Q.Find the mean deviation about the median for the following distribution. Marks: $10\text{-}20, 20\text{-}30, 30\text{-}40, 40\text{-}50, 50\text{-}60, 60\text{-}70$; No. of students: $3, 4, 7, 8, 2, 1$.

Find the median class and median, then compute $\dfrac{1}{N}\sum f_i |x_i - M|$ using class midpoints.

Data:

$N = 3+4+7+8+2+1 = 25$

Step 1: Find the median.

Cumulative frequencies: $3, 7, 14, 22, 24, 25$.

$N/2 = 12.5$, and the cumulative frequency first exceeds this in the class $30\text{-}40$ (cf just before it $=7$, $f=7$, $h=10$, $l=30$).

$M = l + \dfrac{N/2 - cf}{f} \times h = 30 + \dfrac{12.5-7}{7}\times10 = 30 + \dfrac{55}{7} \approx 37.86$

Step 2: Compute $|x_i - M|$ for each midpoint $x_i$.

Midpoints: $15, 25, 35, 45, 55, 65$

| $x_i$ | $f_i$ | $|x_i - 37.86|$ | $f_i|x_i-M|$ |

|---|---|---|---| …