### 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.