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: Understanding the Question:
This question is from "Statistics".
We are given a grouped frequency distribution with two missing frequencies, $x$ and $y$.
The total of all frequencies is $68$, and the median of the distribution is $137$.
We need to set up two simultaneous linear equations to solve for the values of $x$ and $y$.

Step 2: Key Formula or Approach:
1. Compute the cumulative frequency (CF) for each class interval.
2. Equate the sum of the frequencies to 68 to get the first linear equation.
3. Identify the median class (the class interval containing the median value of 137).
4. Use the Median Formula:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \] where:

• $L$ is the lower limit of the median class.

• $N$ is the total frequency ($68$).

• $CF$ is the cumulative frequency of the class preceding the median class.

• $f$ is the frequency of the median class.

• $h$ is the class size ($20$).

Step 3: Detailed Explanation:

• Build the Cumulative Frequency (CF) table:

• From the table, the total frequency is $47 + x + y$. We are given that the total is $68$:
\[ 47 + x + y = 68 \implies x + y = 21 \quad \text{--- (Equation 1)} \]

• The given Median is $137$. Since $137$ lies in the interval $125 - 145$, the

median class is $125 - 145$.
Identify the parameters for this class:
- $L = 125$
- $f = 20$
- $CF = 9 + x$ (from the preceding class)
- $h = 20$
- $N = 68 \implies \frac{N}{2} = 34$

• Apply the Median Formula:
\[ 137 = 125 + \left( \frac{34 - (9 + x)}{20} \right) \times 20 \]

• Simplify the equation by canceling out the common term 20:
\[ 137 - 125 = 34 - 9 - x \] \[ 12 = 25 - x \] \[ x = 25 - 12 = 13 \]

• Substitute $x = 13$ into Equation 1 to find $y$:
\[ 13 + y = 21 \implies y = 8 \]

Step 4: Final Answer:
The missing frequencies are $x = 13$ and $y = 8$.