Question

Solve any one of the following internal choices (a) or (b):
32(a) The median of the following data is 137. Find the values of x and y, given that total of frequencies is 68.

Hint:

Always double-check that the calculated values of $x$ and $y$ are positive integers, as frequencies must always be whole numbers!
If you obtain a fraction or a negative number, re-check your calculations.

Answer 1

Step 1: Build the cumulative frequency table and set up the total-frequency equation, the same first step needed either way.
From the frequency distribution, the running total of frequencies before the last two classes is $47$, so the total frequency equation is:
\[ 47 + x + y = 68 \implies x + y = 21 \quad \text{(Equation 1)} \]

Step 2: Notice that in the median class, each unit of frequency corresponds to exactly one unit of value, and use that directly instead of the full median formula.
The median class is $125$-$145$ (since $137$ falls in it), with frequency $f = 20$ and class width $h = 20$. Because $f$ and $h$ are numerically equal here, every additional observation counted into this class moves the running position forward by exactly $1$ unit of value. This means we can match "how far the median value $137$ is above $L=125$" directly to "how many observations into the median class we must count" without going through the general formula.
\[ 137 - 125 = 12 \]
So we need $12$ observations counted into the median class to reach the median value of $137$.

Step 3: Use this to find the cumulative frequency just before the median class.
Half of the total frequency is $\frac{68}{2} = 34$. This 34th position is reached after covering the cumulative frequency before the median class, plus the $12$ observations found in Step 2:
\[ \text{CF before median class} + 12 = 34 \]
From the table, the cumulative frequency just before the median class is $9 + x$ (built up from the earlier classes). Substituting:
\[ (9 + x) + 12 = 34 \]
\[ x + 21 = 34 \]
\[ x = 13 \]

Step 4: Use Equation 1 to find y.
\[ 13 + y = 21 \implies y = 8 \]

Final Answer:
The missing frequencies are $x = 13$ and $y = 8$. \[ \boxed{x = 13,\ y = 8} \]