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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.

5

7

8

9

10

11

13

Solution 

|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.

Advantages

  • 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.

Disadvantages

  • It cannot be used if fractional data comes into being.
  • Negative values cannot be taken into count.
  • If samples increase, it becomes hectic to recalculate the whole data.
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