Question

The variance of the data \(x_1,x_2,...,x_{30}\) with \(\displaystyle\sum^{50}_{i=1}x_i=650\) and \(\displaystyle\sum^{50}_{i=1}x_i^2=10000\) is

• A. 30
• B. 40
• C. 39
• D. 41
• E. 31

Answer 1

Answer: (E)

We are given the following information:

• \( \sum_{i=1}^{50} x_i = 650 \)
• \( \sum_{i=1}^{50} x_i^2 = 10000 \)
• There are 50 values, so \( n = 50 \)

We can calculate the mean \( \bar{x} \) using the formula:

\[ \bar{x} = \frac{\sum_{i=1}^{50} x_i}{50} = \frac{650}{50} = 13 \]

Next, we use the formula for variance:

\[ \text{Variance} = \frac{1}{n} \left( \sum_{i=1}^{n} x_i^2 - \frac{(\sum_{i=1}^{n} x_i)^2}{n} \right) \] Substitute the known values into the formula: \[ \text{Variance} = \frac{1}{50} \left( 10000 - \frac{650^2}{50} \right) \] First, calculate \( \frac{650^2}{50} \): \[ \frac{650^2}{50} = \frac{422500}{50} = 8450 \] Now, substitute this into the variance formula: \[ \text{Variance} = \frac{1}{50} \left( 10000 - 8450 \right) = \frac{1}{50} \times 1550 = 31 \]

Answer: 31