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Download now on Google PlayMean 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 \(68.3\).