Question

The variance of the following grouped data is:

• A. \( 20 \)
• B. \( 10 \)
• C. \( 19 \)
• D. \( 35 \)

Hint:

For grouped data, the variance \( \sigma^2 \) can be calculated using the formula \( \sigma^2 = \frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2 \), where \( x_i \) is the midpoint of the i-th class, \( f_i \) is the frequency, and \( N = \sum f_i \) is the total frequency.

Answer 1

The Correct Option is C

Step 1: Calculate the midpoint (\(x_i\)) for each class interval.
4-8: \( x_1 = (4+8)/2 = 6 \)
8-12: \( x_2 = (8+12)/2 = 10 \)
12-16: \( x_3 = (12+16)/2 = 14 \)
16-20: \( x_4 = (16+20)/2 = 18 \)

Step 2: Calculate the necessary sums using a table. Let \( f_i \) be the frequency.

Step 3: Calculate the mean (\(\bar{x}\)). \[ \bar{x} = \frac{\sum f_i x_i}{N} = \frac{260}{20} = 13 \]

Step 4: Calculate the variance (\(\sigma^2\)). The formula for variance is \( \sigma^2 = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2 \). \[ \sigma^2 = \frac{3760}{20} - (13)^2 \] \[ \sigma^2 = 188 - 169 \] \[ \sigma^2 = 19 \]

Step 5: Compare the result with the given options. The calculated variance is 19, which matches option (C).