# Standard Deviation For Grouped Data

Standard Deviation, simply stated, is the measure of dispersion of a group of data from its mean. In other words, it measures how much the observations differ from the central mean. Hence standard deviation is an important tool used by statisticians to measure how far or how close are the points in a data group from its mean. The standard deviation of a data is equivalent to the square root of its variance.

It is extremely important while analysing data since once we look at the standard deviation of any data set we can have an idea about the average value of observations and quantitative estimation of the data. In some situations, where you have to find out how much is a particular value is deviating from the average value of the group, finding the mean or the median does not solve the purpose. For instance, if the height of all the boys of a class is measured, and you want to know how much is your height more or less from the class average, then you need to calculate the standard deviation of the data.

The standard deviation is calculated by taking the root of the sum of the squared deviations of the observations from the mean. It is calculated by the formula:

### Standard Deviation For Grouped Data Formula There can be different types of data sets for which the standard deviation might be calculated. For example, the calculation of the standard deviation for grouped data set differs from the ungrouped data set. The grouped data can be divided into two, i.e., discrete data and continuous data. In the case of grouped data, the standard deviation can be calculated using three methods, i.e, actual mean, assumed mean and step deviation method.

For example, let us take the following data set :

 x 0-10 10-20 20-30 30-40 40-50 50-60 60-70 f 1 4 17 45 26 5 2
1. If we calculate using actual mean :

N= 100, ∑fm = 3640, ∑fx= 0, ∑fxd²= 10404

 CI M.P. (m) f fm x= (M-X’) fx fx² 0-10 5 1 5 -31.4 -31.4 985.96 10-20 15 4 60 -21.4 -85.6 1831.84 20-30 25 17 425 -11.4 -193.8 2209.32 30-40 35 45 1575 -1.4 -63.0 88.20 40-50 45 26 1170 8.6 -223.0 1922.96 50-60 55 5 275 18.6 93.0 1729.80 60-70 65 2 100 28.6 57.2 1635.

X’= ∑fm /N = 3640/100

=36.4

Hence,  Standard Deviation = √∑fx²/N – (√∑fx²/N)²

√10404/100

√104.04

=10.2

2. If we calculate using assumed mean :

N=100, Let Assumed Mean (A)= 35

N=100,  ∑fdx = 140,  ∑fd²x = 10600,  ∑fd’x² = 106,

 CI M.P. (m) f dx = m-A fdx fdx² d’x= dx/10 fd’x fd’x² 0-10 5 1 -30 -30 900 -3 -3 9 10-20 15 4 -20 -80 1600 -2 -8 16 20-30 25 17 -10 -170 1700 -1 -17 17 30-40 35 45 0 0 0 0 0 0 40-50 45 26 10 260 2600 1 26 26 50-60 55 5 20 100 2000 2 10 20 60-70 65 2 30 6 1800 3 6 18

S.D.= √∑fdx²/N – (∑fdx²/N)²

√10600/100- (140/100)²

1. If we calculate using Step- Deviation Method

S.D. =  √∑fd’x/N – (∑fd’x/N)² x i

= √106/100- (14/100)² x 10

= √1.06- 0.196 x 10 = √1.404 x10

= 1.02 x 10

=10.2

So using these formulas you can find the Standard Deviation of various types of grouped data. You can easily calculate the standard deviations for any grouped data by using these step-by-step procedures we have provided here.

The standard deviation formula has a wide range of applications in various fields, such as mathematics, statistics, finance, etc. IT also enables us to find the reported margin of error of a data, since it is usually twice the standard deviation. Hence it provides us with a true picture of the polling number.

We hope you have been able to understand the concept of standard deviation through this article and understand the application of the formulas of standard deviation. So now you can find the standard deviation of any grouped data using the three methods we have explained to you.

Here we have also provided you with a video which will help you to understand how to calculate the standard deviation of grouped data.