Key Concepts and Formulas

To find the standard deviation of a set of observations, we rely on a few fundamental statistical definitions and an important algebraic identity.

• Standard Deviation ($\sigma$): A measure of the dispersion or spread of a dataset around its mean. For a population of $N$ observations $a_1, a_2, \ldots, a_N$, the most commonly used formula for standard deviation in JEE is: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} a_i^2}{N} - \left(\frac{\sum_{i=1}^{N} a_i}{N}\right)^2}$ This can also be written using the mean ($\bar{a}$), where $\bar{a} = \frac{\sum_{i=1}^{N} a_i}{N}$: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} a_i^2}{N} - (\bar{a})^2}$
• Mean ($\bar{a}$): The average of the observations. It's calculated as the sum of all observations divided by the number of observations ($N$). $\bar{a} = \frac{\sum_{i=1}^{N} a_i}{N}$
• Algebraic Identity: To find the sum of squares ($\sum a_i^2$) from the sum of observations ($\sum a_i$) and the sum of pairwise products ($\sum_{k<j} a_k a_j$), we use the identity: $\left(\sum_{i=1}^{N} a_i\right)^2 = \sum_{i=1}^{N} a_i^2 + 2 \sum_{\forall k < j} a_k a_j$

Our goal is to calculate $\sigma$ using these formulas, which requires us to first determine the mean ($\bar{a}$) and the sum of squares ($\sum a_i^2$).

Step-by-Step Solution

We are given $N=10$ observations $a_1, a_2, \ldots, a_{10}$ with the following information:

1. Sum of observations: $\sum_{k=1}^{10} a_k = 50$
2. Sum of products of distinct pairs: $\sum_{\forall k < j} a_k \cdot a_j = 1100$

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Step 1: Calculate the Mean ($\bar{a}$) of the Observations

Why this step? The mean is a direct component of the standard deviation formula and is straightforward to calculate from the given sum of observations.

The mean is defined as the sum of observations divided by the number of observations. Given:

• Number of observations, $N = 10$.
• Sum of observations, $\sum_{k=1}^{10} a_k = 50$.

Now, we calculate the mean: $\bar{a} = \frac{\sum_{k=1}^{10} a_k}{N}$ $\bar{a} = \frac{50}{10}$ $\bar{a} = 5$ The mean of the given observations is 5.

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Step 2: Calculate the Sum of Squares ($\sum a_k^2$)

Why this step? The sum of squares ($\sum a_k^2$) is the other crucial component needed for the standard deviation formula. It is not directly given, but we can derive it using the algebraic identity that connects the sum of observations, the sum of squares, and the sum of pairwise products.

The key algebraic identity is: $\left(\sum_{i=1}^{N} a_i\right)^2 = \sum_{i=1}^{N} a_i^2 + 2 \sum_{\forall k < j} a_k a_j$ This identity comes from expanding the square of the sum of numbers.

Let's plug in the given values into this identity:

• We know $\sum_{k=1}^{10} a_k = 50$.
• We know $\sum_{\forall k < j} a_k a_j = 1100$.

Substitute these values: $(50)^2 = \sum_{k=1}^{10} a_k^2 + 2(1100)$ $2500 = \sum_{k=1}^{10} a_k^2 + 2200$ Now, isolate $\sum_{k=1}^{10} a_k^2$: $\sum_{k=1}^{10} a_k^2 = 2500 - 2200$ $\sum_{k=1}^{10} a_k^2 = 300$ The sum of the squares of the observations is 300.

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Step 3: Substitute Values into the Standard Deviation Formula

Why this step? Having calculated both the mean ($\bar{a}$) and the sum of squares ($\sum a_k^2$), we now have all the necessary components to directly apply the standard deviation formula.

Recall the standard deviation formula: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} a_i^2}{N} - (\bar{a})^2}$ Let's substitute the values we found:

• $N = 10$
• $\sum_{k=1}^{10} a_k^2 = 300$
• $\bar{a} = 5$

$\sigma = \sqrt{\frac{300}{10} - (5)^2}$ $\sigma = \sqrt{30 - 25}$ $\sigma = \sqrt{5}$

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Common Mistakes & Tips

• Confusing Sum of Squares and Square of Sums: A very common error is to mix up $(\sum a_k)^2$ (the square of the total sum) with $\sum a_k^2$ (the sum of individual squares). The algebraic identity clearly distinguishes them.
• Incorrect Algebraic Identity: Ensure you use the correct identity $(\sum a_i)^2 = \sum a_i^2 + 2 \sum_{k<j} a_k a_j$.
• Calculation Errors: Double-check arithmetic, especially squaring and subtraction steps.
• Formula for Standard Deviation: Remember the formula for standard deviation and its relation to variance ($\sigma^2 = \frac{\sum a_i^2}{N} - \bar{a}^2$).

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Summary

To find the standard deviation, we first calculated the mean using the sum of observations. Then, we used a crucial algebraic identity to find the sum of the squares of the observations from the given sum of observations and the sum of pairwise products. Finally, we substituted these values into the standard deviation formula to arrive at the result. The mean was found to be 5, the sum of squares 300, leading to a standard deviation of $\sqrt{5}$.

The final answer is $\boxed{\sqrt{5}}$, which corresponds to option (D).