Here's a clear, educational, and well-structured solution to the problem:

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Key Concepts and Formulas

1. Standard Deviation (S.D.): The standard deviation ($\sigma$) measures the dispersion of a dataset around its mean. It is the square root of the variance. For a set of $n$ observations $x_1, x_2, \dots, x_n$, the variance ($\sigma^2$) is given by: $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$ where $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$ is the mean of the data. The standard deviation is $\sigma = \sqrt{\sigma^2}$.

2. Alternative Formula for Variance: This formula is often more convenient for calculations, especially when sums of $x_i$ and $x_i^2$ are known or easily derivable: $\sigma^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 - (\bar{x})^2$ This can be concisely written as $\sigma^2 = \overline{x^2} - (\bar{x})^2$, where $\overline{x^2}$ is the mean of the squares of the observations.

3. Property of Standard Deviation under Change of Origin (Translation): If a constant value '$A$' is added to or subtracted from each observation in a dataset, the standard deviation of the dataset remains unchanged. That is, if a new variable $y_i = x_i - A$ (or $y_i = x_i + A$) is defined, then the standard deviation of $y_i$ is the same as the standard deviation of $x_i$. $\text{S.D.}(x_i) = \text{S.D.}(y_i)$ This property is fundamental for simplifying this problem.

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Step-by-Step Solution

1. Understand the Goal and Analyze Given Information

• We are given information about a dataset of $n=9$ items, $x_1, x_2, \dots, x_9$.
• We have two sums:
  • $\sum\limits_{i = 1}^9 {\left( {{x_i} - 5} \right)} = 9$
  • $\sum\limits_{i = 1}^9 {{{\left( {{x_i} - 5} \right)}^2}} = 45$
• Our objective is to calculate the standard deviation of the original items $x_1, x_2, \dots, x_9$.

2. Simplify the Problem using a Change of Origin

• What we are doing: We introduce a new variable $y_i$ by transforming the original variable $x_i$.
• Why this step: Both given sums involve the term $(x_i - 5)$. This structure directly points to using the property of standard deviation under a change of origin. By defining $y_i = x_i - 5$, we simplify the expressions we need to work with, making calculations much more manageable. Moreover, the standard deviation of $y_i$ will be identical to that of $x_i$.
• Let $y_i = x_i - 5$.
• Now, we can rewrite the given sums in terms of $y_i$: $\sum\limits_{i = 1}^9 {y_i} = 9$ $\sum\limits_{i = 1}^9 {{y_i}^2} = 45$

3. Calculate the Mean of the Transformed Data ($y_i$)

• What we are doing: We calculate the arithmetic mean of the new dataset $y_i$.
• Why this step: The mean of $y_i$ ($\bar{y}$) is a necessary component for calculating the variance of $y_i$ using the alternative formula.
• The mean of $y_i$ is given by: $\bar{y} = \frac{\sum\limits_{i = 1}^n {y_i}}{n}$
• Substitute the values: $n=9$ and $\sum y_i = 9$: $\bar{y} = \frac{9}{9}$ $\bar{y} = 1$

4. Calculate the Variance of the Transformed Data ($y_i$)

• What we are doing: We calculate the variance of the transformed dataset $y_i$.
• Why this step: The standard deviation is the square root of the variance. We use the alternative formula for variance because we have readily available values for $\sum y_i^2$ and $\bar{y}$.
• Using the alternative formula for variance: $\sigma_y^2 = \frac{1}{n} \sum_{i=1}^n y_i^2 - (\bar{y})^2$
• Substitute the known values: $n=9$, $\sum y_i^2 = 45$, and $\bar{y} = 1$: $\sigma_y^2 = \frac{45}{9} - (1)^2$ $\sigma_y^2 = 5 - 1$ $\sigma_y^2 = 4$

5. Calculate the Standard Deviation of the Transformed Data ($y_i$)

• What we are doing: We find the standard deviation of $y_i$ by taking the square root of its variance.
• Why this step: This brings us closer to our final answer, as S.D.$_y$ is directly related to S.D.$_x$.
• The standard deviation of $y_i$ is: $\text{S.D.}_y = \sqrt{\sigma_y^2}$ $\text{S.D.}_y = \sqrt{4}$ $\text{S.D.}_y = 2$ (Note: Standard deviation is always non-negative).

6. Relate Back to the Original Data ($x_i$)

• What we are doing: We apply the property of standard deviation under a change of origin to find the standard deviation of $x_i$.
• Why this step: This is the final and crucial step that connects our calculations for the transformed data back to the original question about $x_i$. It avoids the need to calculate $\bar{x}$ and $\sum x_i^2$ directly, which would be more cumbersome.
• Since $y_i = x_i - 5$, and standard deviation is invariant under a change of origin (addition or subtraction of a constant), the standard deviation of $x_i$ is the same as the standard deviation of $y_i$. $\text{S.D.}_{x} = \text{S.D.}_{y}$
• Therefore, $\text{S.D.}_{x} = 2$

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Common Mistakes & Tips

• Misapplying Variance Formulas: Ensure you use the correct formula for variance. A common error is using $\sum x_i^2 - (\sum x_i)^2$ instead of $\frac{1}{n} \sum x_i^2 - (\frac{1}{n} \sum x_i)^2$.
• Forgetting the Square Root: Remember that variance is $\sigma^2$, and standard deviation is $\sigma = \sqrt{\sigma^2}$. Don't forget to take the square root at the end.
• Ignoring the Change of Origin Property: This property is a huge time-saver. Trying to calculate $\bar{x}$ and $\sum x_i^2$ directly from the given sums would be much more involved and prone to errors.
• Misinterpreting "n-1" vs "n": For JEE, unless specified as "sample standard deviation", always use 'n' in the denominator for variance and standard deviation calculations.

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Summary

This problem effectively tests the understanding of standard deviation calculation and, more importantly, the property of standard deviation remaining invariant under a change of origin. By introducing a transformed variable $y_i = x_i - 5$, we simplified the given sums to $\sum y_i = 9$ and $\sum y_i^2 = 45$. We then calculated the mean of $y_i$ as $\bar{y}=1$. Using the alternative variance formula, $\sigma_y^2 = \frac{1}{n} \sum y_i^2 - (\bar{y})^2$, we found $\sigma_y^2 = 4$. Taking the square root, S.D.$_y = 2$. Due to the change of origin property, S.D.$_x$ is equal to S.D.$_y$.

The final answer is $\boxed{2}$, which corresponds to option (D).