Question

If the median of the distribution given below is 28.5, find the values of x and y.

Class interval	Frequency
0 - 10	5
10 - 20	x
20 - 30	20
30 - 40	15
40 - 50	y
50 - 60	5
Total	60

Answer 1

The cumulative frequency for the given data is calculated as follows.  

Class interval	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
Total	60

From the table, it can be observed that n = 60 

45 + x + y = 60     or    x + y = 15 ……………………….(1)  
Median of the data is given as 28.5 which lies in interval 20 − 30.  

Therefore, median class = 20 − 30 
Lower limit (\(l\)) of median class = 20   
Cumulative frequency (\(cf\)) of class preceding the median class = 5 + x 
Frequency (\(f\)) of median class = 20
Class size (\(h\)) = 10  

Median = \(l + (\frac{\frac{n}2 - cf}{f})\times h\)

28.5 = \(20 + [\frac{\frac{60}2 - (5 +x)}{20}]\times 10\)

8.5 = \((\frac{25 - x}2)\)

17 = 25 - x
 x = 8

From equation (1), 
8 + y = 15 
y = 7 

Hence, the values of x and y are 8 and 7 respectively.

Answer 2

In this formula for the median:

\(\text{Median} = l + \left[ \frac{\frac{n}{2} - \text{cf}}{f} \right] \times h\)

Where:

• l is the lower limit of the median class
• n is the number of observations
• \(\text{cf}\) is the cumulative frequency of the class preceding the median class
• f is the frequency of the median class
• h is the class size or width of the median class

Given:

• \(n=60\) hence \(\frac{n}{2} = 30\)
• The median class is 20-30 with a cumulative frequency of \(25+x\)
• The lower limit of the median class is 20
• \(\text{cf} = 5 + x\)
• \(f = 20\)
• \(h=10\)

Using the median formula: \(28.5 = 20 + \left[ \frac{30 - (5 + x)}{20} \right] \times 10\)

Simplify inside the brackets: \(28.5 = 20 + \left[ \frac{30 - 5 - x}{20} \right] \times 10\)
\(28.5 = 20 + \left[ \frac{25 - x}{20} \right] \times 10\)
\(28.5 = 20 + \left( \frac{25 - x}{2} \right)\)

Subtract 20 from both sides: \(8.5 = \frac{25 - x}{2}\)

Multiply both sides by 2: \(17=25−x\)

Solve for x:
\( x=25−17\)
\( x=8\)

Now, using the cumulative frequency, find the value of \(x+y: \)
\(60=5+20+15+5+x+y\)
\( 60=45+x+y\)
\(60−45=x+y\)
\( 15=x+y\)

Substitute \( x=8:\)
\( y=15−x\)
\( y=15−8\)
 \(y=7\)

Thus, x=8 and y=7.