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: Solve the two equations by elimination instead of substitution.
Once the mean and total-frequency conditions are turned into two linear equations in $x$ and $y$, there is more than one way to solve them. Here we eliminate one variable directly by lining up the equations, instead of substituting one equation into the other.

Step 2: Form the same two equations from the given data.
Since the sum of all frequencies is 25 and the known frequencies add up to 17:
\[ x + y = 8 \quad \text{...(i)} \]
Since the mean is 35, working through $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ with the class marks 15 and 45 attached to the unknown frequencies $x$ and $y$ gives, after simplifying:
\[ x + 3y = 18 \quad \text{...(ii)} \]

Step 3: Eliminate $x$ by subtracting equation (i) from equation (ii).
Both equations already have $x$ with the same coefficient 1, so we can subtract them directly without multiplying either one first:
\[ (x + 3y) - (x + y) = 18 - 8 \]
\[ 2y = 10 \]

Step 4: Solve for $y$.
\[ y = \frac{10}{2} = 5 \]

Step 5: Substitute back into equation (i) to get $x$.
\[ x + 5 = 8 \implies x = 3 \]

Step 6: Final answer.
The values of the missing frequencies are $x = 3$ and $y = 5$.
\[ \boxed{x = 3, \ y = 5} \]