Theory:

The standard deviation of an ungrouped data can be calculated using one of the following methods:
  • Direct Method:
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:
 
\(\sigma = \sqrt{\frac{\sum x_{i}^{2}}{n}- \left(\frac{\sum x_{i}}{n}\right)^2}\)
  • Mean Method:
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:
 
\(\sigma\) \(=\) \(\sqrt{\frac{\sum d_{i}^{2}}{n}}\) where \(d_{i} = x_{i} - \overline{x}\)
  • Assumed Mean Method:
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:
 
\(\sigma = \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\)
  • Step Deviation Method:
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:
 
\(\sigma = c \times \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\)
Important!
  • When each values of the observation is added or subtracted by a fixed constant then the standard deviation remains the same.
  • When each values of the observation is multiplied or divided by a fixed constant then the standard deviation is also multiplied or divided by the same constant.