Standard Deviation

The standard deviation is defined as the positive square root of the mean of the square deviations taken from the 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 formula 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 dominant role in the study of variations in data. It is a very widely used measure of dispersion. It stands like a tower among measures of dispersion. As far as important statistical tools are concerned, the most important tool is the mean $$\overline X $$ and the second most 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 various statistical inferences.

However some alternative methods are also available to compute the standard deviation. These alternative methods simplify the computation. However, in discussing these methods we will focus only on sample data, because sample data rather than the whole population is most interesting to statisticians.


Actual Mean Method

In applying this method first of all we compute the arithmetic mean of the given data, either ungrouped or grouped. Then we 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 the direct method


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

For Ungrouped Data
For Grouped Data
$$S = \sqrt {\frac{{\sum {D^2}}}{n} – {{\left( {\frac{{\sum D}}{n}} \right)}^2}} $$
$$S = \sqrt {\frac{{\sum f{D^2}}}{{\sum f}} – {{\left( {\frac{{\sum fD}}{{\sum f}}} \right)}^2}} $$

Here $$D = X – A$$ and $$A$$ are any assumed mean other than zero. This method is also known as the 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 calculations by taking a common factor or divisor from the given data, the formulas for computing the standard deviation are:

For Ungrouped Data
For Grouped Data
$$S = \sqrt {\frac{{\sum {U^2}}}{n} – {{\left( {\frac{{\sum U}}{n}} \right)}^2}} \times c$$
$$S = \sqrt {\frac{{\sum f{U^2}}}{{\sum f}} – {{\left( {\frac{{\sum fU}}{{\sum f}}} \right)}^2}} \times c{\text{ or }}h$$

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