Question

The variance of the data \[ 4,\;7,\;8,\;10,\;13,\;16,\;19 \] is

• A. \(15\)
• B. \(24\)
• C. \(25\)
• D. \(28\) \bigskip

Hint:

For discrete data, \[ \sigma^2=\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2. \] This formula is usually faster than computing each deviation \((x-\bar{x})^2\) separately.

Answer 1

The Correct Option is B

Concept: Variance can be computed as \(\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}\). Find the mean, compute deviations, square them, and average.

Step 1: Calculate the mean. Data: 4,7,8,10,13,16,19. Sum = 77, \(n=7\), \(\bar{x}=11\).

Step 2: Compute squared deviations. \((4-11)^2=49\), \((7-11)^2=16\), \((8-11)^2=9\), \((10-11)^2=1\), \((13-11)^2=4\), \((16-11)^2=25\), \((19-11)^2=64\). Sum = 49+16+9+1+4+25+64 = 168.

Step 3: Divide by \(n\). \(\sigma^2 = \frac{168}{7} = 24\).

Step 4: Write the final answer. \(\boxed{24}\)