6. Methods to calculate the standard deviation of grouped data

The standard deviation of an grouped data (either discrete or continuous) can be calculated using one of the following methods:

Let $x_{1}, x_{2}, x_{3}, … , x_{n}$ be the given data for $n$ observations.

 

And, $\overline{x}$ is the mean of the $n$ observations.

 

Let $f_{i}$ be the frequency values of the corresponding data values $x_{i}$ and $d_{i} = x_{i} - \overline{x}$ (deviations from  mean).

 

Then, the formula to calculate the standard deviation by mean method is given by:

 

If the mean of the given data is not an integer, then use the assumed mean method to find the standard deviation.

 

Let $x_{1}, x_{2}, x_{3}, … , x_{n}$ be the given data and $\overline{x}$ be its mean.

 

Let $f_{i}$ be the frequency values of the corresponding data values $x_{i}$.

 

Let $d_{i}$ be the deviation of each observation $x_{i}$ from the assumed mean $A$ where $A$ is the middle most value of the given data. That is, $d_{i} = x_{i} - A$.

 

Then, the formula to calculate the standard deviation by assumed mean method is given by:

 

$\sigma = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right)^2}$ where $N = \sum_{i = 1}^{n} f_{i}$.

Let $x_{1}, x_{2}, x_{3}, … , x_{n}$ be the middle values of the given class interval correspondingly and $A$ is its assumed mean.

 

Let $f_{i}$ be the frequency values of the corresponding middle values $x_{i}$.

 

Let $c$ be the width of the class interval.

 

Let $d_{i} = \frac{x_{i} - A}{c}$.

 

Then, the formula to calculate the standard deviation by step deviation method is given by:

 

$\sigma = c \times \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right) ^2}$ where $N = \sum_{i = 1}^{n} f_{i}$.