Let be 10 observations such that and Then the standard deviation of is equal to

Question

Let $a_{1}, a_{2}, \dots, a_{10}$ be 10 observations such that $\sum_{k = 1}^{10} a_{k} = 50$ and $\sum_{\forall k < j} a_{k} \cdot a_{j} = 1100 .$ Then the standard deviation of $a_{1}, a_{2}, \dots, a_{10}$ is equal to [2024]

Smart Achievers · Accepted Answer

(3)

Given, $\sum_{k = 1}^{10} a_{k} = 50$ and $\sum_{\forall k < j} a_{k} \cdot a_{j} = 1100$

$σ^{2} = \frac{\sum a_{k}^{2}}{10} -$ $({Mean)}^{2}$ $= \frac{\sum a_{k}^{2}}{10} - {(\frac{\sum a_{k}}{10})}^{2} = \frac{\sum a_{k}^{2}}{10} - {(\frac{50}{10})}^{2}$

$= \frac{\sum a_{k}^{2}}{10} - 25$ ...(i)

Also, ${(\sum a_{k})}^{2} = \sum a_{k}^{2} + 2 \sum_{\forall k < j} a_{k} a_{j}$

$\Rightarrow 2500 = \sum a_{k}^{2} + 2 (1100) \Rightarrow \sum a_{k}^{2} = 2500 - 2200 = 300$

From (i), we get

$σ^{2} = \frac{300}{10} - 25 = 30 - 25 = 5 \Rightarrow σ = \sqrt{5}$

Solution

Q.
Let $a_{1}, a_{2}, \dots, a_{10}$ be 10 observations such that $\sum_{k = 1}^{10} a_{k} = 50$ and $\sum_{\forall k < j} a_{k} \cdot a_{j} = 1100 .$ Then the standard deviation of $a_{1}, a_{2}, \dots, a_{10}$ is equal to [2024]

1 5

2 $\sqrt{115}$

3 $\sqrt{5}$

4 10

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Solution

Q. Let a1,a2, …, a10 be 10 observations such that ∑k=110ak=50 and ∑∀k<jak·aj=1100. Then the standard deviation of a1,a2, …, a10 is equal to [2024]

1 5

2 115

3 5

4 10

Ans. (3) Given, ∑k=110ak=50 and ∑∀k<jak·aj=1100 σ2=∑ak210-(Mean)2=∑ak210-(∑ak10)2=∑ak210-(5010)2 =∑ak210-25 ...(i) Also, (∑ak)2=∑ak2+2∑∀k<jakaj ⇒2500=∑ak2+2(1100)⇒∑ak2=2500-2200=300 From (i), we get σ2=30010-25=30-25=5⇒σ=5

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Q.
Let $a_{1}, a_{2}, \dots, a_{10}$ be 10 observations such that $\sum_{k = 1}^{10} a_{k} = 50$ and $\sum_{\forall k < j} a_{k} \cdot a_{j} = 1100 .$ Then the standard deviation of $a_{1}, a_{2}, \dots, a_{10}$ is equal to [2024]

2 $\sqrt{115}$

3 $\sqrt{5}$