If the median of the distribution given below is 28.5, find the values of and .
| Class interval | Frequency |
|---|---|
| 0 - 10 | 5 |
| 10 - 20 | |
| 20 - 30 | 20 |
| 30 - 40 | 15 |
| 40 - 50 | |
| 50 - 60 | 5 |
Total | 60 |
x = 8, y = 7
Explanation
This question requires finding unknown frequencies using the median formula. First, we establish a relationship between x and y using the total frequency. Then, we identify the median class based on n/2 and apply the median formula to find x. Finally, we substitute x to find y.
Solution Steps
Step 1: Calculate cumulative frequencies and form first equation.
| Class | Frequency | Cumulative Frequency | | 0-10 | 5 | 5 | | 10-20 | x | 5 + x | | 20-30 | 20 | 25 + x | | 30-40 | 15 | 40 + x | | 40-50 | y | 40 + x + y | | 50-60 | 5 | 45 + x + y |
Given: Total = 60 So, 45 + x + y = 60 x + y = 15 ... (1)
Step 2: Identify the median class.
n = 60, so n/2 = 30 Median = 28.5 lies in class 20-30 This is the median class.
Parameters: l = 20, f = 20, cf = 5 + x, h = 10
Step 3: Apply median formula.
Median = l + [(n/2 - cf)/f] × h
28.5 = 20 + [(30 - (5 + x))/20] × 10
28.5 = 20 + [(25 - x)/20] × 10
28.5 = 20 + (25 - x)/2
8.5 = (25 - x)/2
17 = 25 - x
x = 8
Step 4: Find y using equation (1).
From x + y = 15: 8 + y = 15 y = 7