Question

The standard deviation of the numbers \( -3, 0, 3, 8 \) is

• A. \( \frac{\sqrt{60}}{2} \)
• B. \( \frac{\sqrt{62}}{2} \)
• C. \( \frac{\sqrt{65}}{2} \)
• D. \( \frac{\sqrt{66}}{2} \)
• E. \( \frac{\sqrt{67}}{2} \)

Hint:

Always compute mean first, then deviations to find variance.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
Step 2: Key Formula or Approach:
1. Calculate the mean (\(\mu\)) of the data set. 2. Calculate the variance (\(\sigma^2\)), which is the average of the squared differences from the Mean. The formula for the variance of a population is \(\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}\). 3. The standard deviation (\(\sigma\)) is the square root of the variance.
Step 3: Detailed Explanation:
The data set is \(\{-3, 0, 3, 8\}\). There are N=4 data points. 1. Calculate the mean (\(\mu\)). \[ \mu = \frac{-3 + 0 + 3 + 8}{4} = \frac{8}{4} = 2 \] 2. Calculate the variance (\(\sigma^2\)). We find the squared difference of each data point from the mean:
\( (-3 - 2)^2 = (-5)^2 = 25 \)
\( (0 - 2)^2 = (-2)^2 = 4 \)
\( (3 - 2)^2 = (1)^2 = 1 \)
\( (8 - 2)^2 = (6)^2 = 36 \)
The sum of these squared differences is: \[ \sum(x_i - \mu)^2 = 25 + 4 + 1 + 36 = 66 \] The variance is the sum divided by the number of data points: \[ \sigma^2 = \frac{66}{4} \] 3. Calculate the standard deviation (\(\sigma\)). The standard deviation is the square root of the variance: \[ \sigma = \sqrt{\frac{66}{4}} = \frac{\sqrt{66}}{\sqrt{4}} = \frac{\sqrt{66}}{2} \] Step 4: Final Answer:
The standard deviation is \(\frac{\sqrt{66}}{2}\).