Question

Find the median from the following frequency distribution: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency}
\hline 10 - 25 & 3
25 - 40 & 10
40 - 55 & 20
55 - 70 & 13
70 - 85 & 4
\hline \end{array} \]

Hint:

When calculating the median of a frequency distribution, identify the median class by finding where \( \frac{N}{2} \) lies in the cumulative frequency and apply the median formula accordingly.

Answer 1

The median of a frequency distribution is given by the formula:

Median Formula:

\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times h, \]

where:

• \( L \) is the lower boundary of the median class,
• \( F \) is the cumulative frequency of the class before the median class,
• \( f \) is the frequency of the median class,
• \( h \) is the class width,
• \( N \) is the total frequency.

Step 1: Calculate the cumulative frequency

The cumulative frequency is calculated by adding the frequencies sequentially:

Class Interval	Frequency	Cumulative Frequency
10 - 25	3	3
25 - 40	10	13
40 - 55	20	33
55 - 70	13	46
70 - 85	4	50

Step 2: Determine the median class

The total frequency \( N = 50 \). To find the median class, we calculate \( \frac{N}{2} = \frac{50}{2} = 25 \).

The cumulative frequency just greater than or equal to 25 is 33, which corresponds to the class interval \( 40 - 55 \). This is the median class.

Step 3: Apply the median formula

The median class is \( 40 - 55 \), so:

• \( L = 40 \) (the lower limit of the median class),
• \( F = 13 \) (the cumulative frequency of the class before the median class),
• \( f = 20 \) (the frequency of the median class),
• \( h = 15 \) (the class width, which is \( 55 - 40 \)).

Substitute these values into the median formula:

\[ \text{Median} = 40 + \left( \frac{25 - 13}{20} \right) \times 15 = 40 + \left( \frac{12}{20} \right) \times 15 = 40 + \left( 0.6 \right) \times 15 = 40 + 9 = 49. \]

Conclusion:

The median is 49.