# Standard Deviation Formula

Before we derive the standard deviation formula let us first understand the meaning of standard deviation. For a set of data, the measure of dispersion, about mean, when expressed as the positive square root of the variance, is called standard deviation. It measures the dispersion or spread of data. Standard deviation is denoted by:

where xi is the midpoint of the data

is the mean of the distribution

n = number of observation

This is the basic formula to find the standard deviation for a given set of data.

## Standard Deviation For Discrete Data

For discrete frequency distribution of the type:

The standard deviation is given as:

Where   is the mean of the distribution and

## Standard Deviation For Continuous Frequency Distribution

For continuous frequency distribution, the mid-point of each class is considered for calculating the standard deviation. If the frequency distribution of n classes is defined by its mid-point xi with frequency fi, the standard deviation is given by:

Where  is the mean of the distribution and

## Derivation

Now let us try to obtain another standard deviation formula:

Or,

This gives us the standard deviation (σ) as:

This is the standard deviation formula for a given set of observations. But sometimes it happens that the value xi in a given data set or the midpoints of classes in a given frequency distribution is very enormous. In such cases, determination of mean or median or variance becomes lengthy and time-consuming. To solve this problem we make use of step deviation method for simplifying the procedure.

### Simple Method to Find Standard Deviation

Let us discuss this shortcut method for determining variance and standard deviation.

Let us consider the assumed mean as ‘A’ and the width of the class interval be h. Let the step deviation be yi.

The mean of a data set is given by:

Substituting the values of xi from equation (1) into equation (2), we get

Variance of the variable x,

Substituting the values from equation (1) and (3), we have

σx=h2 × variance of variable yi

⇒σx2=h2σy2

⇒σx=hσy—(4)

From equation (3) and (4), we can conclude that:

## Solved Example

Let us look into an example for a better insight.

Example: Find out the mean, variance and standard deviation for the following data representing the age group of employees working in XYZ Company.

Solution: Let the assumed mean A = 30 and h = 10. From the table given above, we can obtain.

Thus,

The variance of the above data can be calculated as:

= (10/100)2 [100 × 160 – (60)2] = (1/100) [16000 – 3600] = (100/100) [160 – 36] = 124

The standard deviation can be given as:

σ = √124 = 11.136