Comparing Center and Spread in High School Statistics
Ms. Chen teaches high school statistics to 26 students. Students are comparing two machines that fill 12-ounce bottles. Machine A produced: 12.0, 12.1, 12.1, 12.2, 12.2, 12.3 oz. Machine B produced: 11.5, 11.9, 12.0, 12.2, 12.5, 12.7 oz. Both have means near 12.15 oz, and most students conclude the machines are equally reliable. When asked which machine they would recommend, 18 of 26 choose either one, citing the mean. A few students say Machine A because its numbers “look closer together,” but cannot quantify the difference. Ms. Chen wants students to use measures of spread, not just center, and to connect variability to a real decision about consistency. The class has dot plots, calculators, and a unit test on summarizing data in two weeks. She has one 50-minute period for the initial investigation.
1. Explain why comparing means alone is insufficient here and identify what the data show about consistency. Include a rough measure of spread.
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The means are nearly identical: Machine A has mean 12.15 oz, and Machine B has mean about 12.13 oz. If students stop there, the machines look equivalent. But the decision is about consistency, and the means conceal very different variability.
Machine A's values range from 12.0 to 12.3, a range of 0.3 oz. Machine B's values range from 11.5 to 12.7, a range of 1.2 oz. A rough mean absolute deviation makes the difference clear. For Machine A, deviations from 12.15 are about 0.15, 0.05, 0.05, 0.05, 0.05, and 0.15, averaging about 0.08 oz. For Machine B, deviations are about 0.65, 0.25, 0.15, 0.05, 0.35, and 0.55, averaging about 0.33 oz. Machine B varies roughly four times as much.
For a bottling company, consistency matters: underfilled bottles violate label claims, and overfilled bottles waste product. A slightly different mean is less important than the risk of extreme fills. The data support Machine A as the more reliable choice, not because its mean is better, but because its spread is smaller.
2. Design a dot-plot investigation using the two data sets. What questions would you ask to connect center, spread, and the decision about consistency?
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Have students construct dot plots for both machines on the same horizontal scale from 11.4 to 12.8 oz. The shared scale is essential; separate scales can make the distributions look similar. Students mark the mean and median on each plot and describe the shape.
Questions should move from noticing to quantifying to deciding:
- “Where is the center of each distribution, and how similar are the centers?”
- “Which dot plot is more spread out? How can you tell without calculating?”
- “What is the range of each? What is the interquartile range?”
- “If the company rejects bottles below 11.8 oz, which machine is more likely to produce rejects? Above 12.4 oz?”
- “Would you rather have a machine that is usually close to 12.15 oz or one that averages 12.15 oz but sometimes fills 11.5 oz? Why?”
The discussion should surface that center and spread answer different questions. The mean answers “What is typical?” but the spread answers “How predictable is it?” The recommendation depends on the cost of variation, which is a statistical and business judgment, not a calculation alone.
3. Write an assessment item involving a third machine with an outlier. Describe the expected reasoning and how you would evaluate students who ignore the outlier.
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Item: “Machine C produced 11.0, 12.1, 12.1, 12.2, 12.2, and 12.3 oz. Calculate the mean and median. Which measure better represents a typical fill? Does the outlier matter for the company's decision? Explain.”
Expected reasoning: the mean is about 11.98 oz, while the median is 12.15 oz. The single low value pulls the mean down, so the median better represents a typical fill. However, the outlier should not simply be discarded: for quality control, a 11.0-oz bottle is a serious underfill, so the company should investigate the cause and consider both typical performance and worst-case risk.
Students who ignore the outlier and report only the mean have not addressed variability. A strong response distinguishes “typical” from “acceptable” and explains that the outlier affects the mean more than the median. Evaluate whether students choose a measure appropriate to the question, justify the choice, and recognize that a resistant measure does not make the outlier irrelevant to the decision.