For 10 observations $x_1, x_2, \dots, x_{10}$, if $\sum_{i=1}^{10} (x_i + 2)^2 = 180$ and $\sum_{i=1}^{10} (x_i - 1)^2 = 90$, then their standard deviation is:

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Question

For 10 observations $x_1, x_2, \dots, x_{10}$, if $\sum_{i=1}^{10} (x_i + 2)^2 = 180$ and $\sum_{i=1}^{10} (x_i - 1)^2 = 90$, then their standard deviation is:

$\sqrt{3}$

$2\sqrt{2}$

The correct answer is

Standard Deviation Calculation Steps

We are given information about 10 observations ($n=10$) and two summation formulas:

$\sum_{i=1}^{10} (x_i + 2)^2 = 180$
$\sum_{i=1}^{10} (x_i - 1)^2 = 90$

We need to find the standard deviation ($\sigma$). First, let's expand the given equations.

Expanding the Summation Formulas

Let $S_1 = \sum_{i=1}^{10} x_i$ and $S_2 = \sum_{i=1}^{10} x_i^2$. Using the algebraic identity $(a \pm b)^2 = a^2 \pm 2ab + b^2$, we expand the terms:

First Equation Expansion: $ \sum_{i=1}^{10} (x_i + 2)^2 = \sum_{i=1}^{10} (x_i^2 + 4x_i + 4) $ $ = \sum_{i=1}^{10} x_i^2 + 4 \sum_{i=1}^{10} x_i + \sum_{i=1}^{10} 4 $ $ = S_2 + 4S_1 + (10 \times 4) = 180 $ $ \implies S_2 + 4S_1 + 40 = 180 $ $ \implies S_2 + 4S_1 = 140 $ (Equation A)
Second Equation Expansion: $ \sum_{i=1}^{10} (x_i - 1)^2 = \sum_{i=1}^{10} (x_i^2 - 2x_i + 1) $ $ = \sum_{i=1}^{10} x_i^2 - 2 \sum_{i=1}^{10} x_i + \sum_{i=1}^{10} 1 $ $ = S_2 - 2S_1 + (10 \times 1) = 90 $ $ \implies S_2 - 2S_1 + 10 = 90 $ $ \implies S_2 - 2S_1 = 80 $ (Equation B)

Solving for Sums ($S_1$ and $S_2$)

Now we have a system of two linear equations with two variables ($S_1$ and $S_2$):

Equation A: $S_2 + 4S_1 = 140$
Equation B: $S_2 - 2S_1 = 80$

Subtract Equation B from Equation A:

$ (S_2 + 4S_1) - (S_2 - 2S_1) = 140 - 80 $ $ 6S_1 = 60 $ $ S_1 = \frac{60}{6} = 10 $

Substitute $S_1 = 10$ into Equation B:

$ S_2 - 2(10) = 80 $ $ S_2 - 20 = 80 $ $ S_2 = 100 $

Calculating Mean and Variance

The mean ($\bar{x}$) of the observations is:

$ \bar{x} = \frac{S_1}{n} = \frac{10}{10} = 1 $

The variance ($\sigma^2$) is calculated using the formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$:

$ \sigma^2 = \frac{S_2}{n} - (\bar{x})^2 $ $ \sigma^2 = \frac{100}{10} - (1)^2 $ $ \sigma^2 = 10 - 1 = 9 $

Determining Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance:

$ \sigma = \sqrt{\sigma^2} = \sqrt{9} $ $ \sigma = 3 $

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