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# Mean deviation and frequency distribution

A large data set is represented in a graphical or tabular form denoting the frequency of occurrence of each set of data. This observation is known as frequency distribution. For example, a class has how many cricketers, tennis players, and badminton players can be grouped into a frequency table to find out other relevant data. This grouping is known as grouped data and can be used to find out mean deviation.

## Mean deviation of a grouped data

In a grouped data, the class intervals are arranged in such a way that they do not have any gaps, and each class has their own frequency.

Formula: - ∑ f | X-X| / ∑ f

where, f is the value of frequency

x is the mean, calculated as (sum of all the values/number of values) = ∑ f x / ∑ f

mid points are calculated as (lower limit + upper limit) / 2.

 Class interval Players (f) Values (x) F(x) Mean dx – x-x Absolute deviation F |dx| = |fx-x| 0-10 5 5 25 -22 22 110 10-20 8 15 120 -12 12 96 20-30 15 25 375 -2 2 30 30-40 16 35 560 8 8 128 40-50 6 45 270 18 18 108 ∑ f  =50 ∑ f x = 1350 ∑f |dx| = 472

X =∑ f x / ∑ f  = 1350/50

X= 27

Mean Deviation=∑ f | X-X| / ∑ f

= 472/50

= 9.44

Hence, mean deviation is 9.44

## Mean deviation of ungrouped data

In ungrouped data, some of the class intervals tend to be missing with irregular frequency distributions amongst them.

Example: - Find the mean deviation of the following ungrouped data.

 X 5 7 8 9 10 11 13

Solution

 X |x-a| 5 3 7 1 8 0 9 1 10 2 11 3 13 5

We need to find the median first. Here total entries are 7, which is an odd number.

Therefore, median = item corresponding to the value of (n+1) /2

= 8/2 = 4th item = 8

Mean deviation = ∑ f |X-Me| / N = ∑ f | D| / N

= 15/7 = 2.14

## Features of mean deviation

• The units of mean deviation are the same as that of the variables.
• They are rigidly defined.
• Their values depend upon each of the entered data.
• It is also known as absolute deviation because the values are absolute.

• It gives a better result as all the values are taken into consideration for calculation.
• It is widely used in economics, businesses, commerce, and related fields
• It can be used to compare two or more series.
• Since the median is least affected by any of the terms, it gives a least affected result if extreme terms are changed.