Question

The variance of the following frequency distribution is

• A. \(264\)
• B. \(88\)
• C. \(84\)
• D. \(90\)

Hint:

For grouped frequency distributions, always prepare columns for \(f\), \(fx\), and \(fx^2\). This minimizes calculation mistakes while finding variance.

Answer 1

The Correct Option is A

Step 1: Recall the variance formula.
For grouped data, the variance is \[ \sigma^2=\frac{\sum f_i x_i^2}{N}-\left(\frac{\sum f_i x_i}{N}\right)^2, \] where $x_i$ are the class marks and $N$ is the total frequency.

Step 2: List the class marks.
The class marks (midpoints) are $5,15,25,35,45,55$ with frequencies $2,2,3,4,1,3$.

Step 3: Find the total frequency.
$N=2+2+3+4+1+3=15$.

Step 4: Find $\sum f x$.
$\sum fx=2(5)+2(15)+3(25)+4(35)+1(45)+3(55)=10+30+75+140+45+165=465$.

Step 5: Find $\sum f x^2$.
$\sum fx^2=2(25)+2(225)+3(625)+4(1225)+1(2025)+3(3025)=50+450+1875+4900+2025+9075=18375$.

Step 6: Plug into the formula.
The mean is $\dfrac{465}{15}=31$. So \[ \sigma^2=\frac{18375}{15}-(31)^2=1225-961=264. \]

Step 7: State the answer.
The variance of the distribution is $264$. \[ \boxed{264} \]