Question

If the median of the frequency distribution given is 28.5, find the values of \( x \) and \( y \) while the total of frequencies is 60.

Hint:

To find the median of a frequency distribution, first identify the median class, then apply the median formula.

Answer 1

We are given the following frequency distribution: \[ \begin{array}{|c|c|} \hline \text{Class-interval} & \text{Frequency}
\hline 0-10 & 5
10-20 & x
20-30 & 20
30-40 & 15
40-50 & y
50-60 & 5
\hline \end{array} \] We are also told that the median of this distribution is 28.5 and the total frequency is 60. Step 1: Calculate total frequency The total frequency is 60, so \[ 5 + x + 20 + 15 + y + 5 = 60. \] Simplifying: \[ 45 + x + y = 60 \implies x + y = 15. \] Step 2: Find cumulative frequencies Now, calculate the cumulative frequencies: - For class \( 0-10 \), cumulative frequency = 5. - For class \( 10-20 \), cumulative frequency = \( 5 + x \). - For class \( 20-30 \), cumulative frequency = \( 5 + x + 20 = 25 + x \). - For class \( 30-40 \), cumulative frequency = \( 25 + x + 15 = 40 + x \). - For class \( 40-50 \), cumulative frequency = \( 40 + x + y \). - For class \( 50-60 \), cumulative frequency = \( 40 + x + y + 5 = 45 + x + y \). Step 3: Median class The median class corresponds to the class in which the cumulative frequency is at least half of the total frequency, i.e., \( \frac{60}{2} = 30 \). The cumulative frequency \( 40 + x \) corresponds to the class \( 20-30 \), so the median class is \( 20-30 \). Step 4: Median formula The median is given by the formula: \[ \text{Median} = L + \frac{\frac{N}{2} - CF}{f} \times h, \]
where:
- \( L \) is the lower boundary of the median class, i.e., 20. - \( N \) is the total frequency, i.e., 60. - \( CF \) is the cumulative frequency before the median class, i.e., \( 25 + x \). - \( f \) is the frequency of the median class, i.e., 20. - \( h \) is the class width, i.e., 10. Substitute the values into the formula: \[ 28.5 = 20 + \frac{30 - (25 + x)}{20} \times 10. \] Simplifying: \[ 28.5 = 20 + \frac{5 - x}{2}. \] Subtract 20 from both sides: \[ 8.5 = \frac{5 - x}{2}. \] Multiply both sides by 2: \[ 17 = 5 - x. \] Solve for \( x \): \[ x = -12. \] Step 5: Find \( y \) From \( x + y = 15 \), substitute \( x = -12 \): \[ -12 + y = 15 \implies y = 27. \]
Conclusion: The values of \( x \) and \( y \) are \( x = -12 \) and \( y = 27 \).