Multiple Choice
Suppose you are shown four histograms, each representing a different data set with the same mean . Which data set is most likely to have the smallest standard deviation ?
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3. Describing Data Numerically
Standard Deviation
Multiple Choice
Regarding measures of variability, which of the following statistics can take on negative values?
A
Interquartile range
B
Variance
C
Sample mean deviation (, sum of deviations from the mean, not absolute value)
D
Standard deviation
Verified step by step guidance1
Understand that measures of variability describe how spread out the data values are in a dataset.
Recall that the interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1), so it is calculated as \(\text{IQR} = Q_3 - Q_1\), which cannot be negative because \(Q_3 \geq Q_1\).
Remember that variance is calculated as the average of the squared deviations from the mean: \(\text{Variance} = \frac{1}{n-1} \sum (x_i - \bar{x})^2\), and since squares are always non-negative, variance cannot be negative.
Note that standard deviation is the square root of variance: \(\text{Standard Deviation} = \sqrt{\text{Variance}}\), so it also cannot be negative.
Recognize that the sample mean deviation, defined as the sum of deviations from the mean without taking absolute values, is \(\sum (x_i - \bar{x})\), which can be negative because deviations can be positive or negative and they are not squared or made absolute.
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