Standard deviation is the most widely used measure of dispersion. It is defined as the positive square root of sum of the square deviation of the variable from its mean divided by total number of observations. In other words it is the positive square root of a quantity called variance. Standard deviation is a very useful measure of dispersion. The importance of standard deviation can be highlighted with help of simple example. For example company A has average annual profit of $10000 with a standard deviation of $3000 and company B has average annual profit of $10000 with standard deviation of $6000. Now because the standard deviation of company A is only 3000 which indicates that the annual returns of company A fluctuate from $7000 to $13000 (10000-3000=7000, 10000+3000=13000) whereas annual returns of company B fluctuate from $4000 to $16000. Therefore company A is more stable and less risky than company B.
The calculation of standard deviation for ungrouped data is different from the grouped data. Similarly there is also difference in the formulas for sample data and population data. These formulas are given below.
Formulas for Ungrouped Data
Formulas for Grouped Data
Calculation of Standard Deviation for Ungrouped Data
Problem: Following are the number of students enrolled in English coaching program each month for the last 7 months 24, 21, 29, 18, 26, 20, and 23. Find standard deviation by taking data (a) as sample (b) as population.
Calculation of Standard Deviation for Grouped Data
Problem: Following are the observations showing the age of 50 employees working in a whole sale center. Find standard deviation by taking data (a) as sample (b) as population