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: Understanding the Question:
We are given a cumulative frequency distribution of the "more than" type.
We need to:
1. Convert this cumulative table into a standard grouped frequency distribution.
2. Compute the Median and the Mode of the marks.

Step 2: Key Formula or Approach:
Convert "more than" cumulative frequency to class intervals by subtracting consecutive frequencies:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \]
\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]

Step 3: Detailed Explanation:

• Convert the table to standard frequency distribution:
Total number of students, $N = 80$.

• Calculate the Median:
We have $\frac{N}{2} = \frac{80}{2} = 40$.
The cumulative frequency just greater than 40 is 52, which corresponds to the class interval $50 - 60$.
Therefore, the Median Class is $50 - 60$.
Parameters:
- Lower limit of median class, $L = 50$
- Cumulative frequency of preceding class, $cf = 37$
- Frequency of median class, $f = 15$
- Class width, $h = 10$
Apply the formula:
\[ \text{Median} = 50 + \left( \frac{40 - 37}{15} \right) \times 10 \]
\[ \text{Median} = 50 + \left( \frac{3}{15} \right) \times 10 \]
\[ \text{Median} = 50 + 2 = 52 \]

• Calculate the Mode:
The highest frequency is $15$, which lies in the class interval $50 - 60$.
Therefore, the Modal Class is $50 - 60$.
Parameters:
- Lower limit of modal class, $L = 50$
- Frequency of modal class, $f_1 = 15$
- Frequency of preceding class, $f_0 = 12$
- Frequency of succeeding class, $f_2 = 12$
- Class width, $h = 10$
Apply the formula:
\[ \text{Mode} = 50 + \left( \frac{15 - 12}{2(15) - 12 - 12} \right) \times 10 \]
\[ \text{Mode} = 50 + \left( \frac{3}{30 - 24} \right) \times 10 \]
\[ \text{Mode} = 50 + \left( \frac{3}{6} \right) \times 10 \]
\[ \text{Mode} = 50 + 5 = 55 \]


Step 4: Final Answer:
The median of the data is 52 and the mode of the data is 55.