Question

In 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 find the number of observations lying between

\[\bar{ X } \]  − M.D. and

\[\bar{ X } \]  + M.D, where M.D. is the mean deviation from the mean.

Answer 1

Let  \[ \bar {x} \] be the mean of the data set.

\[\bar{ x } = \frac{34 + 66 + 30 + 38 + 44 + 50 + 40 + 60 + 42 + 51}{10} = 45 . 5\]

\[MD = \frac{1}{n} \sum^n_{i = 1} \left| d_i \right|, \text{ where}  \left| d_i \right| = \left| x_i - x \right|\]

\[x_i\]	\[\left| d_i \right| = \left| x_i - 45 . 5 \right|\]
34	11.5
66	20.5
30	15.5
38	7.5
44	1.5
50	4.5
40	5.5
60	14.5
42	3.5
51	5.5
Total	90

\[MD = \frac{1}{10} \times 90 = 9\]

\[\bar{ x } - M . D . = 45 . 5 - 9 = 36 . 5\]

\[Also, \bar { x } + M . D . = 45 . 5 + 9 = 54 . 5\]

Hence, there are 6 observations between 36.5 and 54.5.