Standard Deviation Formula

Standard Deviation Formula

To understand 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 is denoted by

Standard Deviation Formula

For discrete frequency distribution of the type:

Frequency Distribution

The standard deviation is given as:

Standard Deviation Formula

Where \(\overline{x}\)  is the mean of the distribution and

Standard Deviation Formula

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

Standard Deviation Formula

Where \(\overline{x}\)  is the mean of the distribution and

Mean

Now let us try to obtain another standard deviation formula:

Mean of the Distribution

Or,

Standard Deviation Formula

This gives us the standard deviation (σ) as:

Standard Deviation Formula

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 hefty and time-consuming. To solve this problem we make use of step deviation method for simplifying the procedure.

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.

Standard Deviation Formula

The mean of a data set is given by:

Standard Deviation Formula

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

Standard Deviation Formula

Variance of the variable x,

Standard Deviation Formula

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

Standard Deviation Formula

σx=h2 × variance of variable yi

⇒σx2=h2σy2

⇒σx=hσy—(4)

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

Standard Deviation Formula

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.

mean, variance and standard deviation

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

mean, variance and standard deviation -solution

sd18

Thus,

Standard Deviation Formula

The variance of the above data can be calculated as:

Variance

The standard deviation can be given as:

σ=√176=13.266

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Practise This Question

Mean of goals scored by Ronaldo and Messi for each year for the past 5 years is 48 and 50. Standard deviation for each of them is 6 and 4 respectively. Who is more consistent?