FIND MEAN MEDIAN AND MODE OF THE GROUPED DATA

About "Find mean median and mode of grouped data"

Find mean median and mode of grouped data :

Here we are going to see how to find mean median and mode of grouped data.

Mean :

Arithmetic mean (AM) is one of the measures of central tendency which can be defined as the sum of all observations divided by the number of observations. 

Median :

Median is defined as the middle value of the data when the data is arranged in ascending or descending order.

Mode :

If a set of individual observations are given, then the mode is the value which occurs most often.

Let us look into some example problems to understand how to find mean, median and mode of the grouped data.

Example 1 :

Find the mean, median and mode for the following frequency table:

x

10

20

25

30

37

55

f

5

12

14

15

10

4

Solution :

Arithmetic mean  =  ∑fx / N  

x

10

20

25

30

37

55

f

5

12

14

15

10

4

N  =  60

fx

50

240

350

450

370

220

∑fx  =  1680

Arithmetic mean  =  ∑fx / N  =  1680 / 60

  =  28

Hence the required arithmetic mean for the given data is 28.

Median :

x


10

20

25

30

37

55

f


5

12

14

15

10

4

Cumulative

frequency

5

5 + 12  =  17

17 + 14  =  31

31 + 15  =  46

46 + 10  =  56

56 + 4  =  60

Here, the total frequency, N = ∑f = 60

N/2  =  60 / 2  =  30

The median is (N/2)th value = 30th value.

Now, 30th value occurs in the cumulative frequency 31, whose corresponding x value is 25.

Hence, the median = 25.

Mode :

By observing the given data set, the number 30 occurs more number of times. That is 15 times.

Hence the mode is 30.

Mean  =  28

Mode  =  25 and

Mode  =  30.

Example 2 :

Find the mean, median and mode for the following frequency table:

x

19

21

23

25

27

29

31

f

13

15

20

18

16

17

13

Solution :

To find arithmetic mean for this problem, let us use assumed mean method.

Here A  =  25

x

19

21

23

25

27

29

31

f

13

15

20

18

16

17

13

N  =  112

d  =  x - A

-6

-4

-2

0

2

4

6

fd

-78

-60

-40

0

32

68

78

∑fd  =  0

Arithmetic mean  =  A + [∑fd / N] 

  =  25 + (0/112)

  =  25 + 0

=  25

Hence the required arithmetic mean for the given data is 25.

Median :

x


19

21

23

25

27

29

31

f


13

15

20

18

16

17

13

Cumulative

frequency

13

13 + 15  =  28

28 + 20  =  48

48 + 18  =  66

66 + 16  =  82

82 + 17  =  99

99 + 13  =  112

Here, the total frequency, N = ∑f = 112

N/2  =  112 / 2  =  61

The median is (N/2)th value = 61th value.

Now, 61th value occurs in the cumulative frequency 25, whose corresponding x value is 25.

Hence, the median = 25.

Mode :

By observing the given data set, the number 23 occurs more number of times. That is 20 times.

Hence the mode is 23.

Mean  =  25

Mode  =  25

Mode  =  23.

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