Question 14·Medium·One-Variable Data Distributions; Measures of Center and Spread
The results of two independent surveys are shown in the table below.
Men's Height
| Group | Sample size | Mean (centimeters) | Standard deviation (centimeters) |
|---|---|---|---|
| A | 2,500 | 186 | 12.5 |
| B | 2,500 | 186 | 19.1 |
Which statement is true based on the table?
For questions about data sets described by summary statistics, first scan which statistics are the same and which differ between groups. Recall the roles of each measure—mean for center, median for middle (only if given), standard deviation for spread—and then eliminate choices that make claims about values you are not given (like maximums or medians). Finally, connect any differing statistic, especially standard deviation, directly to the wording of the remaining answer choices to select the one that is supported by the table.
Hints
Focus on which numbers actually differ
Look at the table and identify which column entries are the same for both groups and which ones are different.
Think about what each column represents
Ask yourself: Which column tells you about the average height, which would tell you about the middle height, and which tells you how spread out the heights are?
Check what information you do NOT have
Do you see any information about the tallest person in each group or the median height? Can you safely make claims about those if the table doesn’t show them?
Connect the differing statistic to the wording of the choices
The only value that differs between the groups measures how much the data vary around the mean. Look for the answer choice that talks about that idea.
Desmos Guide
Enter and compare the standard deviations
Type 12.5 on one line and 19.1 on another line in Desmos (or define them as A=12.5 and B=19.1). Visually or numerically compare which value is larger; the larger value corresponds to the group with the greater spread of heights.
Relate the larger value to the concept of spread
Remember that standard deviation measures how spread out data are around the mean. Once you see which group’s standard deviation is larger in Desmos, match that idea—greater spread of heights—to the appropriate answer choice in the question.
Step-by-step Explanation
Compare what is the same and what is different
Look at the table columns for Groups A and B:
- Sample size: both are 2,500.
- Mean: both are 186 centimeters.
- Standard deviation: Group A is 12.5, Group B is 19.1.
So the only difference shown between the two groups is the standard deviation.
Recall what each statistic tells you
- Mean is the average height.
- Median is the middle height when all heights are ordered (but the table does not give the median).
- Standard deviation measures how spread out the data are around the mean: a larger standard deviation means the data values vary more from the mean (greater spread).
- The tallest participant (maximum value) is not given; you cannot tell it from the mean and standard deviation alone.
Test each answer choice against the information
Go through each choice using these facts:
- Choice A (identical data sets): If the data sets were identical, all summary statistics would match. But the standard deviations are different (12.5 vs. 19.1), so the data sets cannot be identical.
- Choice B (tallest participant in Group B): The table does not show the maximum height for either group, so you cannot know which group had the tallest person.
- Choice D (larger median for Group B): The table does not give the median for either group. Even though the means are equal, that tells you nothing definite about the medians.
- Choice C (difference in spread): The only statistic that differs is the standard deviation, which directly measures spread of the data.
Use the standard deviations to find the true statement
Group A has standard deviation 12.5 and Group B has standard deviation 19.1, and 19.1 is larger than 12.5. Since standard deviation measures spread, Group B’s heights are more spread out around the mean than Group A’s heights.
Therefore, the true statement is: “The heights of the men in Group B had a larger spread than the heights of the men in Group A.”