# Standard Deviation

The standard deviation of a set of values is the root mean square deviation, when deviations are taken from the A.M. of observations. Root mean square deviation means that we take deviations of observations from the A.M., then square those deviations followed by arithmetic mean of the squared values and finally taking its square root.

If a variable *x* assumes *n* values *x*_{1,} *x*_{2,} *x*_{3, }â€¦ *x _{n}*

_{,}then its standard deviation is given by

The above relation can be simplified as

For grouped frequency distribution, the standard deviation is given by

The above relation can be simplified as

In the above relations

Example

What is the standard deviation of 5, 5, 9, 9, 9, 10, 5, 10, 10?

Solution

x |
5 | 9 | 10 |

f |
3 | 3 | 3 |

x |
f |
|||

5 | 3 | 3 | 9 | 27 |

9 | 3 | â€“1 | 1 | 3 |

10 | 3 | â€“2 | 4 | 12 |

*Variance = S*

^{2}

Example

What is the coefficient of variation of the following numbers?

53, 52, 61, 60, 64

53, 52, 61, 60, 64

Solution

# Properties of Standard Deviation

- If all the observations assumed by a variable are constant,
*i.e.*, equal, then*S*= 0. This means that if all the values taken by a variable*x*are*k*, then*S*= 0. This result applies to range as well as mean deviation. **Standard deviation**remains unaffected due to a change of origin, but is affected in the same ratio due to a change of scale.*x*and*y*related as*y*=*a*+*bx*for any two constants*a*and*b*, then*S*of*y*is given by*Sy*= |*b*|*Sx*- If there are two groups containing
*n*_{1}and*n*_{2}observations,*as respective A.M.s,**S*_{1}and*S*_{2}as respective S.D.s, then the combined S.D. is given by

# Merits

- It is based on all the observations.
- It is rigidly defined.
- It has a greater mathematical significance and is capable of further mathematical treatments.
- It represents the true measurements of dispersion of a series.
- It is least affected by fluctuation of sampling.
- It is not reliable and dependable measure of dispersion.
- It is extremely useful in correlation, etc.

# Demerits

- It is difficult to compute unlike other measures of dispersion.
- It is not simple to understand.
- It gives more weighted to extreme values.
- It consumes much time and labour while computing it.

**Note:** The standard deviation of first n natural numbers is

When comparing the coefficient of variation of two or more objects, the one having the lesser coefficient of variation is said to be more consistent.