STATISTICS-Median Calculation-EXERSCISE-13.3-Q 2. If the median of the distribution given below is 28.5, find the values of x and y.AP/TS/NCERT

Median Calculation with Fractions

Problem Statement

The median of the following distribution is 28.5. Find the values of \( x \) and \( y \).

Class Interval	Frequency
0 - 10	5
10 - 20
20 - 30	20
30 - 40	15
40 - 50
50 - 60	5
Total	60

Step-by-Step Solution

1 We know that the total frequency is 60:

5 + x + 20 + 15 + y + 5 = 60

Simplifying:

x + y + 45 = 60 ⇒ x + y = 15

2 The median is 28.5, which lies in the class interval 20-30.

So, the median class is 20-30 with:

• Lower boundary (L) = 20
• Frequency (f) = 20
• Cumulative frequency before median class (cf) = 5 + x
• Class width (h) = 10
• Total observations (n) = 60

3 Using the median formula:

Median = L +

(n/2 - cf)

f

× h

Substituting the known values:

28.5 = 20 +

(30 - (5 + x))

20

× 10

28.5 = 20 +

(25 - x)

20

× 10

4 Solving the equation:

28.5 - 20 =

(25 - x)

20

× 10

8.5 =

(25 - x)

2

17 = 25 - x ⇒ x = 8

5 Now substitute x = 8 into the equation from step 1:

x + y = 15 ⇒ 8 + y = 15 ⇒ y = 7

6 Final Answer:

x = 8, y = 7