Question

If m and M denote the mean deviations about mean and about median respectively of the data 20, 5, 15, 2, 7, 3, 11, then the mean deviation about the mean of m and M is:

• A. $\frac{1}{7} $
• B. $\frac{38}{7}$
• C. $\frac{36}{7}$
• D. $\frac{37}{7}$

Hint:

Remember the formulas for mean deviation about mean and median: Mean deviation about mean = $\frac{\sum |x_i - \bar{x}|}{n}$ Mean deviation about median = $\frac{\sum |x_i - \text{median}|}{n}$

Answer 1

The Correct Option is A

Step 1: Arrange the data in ascending order.
The given data is 20, 5, 15, 2, 7, 3, 11.
Arranging in ascending order: 2, 3, 5, 7, 11, 15, 20.

Step 2: Calculate the mean.
Mean (\(\bar{x}\)) = \(\frac{2+3+5+7+11+15+20}{7} = \frac{63}{7} = 9\).

Step 3: Calculate the mean deviation about the mean (m).
m = \(\frac{\sum |x_i - \bar{x}|}{n}\)
m = \(\frac{|2-9| + |3-9| + |5-9| + |7-9| + |11-9| + |15-9| + |20-9|}{7}\)
m = \(\frac{7 + 6 + 4 + 2 + 2 + 6 + 11}{7} = \frac{38}{7}\).

Step 4: Calculate the median.
Since there are 7 data points, the median is the middle value, which is 7.

Step 5: Calculate the mean deviation about the median (M).
M = \(\frac{\sum |x_i - \text{median}|}{n}\)
M = \(\frac{|2-7| + |3-7| + |5-7| + |7-7| + |11-7| + |15-7| + |20-7|}{7}\)
M = \(\frac{5 + 4 + 2 + 0 + 4 + 8 + 13}{7} = \frac{36}{7}\).

Step 6: Calculate the mean of m and M.
Mean of m and M = \(\frac{m + M}{2} = \frac{\frac{38}{7} + \frac{36}{7}}{2} = \frac{\frac{74}{7}}{2} = \frac{74}{14} = \frac{37}{7}\).

Step 7: Calculate the mean deviation about the mean of m and M.
Mean of m and M = \(\frac{37}{7}\).
Mean deviation about the mean of m and M = \(\frac{\left|\frac{38}{7} - \frac{37}{7}\right| + \left|\frac{36}{7} - \frac{37}{7}\right|}{2} = \frac{\left|\frac{1}{7}\right| + \left|\frac{-1}{7}\right|}{2} = \frac{\frac{1}{7} + \frac{1}{7}}{2} = \frac{\frac{2}{7}}{2} = \frac{2}{14} = \frac{1}{7}\).
Therefore, the mean deviation about the mean of m and M is \(\frac{1}{7}\).