The mean deviation about the mean for the data gathered x_i 5 7 9 10 12 15 f_i 8 6 2 2 2 6…

Question

The mean deviation about the mean for the data

$\begin{gathered} {x}_{i} \quad  5 \quad  7 \quad  9 \quad  10 \quad  12 \quad  15 \ {f}_{i} \quad  8 \quad  6 \quad  2 \quad  2 \quad  2 \quad  6 \end{gathered}$

Accepted Answer

For a discrete frequency distribution, the mean deviation about the mean is

$=\frac{\sum {f}_{i}|{x}_{i}-\vec{x}|}{\sum {f}_{i}}$

First, find the mean $\vec{x}$.

Calculate $\sum {f}_{i}$ and $\sum {f}_{i}{x}_{i}$: $\begin{gathered} {x}_{i} \quad  {f}_{i} \quad  {f}_{i}{x}_{i} \ 5 \quad  8 \quad  40 \ 7 \quad  6 \quad  42 \ 9 \quad  2 \quad  18 \ 10 \quad  2 \quad  20 \ 12 \quad  2 \quad  24 \ 15 \quad  6 \quad  90 \end{gathered}$

Now,

$\sum {f}_{i}=8+6+2+2+2+6=26$

and

$\sum {f}_{i}{x}_{i}=40+42+18+20+24+90=234$

So, the mean is

$\vec{x}=\frac{\sum {f}_{i}{x}_{i}}{\sum {f}_{i}}=\frac{234}{26}=9$

Calculate $\sum {f}_{i}|{x}_{i}-\vec{x}|$: Since $\vec{x}=9$, we find $|{x}_{i}-9|$:

$\begin{gathered} {x}_{i} \quad  {f}_{i} \quad  |{x}_{i}-9| \quad  {f}_{i}|{x}_{i}-9| \ 5 \quad  8 \quad  4 \quad  32 \ 7 \quad  6 \quad  2 \quad  12 \ 9 \quad  2 \quad  0 \quad  0 \ 10 \quad  2 \quad  1 \quad  2 \ 12 \quad  2 \quad  3 \quad  6 \ 15 \quad  6 \quad  6 \quad  36 \end{gathered}$

Now,

$\sum {f}_{i}|{x}_{i}-\vec{x}|=32+12+0+2+6+36=88$

Find mean deviation about the mean: $=\frac{88}{26}=\frac{44}{13}$

So, the correct answer is

$\frac{44}{13}$

Hence, Option C is correct.

Solutions · 1