Standard Deviation

The standard deviation is defined as the positive square root of the mean of the square deviations taken from arithmetic mean of the data.

For the sample data the standard deviation is denoted by S and is defined as:

S = \sqrt {\frac{{\sum {{\left( {X - \overline X } \right)}^2}}}{n}}

For frequency distribution the formulas becomes

S = \sqrt {\frac{{\sum f{{\left( {X - \overline X }  \right)}^2}}}{{\sum f}}}


The standard deviation is in the same units as the units of the original observations. If the original observations are in grams, the value of the standard deviation will also be in grams.

The standard deviation plays a dominating role for the study of variation in the data. It is a very widely used measure of dispersion. It stands like a tower among measure of dispersion. As far as the important statistical tools are concerned, the first important tool is the mean \overline X and the second important tool is the standard deviation S. It is based on all the observations and is subject to mathematical treatment. It is of great importance for the analysis of data and for the various statistical inferences.

However some alternative methods are also available to compute standard deviation. The alternative methods simplify the computation. Moreover in discussing these methods we will confirm ourselves only to sample data because sample data rather than whole population confront mostly a statistician.

Actual Mean Method:

In applying this method first of all we compute arithmetic mean of the given data either ungroup or grouped data. Then take the deviation from the actual mean. This method is already defined above. The following formulas are applied:

For Ungrouped Data

For Grouped Data

S = \sqrt {\frac{{\sum {{\left( {X - \overline X }    \right)}^2}}}{n}}

S = \sqrt {\frac{{\sum f{{\left( {X - \overline X }    \right)}^2}}}{{\sum f}}}

  This method is also known as direct method

Assumed Mean Method:
            (a) We use following formulas to calculate standard deviation:

For Ungrouped Data

For Grouped Data

S = \sqrt {\frac{{\sum {D^2}}}{n} - {{\left( {\frac{{\sum D}}{n}}    \right)}^2}}

S = \sqrt {\frac{{\sum f{D^2}}}{{\sum f}} - {{\left( {\frac{{\sum    fD}}{{\sum f}}} \right)}^2}}

Where D = X - A and A is any assumed mean other than zero. This method is also known as short-cut method.

            (b) If A is considered to be zero then the above formulas are reduced to the following formulas:

For Ungrouped Data

For Grouped Data

S = \sqrt {\frac{{\sum {X^2}}}{n} - {{\left( {\frac{{\sum X}}{n}}    \right)}^2}}

S = \sqrt {\frac{{\sum f{X^2}}}{{\sum f}} - {{\left( {\frac{{\sum    fX}}{{\sum f}}} \right)}^2}}

            (c) If we are in a position to simplify the calculation by taking some common factor or divisor from the given data the formulas for computing standard deviation are:

For Ungrouped Data

For Grouped Data

S = \sqrt {\frac{{\sum {U^2}}}{n} - {{\left( {\frac{{\sum U}}{n}}    \right)}^2}} \times c

S = \sqrt {\frac{{\sum f{U^2}}}{{\sum f}} - {{\left( {\frac{{\sum    fU}}{{\sum f}}} \right)}^2}} \times    c{\text{ or }}h

Where U = \frac{{X -  A}}{{h{\text{ or }}c}} = \frac{D}{{h{\text{ or }}c}}, h = Class Interval and c = Common Divisor. This method is also called method of step-deviation.