Arithmetic mean — lesson. Mathematics State Board, Class 10.

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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,

\bar{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:

\bar{X} = \frac{\sum_{i = 1}^{n} x_{i} f_{i}}{\sum_{i = 1}^{n} f_{i}}

Click here! To recall the direct method to find mean.

2. Assumed Mean Method:

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

\bar{X} = A + \frac{\sum_{i = 1}^{n} f_{i} d_{i}}{\sum_{i = 1}^{n} f_{i}}

where \(d_{i} = x_{i} - A\).

Click here! To recall the assumed mean method to find mean.

3. Step Deviation Method:

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

\bar{X} = A + c \times \frac{\sum_{i = 1}^{n} f_{i} d_{i}}{\sum_{i = 1}^{n} f_{i}}

where \(d_{i} = \frac{ x_{i} - A}{c}\).

Click here! To recall the step deviation method to find mean.

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2. Arithmetic mean

Theory: