Q.Find the mean deviation about the median for the following data: 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Concept: Mean Deviation About Median — we find the median of the data, then average the absolute deviations from it.

Step 1: Arrange data in ascending order

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

$n = 12$ (even), so median is the average of the 6th and 7th values:

Median $M = \frac{13 + 14}{2} = 13.5$

Step 2: Compute absolute deviations $|x_i - M|$

$|10-13.5|=3.5,\; |11-13.5|=2.5,\; |11-13.5|=2.5,\; |12-13.5|=1.5,$

$|13-13.5|=0.5,\; |13-13.5|=0.5,\; |14-13.5|=0.5,\; |16-13.5|=2.5,$

$|16-13.5|=2.5,\; |17-13.5|=3.5,\; |17-13.5|=3.5,\; |18-13.5|=4.5$

Step 3: Sum of deviations

$3.5+2.5+2.5+1.5+0.5+0.5+0.5+2.5+2.5+3.5+3.5+4.5 = 28$

Step 4: Mean deviation

$\text{MD} = \frac{28}{12} = \frac{7}{3} \approx 2.33$

Mean Deviation About the Median

The mean deviation about the median is the average of the absolute distances of the observations from the median:

Step-by-Step Solution

1. Arrange the data in ascending order.

There are $n = 12$ observations (even).

2. Find the median.

For an even number of observations the median is the mean of the $6^{\text{th}}$ and $7^{\text{th}}$ values:

3. Absolute deviations $|x_i - 13.5|$.

4. Sum the deviations.

5. Divide by $n = 12$.