### Theory:

Arithmetic Mean or Mean is defined as the sum of all the observations divided by the total number of observations. It is usually denoted by $$\overline X$$.

Therefore, mean $$\overline X = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}$$
Arithmetic mean or mean is the commonly used method to find the average of the given data.
Methods of finding mean
• If the given collection of data is ungrouped, then the mean is obtained by direct method.

Mean $$\overline X$$ $=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

If the number of observations is very long, it is a bit difficult to write them. Hence, we use the Sigma notation $$\sum$$ for summation.

That is, $\overline{X}=\frac{\sum _{i=1}^{n}{x}_{i}}{n}$, where $$n$$ is the total number of observations.
• If the given collection of data is grouped, then the mean is obtained by the following methods.

1. Direct Method:

The formula for finding the arithmetic mean using the direct method is given by:

$\overline{X}=\frac{\sum _{i=1}^{n}{x}_{i}{f}_{i}}{\sum _{i=1}^{n}{f}_{i}}$

2. Assumed Mean Method:

The formula for finding the arithmetic mean using the assumed mean method is given by:

$\overline{X}=A+\frac{\sum _{i=1}^{n}{f}_{i}{d}_{i}}{\sum _{i=1}^{n}{f}_{i}}$ where  $$d_{i} = x_{i} - A$$.

$\overline{X}=A+c×\frac{\sum _{i=1}^{n}{f}_{i}{d}_{i}}{\sum _{i=1}^{n}{f}_{i}}$ where  $$d_{i} = \frac{ x_{i} - A}{c}$$.