Question

If the median of the following data is 525 and the sum of frequencies is 100, then find values of \( x \) and \( y \): % Frequency Table \[ \begin{array}{|c|c|} \hline Class Interval & Frequency
\hline 0 - 100 & 2
100 - 200 & 5
200 - 300 & x
300 - 400 & 12
400 - 500 & 17
500 - 600 & 20
600 - 700 & y
700 - 800 & 9
800 - 900 & 7
900 - 1000 & 4
\hline \end{array} \]

Hint:

To solve for unknown frequencies, use the cumulative frequency table and the median formula, then solve for \( x \) and \( y \).

Answer 1

We are given that the sum of the frequencies is 100, and the median is 525. To find \( x \) and \( y \), we first calculate the cumulative frequency and use the median formula.

Step 1: Calculate the cumulative frequency.
The cumulative frequency is calculated by adding the frequencies successively: \[ \begin{array}{|c|c|c|} \hline
Class Interval &
Frequency &
Cumulative Frequency
\hline 0 - 100 & 2 & 2
100 - 200 & 5 & 7
200 - 300 & x & 7 + x
300 - 400 & 12 & 19 + x
400 - 500 & 17 & 36 + x
500 - 600 & 20 & 56 + x
600 - 700 & y & 56 + x + y
700 - 800 & 9 & 65 + x + y
800 - 900 & 7 & 72 + x + y
900 - 1000 & 4 & 76 + x + y
\hline \end{array} \]
Step 2: Use the median formula.
The formula for the median is: \[ \text{Median} = L + \frac{\frac{N}{2} - CF}{f} \times h \]
where:
- \( L \) is the lower limit of the median class,
- \( N \) is the total frequency (100 in this case),
- \( CF \) is the cumulative frequency of the class before the median class,
- \( f \) is the frequency of the median class,
- \( h \) is the class width (100 in this case).
We are given that the median is 525. Now, from the cumulative frequency table, the median class will be the class whose cumulative frequency is closest to \( \frac{N}{2} = 50 \). From the table, the class \( 400 - 500 \) has a cumulative frequency of \( 36 + x \). Setting this equal to 50: \[ 36 + x = 50 \quad \Rightarrow \quad x = 14 \]
Step 3: Solve for \( y \).
Now that we know \( x = 14 \), the cumulative frequency of the \( 600 - 700 \) class is \( 56 + 14 + y = 70 + y \). To ensure the total frequency is 100, we can use the equation: \[ 2 + 5 + 14 + 12 + 17 + 20 + y + 9 + 7 + 4 = 100 \] Simplifying: \[ 86 + y = 100 \quad \Rightarrow \quad y = 14 \]
Conclusion:
The values of \( x \) and \( y \) are \( \boxed{14} \) and \( \boxed{14} \), respectively.