Mean Deviation for Continuous Frequency Distribution: Mean Deviation of Grouped Data

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Representation of data in a tabular or graphical form which indicates the frequency (number of times an observation occurs within a particular interval) is known as a frequency distribution.

If the data is huge, for example, if we need to analyze the marks of 200 students, then the representation of such data in a random fashion is not very practical. So, we use the concept of ‘Grouping of Data’ based on class intervals. Let’s calculate mean deviation for the continuous frequency distribution of data.

Mean Deviation for Continuous Frequency Distribution

Mean Deviation for Continuous Frequency Distribution

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Mean Deviation of Grouped Data

In frequency distribution of continuous type, the class intervals or groups are arranged in such a way that there are no gaps between the classes and each class in the table has its respective frequency. The class intervals are chosen in such a way that they must be mutually exclusive and exhaustive.

To understand the concept of continuous frequency distribution let us take the following example:

The following table represents the age group of employees working in a certain company.

Mean Deviation of Grouped Data

Age Group

Number of people

15-25

25

25-35

54

35-45

34

45-55

20

This representation is continuous in nature and the frequency is mentioned according to the class interval.

Steps to Calculate Mean Deviation of Continuous Frequency Distribution

To calculate the mean deviation for continuous frequency distribution, following steps are followed:

Step i) Assume that the frequency in each class is centered at the mid-point. The mean is calculated for these mid-points.

Considering the above example, the mid points are given as:

Steps to Calculate Mean Deviation of Continuous Frequency Distribution

Age group

Number of people

15 - 25

20

25

25 - 35

30

54

35 – 45

40

34

45 – 55

50

20

The mean is calculated by the formula

Step ii) The mean absolute deviation about mean is given by:

The above example can be tabulated as:

The above example can be tabulated as

Age group

Number of People

15 -25

20

25

500

13.684

324.1

25 -35

30

54

1620

3.684

198.936

35 – 45

40

34

1360

6.316

214.744

45 - 55

50

20

1000

16.316

352.32

Now M.A.D.

Note: Sometimes to reduce the complexity, the mean is calculated using Step Deviation Method. The observation which lies in the middle or close to the mid value is considered as the assumed mean. The result obtained is more or less the same. This method reduces the size of the observations and therefore, calculation complexity reduces.

The formula used is:

M.A.D.

Where \(a\) is the assumed mean, \(h\) is the common factor and

Similarly, to calculate the mean deviation about median we need to find out the median of the given set of data with the help of cumulative frequency, which is given as-

Where, is the lower limit of the median class.

is the frequency of median class,

is the width of class

is the cumulative frequency of the preceding class.

The median class is the one whose cumulative frequency is just greater than

We find the mean deviation about median using the formula:

In the example given the mean deviation about median is given as follows:

In the example given the mean deviation about median is given as follows

Class

Frequency

Cumulative Frequency

Mid – Point

5 - 15

5

5

10

17.42

87.1

15 -25

9

14

20

7.42

66.78

35 - 35

7

21

30

2.58

18.06

35 - 45

3

24

40

12.58

37.74

45 – 55

8

32

50

22.58

180.64

32

390.32

Since Therefore the class is the median class.

The mean deviation about median is