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Missing Frequencies

Example

Find the missing frequency (p) for the following distribution whose mean is 7.68:

x: 3 5 7 9 11 13
f: 6 8 15 p 8 4
Solution​ 
xi fi fixi
3 6 18
5 8 40
7 15 105
9 p 9p
11 8 88
13 4 52
Total N = Σfi = 41 + p Σfix= 303 + 9p


Mean  = 
          

⇒ 7.68 = 

=>7.68 (41+p) = 303 + 9p

                ⇒ p = 9 


 

Example

 Find the value of p, if the mean of the following distribution is 20.

x 15 17 19 20 + p 23
f 2 3 4 5p 6
Solution
xi fi xfi
15 2 30
17 3 51
19 4 76
20 + p 5p 100p + 5p2
23 6 138
Total N = Σfi = 15 + 5p Σfix= 295 + 100p + 5p2


 =           ⇒ 20 =   
          ⇒ 300 + 100p = 295 + 100 p + 5p2

         
⇒ 5p= 5

            p = ±1.

           ⇒ p = 1           (Since the frequency cannot be negative).

 

 

Example

Candidates of four schools appeared in a science test. The data is given in the table:

Schools No. of Candidates Average scores
A 60 75
B 48 80
C Not available 55
D 40 50

If the average score of the candidates of all the four schools is 66, find the number of candidates that appeared from school 'C'.
 

Solution

Let the number of candidates that appeared from school 'C' be p.

Schools Average 
Score xi
No. of Candidates fi fixi
A 75 60 4500
B 80 48 3840
C 55 P 55P
D 50 40 2000
    Σfi = 148 + p Σfix= 10340 + 55P



 66(148 + p) = 10340 + 55p

   10340 - 9765 = 66p - 55p

                572 = 10p   
                (or) p = 57.2.


 

Example

The mean of the following frequency distribution is 50. Frequencies f1 and f2 are unknown. Find the missing frequencies:

Class-Interval Frequency
0 - 20 17
20 - 40 f1
40 - 60 32
60 - 80 f2
80 - 100 19
Total 120
Solution

Mean x =50

Class - Interval Mid-Value xi Frequency fi fi xi
0 - 20 10 17 170
20 - 40 30 f1 30f1
40 - 60 50 32 1600
60 - 80 70 f2 70f2
80 - 100 90 19 1710
Total   68 + f+ f= 120 3480 + 30f+ 70f2

Clearly, 68 + f+ f= 120
or f+ f= 52----(1)
Mean (x) = 

⇒  50 = 
⇒ 6000 = 3480 + 30f+ 70f2
⇒ 3f+ 7f= 252----(2)
Solving eqs (1) and (2), we get
f= 28 and f= 24.
Hence missing frequencies are 28 and 24.





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