Median Calculation for the Given Distribution

The median is the middle value in a dataset that separates the higher half from the lower half. For a grouped frequency distribution, we calculate the median using a specific formula after identifying the median class.

Distribution Data and Cumulative Frequency

First, let's organize the given data and calculate the cumulative frequency ($CF$) for each class interval.

Steps to Find the Median

1. Calculate Total Frequency ($N$):

   Sum all the frequencies to find the total number of observations.

   $ N = 3 + 13 + 10 + 3 + 1 = 30 $

2. Determine the Median Position:

   The position of the median value is found by $\frac{N}{2}$.

   $ \text{Median Position} = \frac{N}{2} = \frac{30}{2} = 15 $

   This means the median is the value of the 15th observation.

3. Identify the Median Class:

   Look at the cumulative frequency ($CF$) column. The median class is the class interval where the cumulative frequency first equals or exceeds the median position (15).

   • The $CF$ for the 120-170 class is 3 (less than 15).
   • The $CF$ for the 170-220 class is 16 (greater than or equal to 15).

   Therefore, the **median class** is 170-220.

4. Apply the Median Formula:

   The formula to calculate the median for grouped data is:

   $ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times w $

   Where:

   • $L$ = Lower limit of the median class
   • $N$ = Total frequency
   • $CF$ = Cumulative frequency of the class *preceding* the median class
   • $f$ = Frequency of the median class
   • $w$ = Width of the median class
5. Substitute Values and Calculate:

   From our data:

   • $L = 170$
   • $\frac{N}{2} = 15$
   • $CF = 3$ (the $CF$ of the class 120-170)
   • $f = 13$ (the frequency of the median class 170-220)
   • $w = 170 - 120 = 50$

   Now, substitute these values into the formula:

   $ \text{Median} = 170 + \left( \frac{15 - 3}{13} \right) \times 50 $

   $ \text{Median} = 170 + \left( \frac{12}{13} \right) \times 50 $

   $ \text{Median} = 170 + \frac{600}{13} $

   $ \text{Median} \approx 170 + 46.15 $

   $ \text{Median} \approx 216.15 $

The calculation shows the median value to be approximately 216.15 based on the standard formula and the provided distribution data.