### Theory:

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 $$\overline X = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}$$ $=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}$

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, $\overline{X}=\frac{\sum _{i=1}^{n}{x}_{i}}{n}=\frac{\sum x}{n}$, where $$n$$ is the total number of observations.
Arithmetic mean or mean is the commonly used method to find the average of the given data.
Example:
The marks scored in science by $$10$$ students in a class are: $$55$$, $$60$$, $$92$$, $$100$$, $$45$$, $$78$$, $$85$$, $$50$$, $$48$$, $$70$$. Find the mean of the given marks.

Solution:

The mean of arithmetic mean can be determined by

$$\overline X = \frac{\sum x}{n}$$

$$= \frac{55 + 60 + 92 + 100 + 45 + 78 + 85 + 50 + 48 + 70}{10}$$

$$= \frac{683}{10}$$

$$\overline X = 68.3$$

Therefore, the mean of the given data is $$68.3$$.
Assumed mean method
In some problems, we make the problems easier by assuming a number would be the correct answer. This guessed number is called as assumed mean.
Example:
Let us consider the above example.

Let us assume that $$45$$ is the assumed mean. Now, we shall find the differences between the assumed mean of each mark.

Thus, we have:

$$55 - 45 = 10$$

$$60 - 45 = 15$$

$$92 - 45 = 47$$

$$100 - 45 = 55$$

$$45 - 45 = 0$$

$$78 - 45 = 33$$

$$85 - 45 = 40$$

$$50 - 45 = 5$$

$$48 - 45 = 3$$

$$70 - 45 = 25$$

The average of the differences $$= \frac{10 + 15+ 47 + 55 + 0 + 33 + 40 + 5 + 3 + 25}{10}$$ $$= \frac{233}{10}$$ $$= 23.3$$

Let us add the mean difference to the assumed mean to get the correct mean.

Correct mean $$=$$ Assumed mean $$+$$ Mean difference

Correct mean $$= 45 + 23.3$$ $$= 68.3$$

Therefore, the mean of the above given data is $$68.3$$.