Question

Consider the data $x_{1},x_{2},\dots,x_{10}$. If $\sum_{i=1}^{10}(|x_{i}-\overline{x}|)^{2}=662,$ where $\overline{x}$ is the mean, then the standard deviation is approximately, equal to

• A. 6.452
• B. 9.126
• C. 8.136
• D. 9.145
• E. 7.111

Hint:

Math Tip: If you don't have a calculator, use the estimation method! $\sqrt{64} = 8$. The difference is $2.2$. Using the linear approximation $\sqrt{x + dx} \approx \sqrt{x} + \frac{dx}{2\sqrt{x}}$, we get $8 + \frac{2.2}{16} = 8 + 0.1375 = 8.1375$. This perfectly aligns with option C!

Answer 1

The Correct Option is C

Concept:
Statistics - Variance and Standard Deviation.
The variance ($\sigma^2$) of a dataset is the average of the squared deviations from the mean: $$ \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \overline{x})^2}{N} $$ The standard deviation ($\sigma$) is the square root of the variance.
Step 1: Identify the given parameters.

• The sum of squared deviations: $\sum_{i=1}^{10} (x_i - \overline{x})^2 = 662$
• The number of observations: $N = 10$

Step 2: Calculate the variance ($\sigma^2$).
Substitute the values into the variance formula: $$ \sigma^2 = \frac{662}{10} $$ $$ \sigma^2 = 66.2 $$
Step 3: Calculate the standard deviation ($\sigma$).
Take the square root of the variance to find the standard deviation: $$ \sigma = \sqrt{66.2} $$
Step 4: Estimate the square root value.

• We know $8^2 = 64$ and $9^2 = 81$.
• Since $66.2$ is very close to $64$, the square root must be slightly greater than $8$.
• Checking the options, $8.136$ is the only logical fit.

$$ \sigma \approx 8.136 $$