Given the data set $2$, $4$, $4$, $4$, $5$, $5$, $7$, $9$, what is the variance and what is the standard deviation, rounded to the nearest whole number?

• A. Variance: $5$; Standard deviation: $2$
• B. Variance: $4$; Standard deviation: $2$
• C. Variance: $2$; Standard deviation: $4$
• D. Variance: $3$; Standard deviation: $3$

Answer: (B)

Verified step by step guidance

1. First, calculate the mean (average) of the data set. The mean \( \bar{x} \) is found by summing all the data points and dividing by the number of points: \( \bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} \).
2. Next, find the squared differences between each data point and the mean. For each data point \( x_i \), compute \( (x_i - \bar{x})^2 \).
3. Then, calculate the variance by taking the average of these squared differences. Since this is a sample, use the formula for sample variance: \( s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \), where \( n \) is the number of data points.
4. After finding the variance, compute the standard deviation by taking the square root of the variance: \( s = \sqrt{s^2} \).
5. Finally, round both the variance and the standard deviation to the nearest whole number as requested.