5. Methods to calculate the standard deviation of ungrouped data

The standard deviation of an ungrouped data 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.

 

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

 

$\sigma = \sqrt{\frac{\sum x_{i}^{2}}{n}- \left(\frac{\sum x_{i}}{n}\right)^2}$

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.

 

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 $d_{i}$ 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:

 

$\sigma = \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}$

Let $x_{1}, x_{2}, x_{3}, … , x_{n}$ be the given data and $A$ is its assumed mean.

 

Let $c$ be the common divisor of $x_{i} - A$.

 

Let $d_{i} = \frac{x_{i} - A}{c}$.

 

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

 

$\sigma = c \times \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}$