Given the data set $1,2,4,5$, what is the variance of the data?

• A. $3.5$
• B. $1.5$
• C. $2.9167$
• D. $2.5$

Answer: (C)

Verified step by step guidance

1. First, calculate the mean (average) of the data set. The mean \( \bar{x} \) is given by the formula: \[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \] where \( x_i \) are the data points and \( n \) is the number of data points.
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, sum all the squared differences obtained in the previous step: \[ \sum_{i=1}^n (x_i - \bar{x})^2 \]
4. Since this is a sample variance calculation, divide the sum of squared differences by \( n - 1 \) (where \( n \) is the number of data points) to get the sample variance \( s^2 \): \[ s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1} \]
5. Finally, interpret the result as the variance of the data set, which measures the average squared deviation from the mean.