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Mean of Grouped Data

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description

  • Computation of Measures of Central Tendency - Mean
  • Direct Method of Mean
  • Assumed Mean Method 
  • Step Deviation Method for Mean
  • Step-deviation method

notes

To calculate the mean of grouped data, the first step is to determine the midpoint (also called a class mark) of each interval, or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.

Average of any observation is known as mean. In this chapter there are three methods to find mean-

1) Direct method

2) Assumed mean method

3) Step deviation method

We will take a common example to understand these three methods

1) Direct method-

Class interval

10-25

25-40

40-55

55-70

70-85

85-100

No. of students

2

3

7

6

6

6

CI- class interval

fi- Number of students

xi= class mark= `"Upper class limit+ Lower class limit"/2`

means summation, which means the total

 

CI

fi

xi

fixi

10-25

2

(10+25)/2= 17.5

35

25-40

3

32.5

97.5

40-55

7

47.5

332.5

55-70

6

62.5

375

70-85

6

77.5

465

85-100

6

92.5

555

 

fi= 30

 

fixi= 1860

 

Mean through direct method= `bar(x)`= `(sum "fixi")/ (sum "fi")`

Mean through direct method= `1860/30= 62`

2) Assumed mean method-

Mean through assumed mean method= `bar(x)`= ` a+(sum "fidi")/(sum "fi")`

where a= assumed mean i.e any value of xi

di= deviation= xi-a

 

CI

fi

xi

fixi

di=xi-a

fidi

10-25

2

17.5

35

-30

-60

25-40

3

32.5

97.5

-15

-45

40-55

7

47.5

332.5

0

0

55-70

6

62.5

375

15

90

70-85

6

77.5

465

30

180

85-100

6

92.5

555

45

270

 

fi= 30

 

fixi= 1860

 

fidi= 435

Let a= 47.5

Mean through assumed mean method= `bar(x)`= ` a+(sum "fidi")/(sum "fi")`

                                                                    = 47.5+ 435/30

                                                                    = 47.5+ 14.5

Mean through assumed mean method= 62

3) Step deviation method-

Mean through step deviation method= `bar(x)`= `a+ (sum "fiui")/(sum "fi") xx h`

where, ui= modified class mark= `(di)/h`

h= Class size

CI

fi

xi

fixi

di=xi-a

fidi

ui=di/h

fiui

10-25

2

17.5

35

-30

-60

-2

-4

25-40

3

32.5

97.5

-15

-45

-1

-3

40-55

7

47.5

332.5

0

0

0

0

55-70

6

62.5

375

15

90

1

6

70-85

6

77.5

465

30

180

2

12

85-100

6

92.5

555

45

270

3

18

 

fi= 30

 

fixi= 1860

 

fidi= 435

 

fiui= 29

 

Mean through step divation method= `bar(x)`= `a+ (sum "fiui")/(sum "fi") xx h`

 

                                                                 =`47.5+ (29/30) xx 5`

 

                                                                 =` 47.5+ 14.5`

 

Mean through step deviation= `bar(x)`= 62

As you can see, the mean obtained is same i.e 62 from any of the method.

 

 

Shaalaa.com | Statistics part 4 (Direct Method for mean)

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