Question

The marks obtained by 80 students of class X in a mock test of Mathematics are given below in the table. Find median and the mode of the data :

Hint:

When converting a cumulative table, make sure the sum of your calculated frequencies equals the total frequency given in the first category ($N = 80$).
This acts as an immediate check on your subtraction!

Answer 1

Step 1: Locate the median class from the cumulative frequencies.
With $N=80$, we need $\frac{N}{2}=40$. The class $50\text{-}60$ is the first one whose cumulative frequency (52) exceeds 40, with the cumulative frequency just before it being $cf=37$, frequency $f=15$, lower limit $L=50$, and class width $h=10$.
Step 2: Apply the median formula for grouped data.
\[ \text{Median}=L+\left(\dfrac{\tfrac{N}{2}-cf}{f}\right)\times h = 50+\left(\dfrac{40-37}{15}\right)\times10 = 50+2=52 \]
Step 3: The same class also holds the highest frequency, so find the mode there.
With modal class $50\text{-}60$, $f_1=15$, $f_0=12$, $f_2=12$, $L=50$, $h=10$: \[ \text{Mode}=L+\left(\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h = 50+\left(\dfrac{15-12}{30-12-12}\right)\times10 = 50+5=55 \]
\[ \boxed{\text{Median}=52,\ \text{Mode}=55} \]