Interpreting a Class Data Set
A class measures the boiling point of water five times and records 98.0 °C, 98.2 °C, 97.9 °C, 98.1 °C, and 104.0 °C. The accepted value at their altitude is about 98 °C. One student wants to average all five values and report 99.24 °C. Another wants to throw out the 104.0 °C reading because 'it's wrong.' A third says they should report the median. The teacher asks the class to decide how to handle the outlier and what the spread of the other four values says about their technique.
1. Compare the mean of all five values with the mean of the four consistent values, and explain how the outlier affects each.
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All five: (98.0 + 98.2 + 97.9 + 98.1 + 104.0) ÷ 5 = 496.2 ÷ 5 = 99.24 °C.
Four consistent values: (98.0 + 98.2 + 97.9 + 98.1) ÷ 4 = 392.2 ÷ 4 = 98.05 °C.
Effect of the outlier: The single 104.0 °C reading pulls the mean up by about 1.2 °C, even though four of five readings cluster within 0.3 °C of 98 °C. The mean is sensitive to extreme values; the median (98.1 °C) is more resistant.
Reasoning: Reporting 99.24 °C misrepresents the class's typical measurement. The four consistent values agree closely with the accepted ~98 °C, which is meaningful evidence about their technique.
2. Should the class discard the 104.0 °C reading? Justify a defensible procedure.
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Do not discard it just because it is inconvenient. Removing data to get a preferred answer is a serious error.
Defensible procedure:
- Investigate first - check the lab notebook for a known cause (e.g., thermometer touching the beaker bottom, reading taken during a surge, misread scale). A documented procedural error justifies exclusion.
- If no cause is found, keep the value but flag it as an outlier and report both the full-data result and the robust statistic (median), noting the discrepancy.
- Never exclude a point solely because it disagrees with the others.
Reasoning: Outliers can be real signals (a flaw in the setup) or errors. The decision must rest on evidence about the cause, not on which answer is preferred.
3. The four consistent readings span only 0.3 °C. What does that tell you about precision, and what does it *not* tell you about accuracy?
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Precision: A 0.3 °C spread across four readings indicates good precision - the measurements are repeatable and close to one another.
Accuracy: Precision alone does not guarantee accuracy. If the thermometer were mis-calibrated (say, reading 1 °C high), all four values would cluster tightly but be wrong. Here the cluster near 98 °C matches the accepted altitude-adjusted value, which supports accuracy - but that conclusion depends on trusting the reference value, not on the tight spread itself.
Reasoning: Precision is about consistency among measurements; accuracy is about closeness to the true value. Tight clustering shows the first, not automatically the second. A systematic error could shift every reading while keeping them tightly grouped.