Question

The median of the following data is 32.5, find the missing frequencies \(x\) and \(y\) :

Hint:

When dealing with missing frequencies, always form two equations: one from the total sum and one from the Median/Mean/Mode formula given. This ensures you can solve for both variables.

Answer 1

Step 1: Finding Total Frequency:
Given total frequency = 40

x + 5 + 9 + 12 + y + 3 + 2 = 40
x + y + 31 = 40
x + y = 9 …(1)

Step 2: Identifying the Median Class:
Total N = 40
N/2 = 20

Median = 32.5
So median class = 30 – 40

For this class:
l = 30
f = 12
h = 10

Cumulative frequency before median class:
cf = x + 5 + 9
cf = x + 14

Step 3: Applying Median Formula:
Median formula:
M = l + [(N/2 − cf) / f] × h

32.5 = 30 + [(20 − (x + 14)) / 12] × 10

32.5 − 30 = [(20 − x − 14) / 12] × 10
2.5 = [(6 − x) / 12] × 10

Multiply both sides by 12/10:
2.5 × 12/10 = 6 − x
3 = 6 − x
x = 3

Step 4: Finding y:
From equation (1):
x + y = 9
3 + y = 9
y = 6

Final Answer:
The missing frequencies are:
x = 3
y = 6