Question

The mean of the following frequency distribution is 35. Find the values of $x$ and $y$, if the sum of frequencies is 25:

Hint:

For grouped data with symmetric class intervals, you can also use the Assumed Mean Method or Step-Deviation Method.
Using these shortcut methods keeps the numbers smaller and makes solving the equations much easier.

Answer 1

Step 1: Understanding the Question:
The topic is Statistics, specifically finding missing frequencies in a grouped frequency distribution given the Mean and the total frequency.
We are given a grouped table with two missing frequencies, denoted as $x$ and $y$.
We need to set up two linear equations to find their exact values.

Step 2: Key Formula or Approach:
The formula for the Mean ($\bar{x}$) of grouped data using the Direct Method is:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
where:

• $f_i$ represents the frequency of the $i$-th class.

• $x_i$ represents the class mark (midpoint) of the $i$-th class, calculated as:
\[ x_i = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]

We will form our first equation from the sum of frequencies, and our second equation from the mean formula.

Step 3: Detailed Explanation:

• Find the class marks ($x_i$) and construct the calculation table:

• Form the first equation using the given total frequency $\sum f_i = 25$:
\[ 17 + x + y = 25 \]
\[ x + y = 8 \implies y = 8 - x \quad \text{--- (Equation 1)} \]

• Form the second equation using the given Mean $\bar{x} = 35$:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
\[ 35 = \frac{605 + 15x + 45y}{25} \]
Multiply both sides by 25:
\[ 35 \times 25 = 605 + 15x + 45y \]
\[ 875 = 605 + 15x + 45y \]
Subtract 605 from both sides:
\[ 15x + 45y = 875 - 605 \]
\[ 15x + 45y = 270 \]
Divide the entire equation by 15 to simplify:
\[ x + 3y = 18 \quad \text{--- (Equation 2)} \]

• Substitute Equation 1 into Equation 2:
\[ x + 3(8 - x) = 18 \]
\[ x + 24 - 3x = 18 \]
\[ -2x + 24 = 18 \]
\[ -2x = 18 - 24 \]
\[ -2x = -6 \implies x = 3 \]

• Calculate $y$ using Equation 1:
\[ y = 8 - x = 8 - 3 = 5 \]


Step 4: Final Answer:
The values of the missing frequencies are $x = 3$ and $y = 5$.