Mean ( Direct method)

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Mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations.
If x1, x2,… xn, are observations with respective frequencies f1, f2,, . . ., fn then this means observation x1, occurs f1 times, x2, occurs f2, times, and so on. Now, the sum of the values of all the observations = f1 x1 + f2 x2 + . . . + fn xn, and the number of observations = f1 + f2 + . . . + fn.

          f1 x1 + f2 x2 + . . . + fn xn
x = ---------------------------------
           f1 + f2 + . . . + fn

This can be written in short way using Greek letter Σ
       Σfi xi
x = ------
         Σ fi

Example 1 :
Weight (in Kgs) 67 70 72 73 75
Number of students 4 3 2 2 1

Find the average weight.
Solution :
Weight (in kg)xi
Frequency (fi)
fixi
67
4
268
70
3
210
72
2
144
73
2
146
75
1
75
N=Σ fi = 12
Σ fixi = 843

       Σfi xi         843
x = ------- = ------- = 70.25 kg
       Σ fi             12

________________________________________________________________
Example 2 :
If the average of the following data is 20.2, find the value of p:
x
10
15
20
25
30
f
6
8
P
10
6
Solution :
x
f
f x
10
6
60
15
8
120
20
P
20P
25
10
250
30
6
180
30 + P
610 + 20P

       Σfi xi
x = -------
         Σ fi

20.2 = ( 610 + 20P) / ( 30 + P)
20.2 ( 30 + P ) = 610 + 20P
606 + 20.2P = 610 + 20P
20.2P – 20P = 610 – 606
0.2 P = 4
∴ P = 4 / 0.2
P = 20
________________________________________________________________
Example 3: Find the average of the following data :
Marks
0 - 10
10 - 20
20 - 30
30 - 40
40 - 50
No.of students
4
10
18
12
6

Solution: First find the class mark,
Class Mark = ( upper class limit + Lower class limit ) /2
Example : Class Mark = ( 10 + 0 ) / 2 = 10 / 2 = 5
Prepare the frequency table :
Class interval
fi
Class Mark (xi)
fi xi
0 - 10
4
5
20
10 - 20
10
15
150
20 -30
18
25
450
30 - 40
12
35
420
40 - 50
6
45
270
Σ fi = 50
Σ fi xi = 1310

       Σfi xi
x = -------
        Σ fi
Mean = (1310 / 50)
= 26.2


• Direct method.
Short cut method.
Step - Deviation method.

From median to measures of central tendency

Statistics

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