Question

Marks obtained by 40 students in a short assessment is given below, where a and b are two missing data.

Marks	5	6	7	8	9
Number of Students	6	a	16	13	b

If the mean of the distribution is 7.2, find a and b.

Answer 1

Marks (x)	Number of students (f)	fx
5	6	30
6	a	6a
7	16	112
8	13	104
9	b	9b
	${\displaystyle \sum f=35+a+b}$	${\displaystyle \sum fx=246+6a+9b}$

It is given that the number of students is 40.

∴ 35 + a + b = 40

${\displaystyle \Rightarrow }$ a + b – 5 = 0    ...(1)

Mean = ${\displaystyle \frac{\sum fx}{\sum f}}$

${\displaystyle \Rightarrow \frac{246+6a+9b}{35+a+b}=7.2}$

${\displaystyle \Rightarrow }$ 246 + 6a + 9b = 7.2(35 + a + b)

${\displaystyle \Rightarrow }$ 246 + 6a + 9b = 252 + 7.2a + 7.2b

${\displaystyle \Rightarrow }$ 0 = 252 – 246 + 7.2a – 6a + 7.2b – 9b

${\displaystyle \Rightarrow }$ 6 + 1.2a – 1.8b = 0

${\displaystyle \Rightarrow }$ 10 + 2a – 3b = 0   ...(2)

Solving equations (1) and (2), we have

5a – 5 = 0

${\displaystyle \Rightarrow }$ a = 1

From (1), we have b = 4

Hence, the values of a and b are 1 and 4 respectively.

Answer 2

Mean = ${\displaystyle \frac{\sum fx}{\sum f}}$ 

${\displaystyle \Rightarrow 7.2=\frac{6\times 5+a\times 6+16\times 7+13\times 8+b\times 9}{6+a+16+13+b}}$

${\displaystyle \Rightarrow 7.2=\frac{246+6a+9b}{35+a+b}}$ 

${\displaystyle \Rightarrow }$ 1.2 – 1.8b = –6  ...(i)  

Total number of students = 6 + a + 16 + 13 + b 

${\displaystyle \Rightarrow }$ 40 = 35 + a + b 

${\displaystyle \Rightarrow }$ a + b = 5  ...(ii) 

Multiple equation (ii) by 1.8 and add it to equation (i)

1.8a + 1.8b = 9
1.2a – 1.8b = –6 
   3a            = 3

${\displaystyle \Rightarrow }$ a = 1 

Substituting = 1 in equation (ii) we get,

1 + b = 5 

${\displaystyle \Rightarrow }$ b = 4

Answer 3

Marks (x)	No. of Students (f)	fx
5	6	30
6	a	6a
7	16	112
8	13	104
9	b	9b
Total	${\displaystyle \sum f=35+a+b}$	${\displaystyle \sum fx=246+6a+9b}$

Now, ${\displaystyle \sum f}$ = 40

35 + a + b = 40

a + b = 5    ...(1)

And ${\displaystyle \overline{X}=\frac{\sum fx}{\sum f}}$

${\displaystyle 7.2=\frac{246+6a+9b}{40}}$

${\displaystyle \Rightarrow }$ 6a + 9b + 246 = 288

${\displaystyle \Rightarrow }$ 6a + 9b = 42

${\displaystyle \Rightarrow }$ 2a + 3b = 14    ...(2)

From (1) and (2), 

a = 1, b = 4