Question

Let \( a_1, a_2, \ldots, a_{10} \) be 10 observations such that   \(\sum_{k=1}^{10} a_k = 50 \quad \text{and} \quad \sum_{1 \leq k < j \leq 10} a_k \cdot a_j = 1100\). Then the standard deviation of \( a_1, a_2, \ldots, a_{10} \) is equal to:

• A. \( 5 \)
• B. \( \sqrt{5} \)
• C. \( 10 \)
• D. \( \sqrt{15} \)

Answer 1

The Correct Option is B

To determine the standard deviation for the observations \( a_1, a_2, \ldots, a_{10} \), the following information is provided:

• \(\sum_{k=1}^{10} a_k = 50\)
• \(\sum_{1 \leq k < j \leq 10} a_k \cdot a_j = 1100\)

These conditions will be utilized to compute the standard deviation.

1. The variance formula for a dataset is expressed as: \(\sigma^2 = \frac{1}{n} \left( \sum_{k=1}^{n} a_k^2 - \frac{1}{n} \left( \sum_{k=1}^{n} a_k \right)^2 \right)\), where \( n = 10 \).
2. The term \(\sum_{k=1}^{10} a_k^2\) must be calculated. The following relationship is known: \((\sum_{k=1}^{10} a_k )^2 = \sum_{k=1}^{10} a_k^2 + 2 \sum_{1 \leq k < j \leq 10} a_k \cdot a_j\).
3. Substituting the given values yields: \(50^2 = \sum_{k=1}^{10} a_k^2 + 2 \cdot 1100\).
4. This simplifies to: \(2500 = \sum_{k=1}^{10} a_k^2 + 2200\).
5. Therefore, \(\sum_{k=1}^{10} a_k^2 = 2500 - 2200 = 300\).
6. The variance formula is then populated: \(\sigma^2 = \frac{1}{10} \left( 300 - \frac{1}{10} \times 2500 \right) = \frac{1}{10} (300 - 250) = \frac{1}{10} \times 50 = 5\).
7. The standard deviation is the square root of the variance: \(\sigma = \sqrt{5}\).

Consequently, the standard deviation for the observations \( a_1, a_2, \ldots, a_{10} \) is \(\sqrt{5}\).