Compare Quantitative Distributions and Relative Position — Worked Solutions

A is balanced around about 13.5 minutes and spans 10–17. B’s center is somewhat higher (median 15), with much broader 8–34-minute spread and a right tail. B’s main values lie 8–20, with a gap to the isolated 34; A has no comparable internal gap or isolated extreme.

B median 15 exceeds A 13.5 by 1.5 minutes; mean 16.5 exceeds 13.5 by 3. B IQR=8 exceeds A 4, and SD 8.124 exceeds 2.449. Boxplots show positions and outliers but do not identify all internal modes or gaps.

z=(x_i−μ)/σ with data score x_i, population mean μ=70, and population SD σ=8. Scores yield z=2,−1,0: two SDs above, one below, and at the mean. Points cancel in the ratio.

100 exceeds 60 absolutely. But A’s score has z=(60−50)/5=2 and B’s has z=(100−80)/20=1, so A stands farther above its own mean in SD units. The reference mean/spread differs.

C is relatively balanced around 7.5 minutes, spans 4–11, and has no isolated extreme. D has median 10, spans 3–35, and is right-tailed with a main group 3–15, a long empty gap after 15, and an isolated 35-minute value. Broad 10-minute bins merge distinct individual positions, so the dots reveal the exact gap and within-bin values more clearly.

C left leaves: stem 0 has 9 8 7 6 5 4; stem 1 has 1 0. D right leaves: 0|3 5 7 9, 1|1 3 5, 2 has none, 3|5. Left stem 1/leaf 1 means 11; right 1|3 means 13 minutes. It confirms C’s compact set and D’s larger range and isolated 35.

C mean/median=7.5, IQR=4, s=√6≈2.449. D mean=12.25, median=10, IQR=8, s=√100.5≈10.025 minutes. Mean difference is 4.75 minutes, median difference 2.5. D’s high tail/outlier pulls its nonresistant mean more strongly than its median.

First claim is false: D includes 3 minutes, below all C observations. Second is supported by IQR 8 vs 4, SD≈10.025 vs 2.449, and range 32 vs 7 minutes. These describe the observed office samples and do not prove the cause of their difference.

Use (x_i−μ)/σ. C12 gives 2 (two SD above), C4 gives −2 (two below), C8 gives 0 (at mean). D17 gives 1 (one above). x_i is individual minutes, μ the appropriate population mean, σ its population SD; minutes cancel.

C12 has z=2 and C10 z=1, so C12 is farther above C’s mean. D17 has z=1, so despite its larger raw time it is less high relative to its mean/SD than C12. Z-scores alone do not give percentile ranks without distribution-shape information.

Sample-based z=(11−7.5)/√6≈1.429, about 1.429 sample SDs above the sample mean. Sample summaries may be used when population parameters are unknown, but this is not a calculation using known full-population μ/σ and is not automatically a percentile.