Standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given observation from their arithmetic mean.

Among all the methods of finding out dispersion, standard deviation is regarded as the best. It is free from the defects which the earlier methods range, quartile deviation and mean deviation suffer. Its value is based upon each and every item of the series and it also take into account algebraic signs. Standard deviation is also known as ‘Root Mean-Square Deviation’ because it is the square root of the arithmetic mean of the square of the deviations. It is denoted by the σ (sigma).

### Individual Series:

The standard deviation of the set of n observation is given by

### Discrete and Continuous Series:

For a discrete and continuous frequency distribution, the standard deviation is given by

Where N is the total number of observations of the given data. In case of continuous distribution x is taken as the mid-value of the corresponding class.

If the values of the observations and their corresponding frequencies are large, then the calculation of standard deviation is tedious and time consuming. In such cases, we can calculate the standard deviation by taking deviation as follows:

### Individual Series:

When the values of the observations are very large in the individual series, the standard deviation can be calculated by using the following formula,

Where d=x–A  and A is assumed mean.

### Discrete or Continuous Series:

When the values of the observations and the frequencies are large in case of discrete or continuous series, the calculation of standard deviation can be made easy by using the following formula. Where d=x–A  and A is assumed mean.

When the value of variables or mid-values have some common factor, the calculation of the standard deviation can be made still easier by taking the deviation as follows:

### Coefficient of Standard Deviation

The relative measure or the coefficient of standard deviation is given by the formula,

Workout Examples

Example 1: The weekly expenditure of 5 families in rupees are given below:
 Family A B C D E Expenditure (Rs.) 300 325 350 375 425

Calculate the standard deviation and its coefficient by:
a.    Direct method
b.    Deviation method

Solution: Here,
a.    Direct method
 Families Expenditure x2 A B C D E 300 325 350 375 425 90000 105625 122500 140625 180625 ∑x = 1775 ∑x2 = 639375

b.    Deviation method
 Family Exp. (x) d = x – A(350) d2 A B C D E 300 325 350 375 425 -50 -25 0 25 75 2500 625 0 625 5625 25 9375

Example 2: Find the standard deviation and its coefficient from the following frequency distribution.

 Marks 20 24 30 35 38 40 No. of students 8 7 10 12 6 3

Solution: Here,

 Marks No. of students fx fx2 20 24 30 35 38 40 8 7 10 12 6 3 160 168 300 420 228 120 3200 4032 9000 14700 8664 4800 N = 46 ∑fx = 1396 ∑fx2 = 44396

Example 3: Following are the marks obtained by students in a test exam.

 Marks 20-30 30-40 40-50 50-60 60-70 70-80 80-90 No. of students 4 6 10 17 11 9 3

Calculate the standard deviation and its coefficient by
a.    Direct method
b.    Deviation method

Solution: Here,
a.    Direct method

 Marks No. of std(f) Mid value(x) fx fx2 20-30 30-40 40-50 50-60 60-70 70-80 80-90 4 6 10 17 11 9 3 25 35 45 55 65 75 85 100 210 450 935 715 675 255 2500 7350 20250 51425 46475 50625 21675 N = 60 ∑fx = 3340 ∑fx2 = 200300

b.    Deviation method

 Marks f Mid value(x) d=(x-55)/10 fd fd2 20-30 30-40 40-50 50-60 60-70 70-80 80-90 4 6 10 17 11 9 3 25 35 45 55 65 75 85 -3 -2 -1 0 1 2 3 -12 -12 -10 0 11 18 9 36 24 10 0 11 36 27 N = 60 ∑fd = 4 ∑fd2 = 144

Example 4: Find the variance and its coefficient from the following data.

 Age in yrs 10 15 20 25 30 35 No. of people 5 7 15 25 10 8

Solution: Here,

 Age (x) No. of people (f) fx fx2 10 15 20 25 30 35 5 7 15 25 10 8 50 105 300 625 300 280 500 1575 6000 15625 9000 9800 N = 70 ∑fx = 1660 ∑fx2 = 42500

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