The arithmetic mean of the following frequency distribution is 53. Find the value of k.

Use the shortcut method to find the mean of given data.
For that,
Let the assumed mean be (A) = 2,
The deviation of values x_i from assumed mean be d_i = x_i – A.
Now to find the mean:
First multiply the frequencies in column (ii) with the value of deviations in column (iv) as f_id_i.

Now add the sum of all entries in column (iv) to obtain

and the sum of all frequencies in the column (ii) to obtain

So,
=

where, N = total number of observations

⇒ 3(72 + k) = -260 + 20k

⇒ 216 + 3k = -260 + 20k

⇒ 476 = 17k

⇒ k = 28