e., compute (X – X bar)^2.
Multiply each of the squared deviations obtained in
step 3 by the corresponding frequency to get f(X – X bar)^2.
Find the sum of the values obtained in
step 4 to get summation f(X – X bar)^2Divide the sum obtained in
step 5 by N = summation f, the total frequency. The positive square root of the value obtained in step 6 gives the standard deviation of the distribution.

Let us take an example for Standard Deviation Calculation that the variable X takes the same constant values, say 8, 8, 8, 8, 8.
Then X bar = 1/5 summation X = (8 + 8 + 8 + 8 + 8)/5 = 40/5 = 8.o, S.D. = sqrt(1/5 summation (X – X bar)^2) =sqrt( 1/5 (0)) = 0.