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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 
1025 
2540 
4055 
5570 
7085 
85100 
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 
1025 
2 
(10+25)/2= 17.5 
35 
2540 
3 
32.5 
97.5 
4055 
7 
47.5 
332.5 
5570 
6 
62.5 
375 
7085 
6 
77.5 
465 
85100 
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= xia
CI 
fi 
xi 
fixi 
di=xia 
fidi 
1025 
2 
17.5 
35 
30 
60 
2540 
3 
32.5 
97.5 
15 
45 
4055 
7 
47.5 
332.5 
0 
0 
5570 
6 
62.5 
375 
15 
90 
7085 
6 
77.5 
465 
30 
180 
85100 
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=xia 
fidi 
ui=di/h 
fiui 
1025 
2 
17.5 
35 
30 
60 
2 
4 
2540 
3 
32.5 
97.5 
15 
45 
1 
3 
4055 
7 
47.5 
332.5 
0 
0 
0 
0 
5570 
6 
62.5 
375 
15 
90 
1 
6 
7085 
6 
77.5 
465 
30 
180 
2 
12 
85100 
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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Shaalaa.com  Statistics part 4 (Direct Method for mean)
Related QuestionsVIEW ALL [20]
The following frequency distribution table shows the amount of aid given to 50 flood affected families. Find the mean of the amount of aid.
Amount of aid
(Thousand rupees)

50  60  60  70  70  80  80  90  90  100 
No. of families  7  13  20  6  4 
The following table gives the frequency distribution of trees planted by different Housing Societies in a particular locality:
No. of Trees  No. of Housing Societies 
1015  2 
1520  7 
2025  9 
2530  8 
3035  6 
3540  4 
Find the mean number of trees planted by Housing Societies by using ‘Assumed Means Method’
The distances covered by 250 public transport buses in a day is shown in the following frequency distribution table. Find the median of the distance.
Distance (km)

200  210  210  220  220  230  230  240  240  250 
No. of buses  40  60  80  50  20 
The measurements (in mm) of the diameters of the head of the screws are given below:
Diameter (in mm)  No. of Screws 
33 — 35  10 
36 — 38  19 
39 — 41  23 
42 — 44  21 
45 — 47  27 
Calculate mean diameter of head of a screw by ‘Assumed Mean Method’.
The measurements (in mm) of the diameters of the head of the screws are given below :
Diameter (in mm)  no. of screws 
33  35  9 
36  38  21 
39  41  30 
42  44  22 
45  47  18 
Calculate the mean diameter of the head of a screw by the ' Assumed Mean Method'.
The weekly wages of 120 workers in a factory are shown in the following frequency distribution table. Find the mean of the weekly wages.
Weekly wages
(Rupees)

0  2000  2000  4000  4000  6000  6000  8000 
No. of workers  15  35  50  20 