Question

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be –3.5. The mean of the given numbers is ______.

Options

• 46.5

• 49.5

• 53.5

• 56.5

Answer 1

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be –3.5. The mean of the given numbers is 56.5.

Explanation:

Given, n = 50, then mean `barx = (sum_(i = 1)^n x_i)/n`

Then, `barx = 1/50 xx sum_(i = 1)^50 x_i`  ...(i)

⇒ `sum_(i = 1)^50 x_i = 50 barx`

Now, subtract each observation from 53, we get a new mean say `barx_("new")`.

∴ `barx_("new") = ((-x_1 + 53) + (-x_2 + 53) + ... + (-x_50 + 53))/50`

⇒ `-3.5 = (-(x_1 + x_2 + ... + x_50) + (53 + 53 + ... + 50  "times"))/50`

⇒ `-3.5 xx 50 = - (x_1 + x_2 + ... + x_50) + 53 xx 50`

⇒ `sum_(i = 1)^50 x_i` = 2650 + 175 = 2825

∴ Mean of 50 observations = `1/50 sum_(i = 1)^50 x_i`  ...`[∵ "Mean" = (sum_(i = 1)^n x_i)/n]`

= `1/50 xx 2825`  

= 56.5    

Hence, the mean of given number is 56.5