Mean Deviation for Grouped Data:

To present the data in the more compressed form we group it and mention the frequency distribution of each such group. These groups are known as class intervals.

Grouping of data is possible in two ways:

- Discrete Frequency Distribution
- Continuous Frequency Distribution

In the upcoming discussion, we will be discussing mean absolute deviation in a discrete frequency distribution.

Let us first know what is actually meant by the discrete distribution of frequency.

As the name itself suggests,by discrete we mean distinct or non-continuous. In such a distribution the frequency (number of observations) given in the set of data is discrete in nature.

If the data set consists of values x1,x2, x3………xn each occurring with a frequency of f1, f2… fn respectively then such a representation of data is known as discrete distribution of frequency.

To calculate the mean deviation for grouped data and particularly for discrete distribution data the following steps are followed:

Step I The measure of central tendency about which mean deviation is to be found out is calculated. Let this measure be a.

If this measure is mean then it is calculated as,

where \(N=\sum_{i=1}^{n}\;f_{i}\)

If the measure is median then the given set of data is arranged in ascending order and then the cumulative frequency is calculated then the observations whose cumulative frequency is equal to or just greater than N/2 is taken as the median for the given discrete distribution of frequency and it is seen that this value lies in the middle of the frequency distribution.

Step II Calculate the absolute deviation of each observation from the measure of central tendency calculated in step (I)

StepIIIThe mean absolute deviation around the measure of central tendency is then calculated by using the formula

If the central tendency is mean then,

In case of median

Let us look into the following examples for a better understanding.

Example: In a foreign language class there are 4 languages and the frequencies of students learning the language and the frequency of lectures per week is given as:

Language | Sanskrit | Spanish | French | English |

No. of students(x_{i}) |
6 | 5 | 9 | 12 |

Frequency of lectures(f_{i}) |
5 | 7 | 4 | 9 |

Calculate the mean deviation about the mean for the given data.

Solution: The following table gives us a tabular representation of data and the calculations

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