In statistics, what does the standard deviation of a data set measure?

• A. The most frequently occurring value in the data set (the mode)
• B. The typical distance of data values from the mean (a measure of spread)
• C. The central value of the data set when ordered (the median)
• D. The total of all data values divided by the number of values (the mean)

Answer: (B)

Verified step by step guidance

1. Understand that the standard deviation is a measure of spread or variability in a data set, not a measure of central tendency like the mean or median.
2. Recall that the mean (average) of the data set is calculated by summing all data values and dividing by the number of values: \(\text{mean} = \frac{\sum x_i}{n}\).
3. Recognize that the standard deviation quantifies how much the data values typically deviate from the mean, giving insight into the dispersion of the data.
4. The formula for the standard deviation (population) is: \(\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}\), where \(x_i\) are data points, \(\mu\) is the mean, and \(n\) is the number of data points.
5. Thus, the standard deviation measures the typical distance of data values from the mean, indicating the spread of the data around the central value.