# Mean of Grouped Data

#### 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.

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Statistics part 4 (Direct Method for mean) [00:04:20]
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