Question

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

• A. \( \frac{15}{2} \)
• B. \(6\)
• C. \( \frac{13}{2} \)
• D. \( \frac{5}{2} \)
• E. \( \frac{11}{2} \)

Hint:

For combined variance problems: First find combined mean, then use the combined variance formula. Never directly average variances.

Answer 1

The Correct Option is C

Concept: For two groups of observations, the combined variance is calculated using: \[ \sigma^2 = \frac{ n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2) }{ n_1 + n_2 } \] where \[ d_1 = \bar{x}_1 - \bar{x} \text{and} d_2 = \bar{x}_2 - \bar{x} \] and \[ \bar{x} \] is the combined mean.

Step 1: Write the given data. For first data set: \[ n_1 = 5, \bar{x}_1 = 2, \sigma_1^2 = 4 \] For second data set: \[ n_2 = 5, \bar{x}_2 = 4, \sigma_2^2 = 5 \]

Step 2: Find the combined mean. \[ \bar{x} = \frac{ n_1\bar{x}_1 + n_2\bar{x}_2 }{ n_1+n_2 } \] \[ = \frac{ 5(2)+5(4) }{ 10 } \] \[ = \frac{ 10+20 }{10} = 3 \] Thus, \[ \bar{x} = 3 \]

Step 3: Find deviations from combined mean. \[ d_1 = 2 - 3 = -1 \] \[ d_2 = 4 - 3 = 1 \] So, \[ d_1^2 = 1 \text{and} d_2^2 = 1 \]

Step 4: Apply combined variance formula. \[ \sigma^2 = \frac{ 5(4+1) + 5(5+1) }{ 10 } \] \[ = \frac{ 5(5)+5(6) }{ 10 } \] \[ = \frac{ 25+30 }{ 10 } \] \[ = \frac{55}{10} = \frac{11}{2} \] Thus, \[ \boxed{ \sigma^2 = \frac{11}{2} } \] Hence, correct option is: \[ \boxed{(E)\ \frac{11}{2}} \]