There are 60 students in a class. The following is the frequency distribution of the marks obtained by the students in a test:

where x is a positive integer. Determine the mean and standard deviation of the marks.

Given: There are 60 students in a class. The frequency distribution of the marks obtained by the students in a test is also given.

To find: the mean and standard deviation of the marks.

It is given there are 60 students in the class, so

∑f_i=60

⇒ (x-2)+x+x²+(x+1)²+2x+x+1=60

⇒ 5x-1 +x²+x²+2x+1=60

⇒ 2x² +7x=60

⇒ 2x² +7x-60=0

Splitting the middle term, we get

⇒ 2x² + 15x – 8x – 60 = 0

⇒ x(2x + 15) – 4(2x + 15) = 0

⇒ (2x + 15) (x – 4) = 0

⇒ 2x + 15 = 0 or x-4=0

⇒ 2x=-15 or x=4

Given x is a positive number, so x can take 4 as the only value.

And let assumed mean, a=3.

Now put x = 4 and a=3 in the frequency distribution table and add other columns after calculations, we get

And we know standard deviation is

Substituting values from above table, we get

⇒ σ=1.12

Hence the standard deviation is 1.12

Now mean is

=2.8

Hence the mean and standard deviation of the marks are 2.8 and 1.12 respectively.