Question

Arrange the following in increasing order on the basis of dispersion of data with given mean, number of data points and sum of squares:
A. \(n=5,\ \sum x^2=100,\ \bar{x}=3\);
B. \(n=5,\ \sum x^2=100,\ \bar{x}=4\);
C. \(n=10,\ \sum x^2=100,\ \bar{x}=2\);
D. \(n=10,\ \sum x^2=100,\ \bar{x}=3\).

• A. \(A,B,C,D\)
• B. \(A,B,D,C\)
• C. \(B,A,D,C\)
• D. \(D,B,C,A\)

Hint:

When \(\sum x^2\), \(n\), and \(\bar{x}\) are given, use \(\sigma^2=\frac{\sum x^2}{n}-\bar{x}^2\).

Answer 1

The Correct Option is D

Concept:
Variance is: \[ \sigma^2=\frac{\sum x^2}{n}-\bar{x}^2 \]

Step 1: For A. \[ \sigma_A^2=\frac{100}{5}-3^2=20-9=11 \]

Step 2: For B. \[ \sigma_B^2=\frac{100}{5}-4^2=20-16=4 \]

Step 3: For C. \[ \sigma_C^2=\frac{100}{10}-2^2=10-4=6 \]

Step 4: For D. \[ \sigma_D^2=\frac{100}{10}-3^2=10-9=1 \]

Step 5: Arrange in increasing order. \[ D=1,\quad B=4,\quad C=6,\quad A=11 \] So: \[ D,B,C,A \] \[ \therefore \text{Correct Answer is (D)} \]