Interpreting and Comparing One-Variable Data Sets

Median and IQR (or the five-number summary), because they are resistant to skew and outliers.

"Both classes had the same mean mark (72), but Class Y's marks were about twice as spread out (s = 13 vs 6), so Class X's marks were much more consistent."

The groups have different sizes; raw frequencies would make the bigger group look taller everywhere. Relative frequencies put both on the same scale.

The median and spread (wait times are usually skewed right), the sample sizes, any outliers, the time period, and whether the patients were similar (how serious their cases were).

Town C: mean ≈ median, roughly symmetric. Town D: mean much greater than median, skewed right (a few high incomes).

The sample is small and not random (friends are likely similar to each other), so it may not represent all teens; self-reported sleep may also be inaccurate.

Shape, Outliers, Centre, Spread.

The median.

The standard deviation.

Not reliably. Use the same scale.

Yes, always (cm, hours, dollars...).

P: IQR 13, fences −7.5 and 44.5: no outliers. Q: IQR 7, fences −2.5 and 25.5: 58 is an outlier (perhaps also other high values). Comparison: "Students at School Q typically have shorter commutes (median 11 vs 18 min) and their times are more consistent (IQR 7 vs 13 min), but School Q has at least one very long commute (58 min). School P's distribution is roughly symmetric (its median is near the middle of its box); School Q's long right whisker and outlier suggest right skew."

With a mean of 42 and s = 61, many values must be far above the mean while none can be below 0: a strong right skew (most people volunteer little, a few volunteer a lot). Report the median and IQR.

"On average, Brand Y batteries lasted slightly longer (11.4 h vs 11.0 h). However, Brand Y was far less consistent (s = 2.3 h vs 0.6 h), and one Brand Y battery lasted only 4.0 h, an outlier. Both distributions were otherwise roughly symmetric. For uses where reliability matters, Brand X is the better choice, although samples of 20 are small."

Incomes are skewed right, so a few very high incomes can raise one province's mean a lot without changing a typical household's income. The median describes a typical household better. Differences in household size and cost of living also matter.

1. Question and data ("we compared samples of 40 brook and 30 rainbow trout"); 2. show the side-by-side box plots; 3. key numbers: medians 30.2 vs 36.05 cm, IQRs 6.9 vs 9.7 cm, one rainbow outlier; 4. conclusion: rainbow trout were typically about 6 cm longer and more variable; 5. limitation: the samples may not represent the whole lake, and follow-up: compare fish of known ages.

Summary statistics compress the data into a few numbers. Two sets can match on those numbers but differ in shape: one could be bimodal with clusters at two values, another a single mound with long tails. A histogram or dot plot reveals the difference. (Statisticians have famous examples of very different data sets with identical summary statistics.)

Both can be accurate. If a number of new, very expensive units were added (or the most expensive units rose sharply), the long right tail would pull the mean up a lot while the median barely moved. The median better describes a typical tenant; the mean reflects the high-end changes. A full report would give both, plus the number of units in each year.

1. The data shows an association, not a cause: students were not randomly assigned to classes, so the classes may differ in prior achievement or other factors. 2. The difference in medians (4 marks) is small compared with the spreads (IQRs 10 and 22), so the classes overlap a lot. 3. One test is a small, single measure of learning; it could reflect timing, absences or the test itself. (Causation versus association is studied in the unit Two-Variable Data Analysis and Validity.)