### Theory:

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

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:

$$\sigma$$ $$=$$ $$\sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}}$$ where $$N = \sum_{i = 1}^{n} f_{i}$$
• 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 $$f_{i}$$ 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 $$N = \sum_{i = 1}^{n} f_{i}$$.
• Step Deviation Method:
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 $$N = \sum_{i = 1}^{n} f_{i}$$.