Concept: Mean Deviation About Median

The mean deviation about the median measures average absolute dispersion from the middle value of the distribution.

Step 1: Find the median

Total frequency $N = 6 + 7 + 15 + 16 + 4 + 2 = 50$, so $\frac{N}{2} = 25$.

Cumulative frequencies: 6, 13, 28, 44, 48, 50. The median class is 20–30 (since cf = 28 first exceeds 25).

Using the median formula:

Step 2: Compute $|x_i - \text{Median}|$ for each class midpoint

| Class | Midpoint $x_i$ | $f_i$ | $|x_i - 28|$ | $f_i |x_i - 28|$ |

|---|---|---|---|---|

| 0-10 | 5 | 6 | 23 | 138 |

| 10-20 | 15 | 7 | 13 | 91 | …

Understanding Mean Deviation About the Median (Grouped Data)

For a continuous frequency distribution we find the median by interpolation within the median class, then take the frequency-weighted average of the absolute deviations of the class marks from that median.

Step-by-Step Solution

1. Class marks, frequencies, and cumulative frequencies

Total $N = 50$, so $\dfrac{N}{2} = 25$.

2. Find the median

The cumulative frequency first exceeds $25$ in the class $20\text{-}30$, so this is the median class:

$L = 20,\quad C = 13,\quad f = 15,\quad h = 10$ …