The director of the City Transportation System is interested in the amount of time required for a bus to make the trip from Downtown Station to City Mall. After collecting data for several months by recording the time it takes to make the trip, she finds that the distribution of times has a standard deviation of 3 minutes. Which of the following is the best interpretation of the standard deviation?
(A) A bus that leaves from Downtown Station typically arrives at City Mall 3 minutes later than the scheduled time.
(B) A bus typically takes about 3 minutes to get from Downtown Station to City Mall.
(C) The time a bus takes to get from Downtown Station to City Mall never varies more than 3 minutes from the mean trip time.
(D) The difference between the actual time a bus takes to get from Downtown Station to City Mall and the mean trip time is, on average, about 3 minutes.
Standards
6.SP.3: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.5c: Summarize numerical data sets in relation to their context, such as by: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
S-ID: Summarize, represent, and interpret data on a single count or measurement variable
Correct answer and commentary
Student performance
The correct answer for this question is Option (D). Measures of variability, including standard deviation, are commonly used for comparing datasets. As students become familiar with their properties, making claims about which dataset has more (or less) variability than another should become a routine exercise. In addition to their use for comparison, however, students should also be able to interpret the meaning of measures of variability in context.
Of the four options presented, (D) is the only correct interpretation for standard deviation. This interpretation can be motivated by the mean absolute deviation (MAD), a conceptual precursor to standard deviation that is introduced to middle school students under the CCSSM.
Option (C) is a common misconception about the interpretation of the standard deviation. This option is incorrect because it prescribes absolute bounds on the variability, e.g. “never varies by more than 3 minutes.” The standard deviation is a statement about average variability from the mean and not maximum variability from the mean. Options (A) and (B) both make statements regarding the typical time of the bus trip, which would be measures of center, and are not interpretations for the standard deviation, which is a measure of spread.