Find the mean deviation for the given data is p, 6, 6, 7, 8, 11, 15, 16, if value of mean of the data is 3 times of the ‘p’.

CONCEPT:

Mean Deviation for ungrouped data:

For ‘n’ observation x₁, x₂………….. x_n, the mean deviation about their mean \(\bar x\) is given by \(M.D= \frac{{\mathop \sum \nolimits_{i = 1}^n \left| {{x_i} - \bar x} \right|}}{N}\).where N is the number of observations

CALCULATIONS:

Given data of numbers are p, 6, 6, 7, 8, 11, 15, and 16.

Mean \(\rm\bar x = \frac{{Sum\;of\;all\;the\;observations}}{{Total\;number\;of\;observations}} = \frac{{p + 6 + 6 + 7 + 8 + 11 + 15 + 16}}{8} = 3p\)

⇒ p + 69 = 24p

⇒  23p = 69                  

∴  p = 3

So given data is 3, 6, 6, 7, 8, 11, 15, 16 and mean is 3p, i.e. 9.

∴ Mean deviation \(= \frac{{\left| {3 - 9} \right| + \left| {6 - 9} \right| + \left| {6 - 9} \right| + \left| {7 - 9} \right| + \left| {8 - 9} \right| + \left| {11 - 9} \right| + \left| {15 - 9} \right| + \left| {16 - 9} \right|}}{8}\)