Unit 1 · Topic 1.9 Beta

Comparisons of the Distributions for One Quantitative Variable

To compare distributions, describe each feature (shape, center, spread, outliers) with a comparing word and numbers, in context.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

Two bus routes get a commuter to work. Route 2 is usually a little faster, but now and then it takes 45 minutes or more. Route 1 is slower on a typical day but very steady. Which is better depends on what you need, and to decide, you have to compare whole distributions: shape, center, spread and outliers, not just one number.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which summary is the length of the box in a boxplot?

  1. The IQR
  2. The range
  3. The median
  4. The standard deviation
Show the answer

The box runs from Q1Q_1 to Q3Q_3.

  • Correct: The IQR:
  • The range:
  • The median:
  • The standard deviation:

2. What does a standard deviation of 4 points tell you about test scores?

  1. Scores typically differ from the mean by about 4 points
  2. The mean is 4
  3. Four students were absent
  4. The highest score is 4 above the median
Show the answer

The standard deviation is a typical distance from the mean.

  • Correct: Scores typically differ from the mean by about 4 points:
  • The mean is 4:
  • Four students were absent:
  • The highest score is 4 above the median:

Part 4 · See it

See it first

Top: parallel boxplots of delivery times in minutes for two restaurants, 15 orders each. Restaurant A has median 32 and a narrow box from 28 to 36. Restaurant B has a lower median, 27, a wider box from 23 to 37, a long right whisker and one high outlier at 60. Bottom: a z-score number line from minus 3 to plus 3 with the mean at 0; a value with z = 1.5 sits 1.5 standard deviations above the mean.
Top: parallel boxplots compare two groups on one scale. Bottom: a z-score measures distance from the mean in standard deviations. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Two groups are measured on the same variablegraph them on one scale, such as parallel boxplots or stacked dotplots
  2. Compare shape, center, spread and unusual featuresuse comparing words (greater, less variable) with numbers for each feature, in context
  3. Two values come from different distributionscompare their positions, not their raw sizes
  4. Subtract the mean and divide by the standard deviationthe z-score puts both on one scale: standard deviations above or below their own mean

Part 6 · Key ideas

Key ideas

  • Compare distributions on every feature: shape, center, spread and outliers, each with a comparing word and numbers, in context.
  • A z-score is z=x−x¯sz = \frac{x - \bar{x}}{s} (or x−μσ\frac{x - \mu}{\sigma} for a population): how many standard deviations a value is above (+) or below (−) the mean. It has no units.
  • Relative position compares values from different groups: z-scores and percentiles do this; raw values do not.
  • The median of combined groups is not the average of their medians.

Part 7 · Misconception

A common mistake

The wrong idea: Route 1’s median is 31 minutes and Route 2’s median is 28 minutes. (Given as a comparison.)

What actually happens: That lists numbers but compares nothing. Say "Route 1 has a greater median travel time (31 minutes) than Route 2 (28 minutes)."

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Travel times on two bus routes

A commuter timed 15 trips on each of two bus routes from her home to work, in minutes. Route 1: 24, 26, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 37, 39. Route 2: 18, 20, 22, 23, 25, 26, 27, 28, 30, 32, 35, 38, 41, 45, 62 (invented data).
10203040506070minutesRoute 1Route 2

1. Which statement compares the centers of the two distributions?

  1. The median travel time for Route 1 is 31 minutes, and the median for Route 2 is 28 minutes
  2. Route 1 has a greater median travel time (31 minutes) than Route 2 (28 minutes)
  3. Route 2 has the greater center, as its trips run up to 62 minutes
  4. The routes have about the same center, as both are about 30 minutes
Show the answer

Compare with a comparing word and numbers: Route 1’s median (31) is greater than Route 2’s (28).

  • The median travel time for Route 1 is 31 minutes, and the median for Route 2 is 28 minutes: Both numbers are right, but a comparison needs a comparing word: which median is greater?
  • Correct: Route 1 has a greater median travel time (31 minutes) than Route 2 (28 minutes): Right: it names both medians and says which is greater, in context.
  • Route 2 has the greater center, as its trips run up to 62 minutes: The longest trip is not the center. Route 2’s median, 28, is the smaller one.
  • The routes have about the same center, as both are about 30 minutes: The medians differ by 3 minutes, and the comparison should say which is greater.

2. Which statement compares the variability of the two routes?

  1. Route 1’s times vary more, since its median of 31 minutes is the greater of the two medians
  2. The routes vary equally, since each has 15 timed trips
  3. Route 2’s times vary less, since its box starts at a lower time
  4. Route 2’s times vary more: its IQR is 15 minutes, against 6 for Route 1
Show the answer

Compare spreads with a measure such as the IQR: 15 minutes for Route 2 against 6 for Route 1, so Route 2 is more variable.

  • Route 1’s times vary more, since its median of 31 minutes is the greater of the two medians: A larger center says nothing about spread.
  • The routes vary equally, since each has 15 timed trips: Equal sample sizes do not mean equal spread.
  • Route 2’s times vary less, since its box starts at a lower time: Where the box starts is position, not spread; Route 2’s box is wider.
  • Correct: Route 2’s times vary more: its IQR is 15 minutes, against 6 for Route 1: Right: Route 2’s box is much wider, 38−23=1538 - 23 = 15 against 34−28=634 - 28 = 6.

3. Which comparison is the most complete?

  1. Route 2 is skewed right with an outlier at 62 minutes, while Route 1 is roughly symmetric with none; Route 1 has the greater median (31 against 28) but much less variability (IQR 6 against 15 minutes)
  2. Route 1’s median is 31 and its IQR is 6; Route 2’s median is 28 and its IQR is 15, and Route 2 also has an outlier at 62 minutes
  3. Route 2 is better, because some of its trips are much faster, and Route 1 is worse, because its trips are slower on most days
  4. Both routes have trips from about 20 to about 40 minutes, so the two distributions are nearly the same
Show the answer

A complete comparison covers shape, center, spread and unusual features, each with a comparing word and numbers, in context.

  • Correct: Route 2 is skewed right with an outlier at 62 minutes, while Route 1 is roughly symmetric with none; Route 1 has the greater median (31 against 28) but much less variability (IQR 6 against 15 minutes): Right: shape, outliers, center and spread are all compared, with numbers and comparing words, in context.
  • Route 1’s median is 31 and its IQR is 6; Route 2’s median is 28 and its IQR is 15, and Route 2 also has an outlier at 62 minutes: The numbers are right, but nothing is compared: which is greater, which varies more?
  • Route 2 is better, because some of its trips are much faster, and Route 1 is worse, because its trips are slower on most days: "Better" and "worse" are judgments; the comparison needs shape, center and spread.
  • Both routes have trips from about 20 to about 40 minutes, so the two distributions are nearly the same: This ignores the outlier, the different medians and the very different spreads.

4. The commuter must reach work within 40 minutes every day. Which route should she choose, based on these trips?

  1. Route 2: its median of 28 minutes is lower than Route 1’s
  2. Route 2: its fastest trip, 18 minutes, beats any Route 1 trip
  3. Route 1: all 15 of its trips took 39 minutes or less
  4. Either route
Show the answer

The claim is about the longest trips, so look at the upper end: Route 1 never went over 39 minutes; Route 2 went over 40 three times.

  • Route 2: its median of 28 minutes is lower than Route 1’s: A lower typical time does not help if some trips are too long; 3 of Route 2’s trips took over 40 minutes.
  • Route 2: its fastest trip, 18 minutes, beats any Route 1 trip: Her need is to never be late, which depends on the longest trips.
  • Correct: Route 1: all 15 of its trips took 39 minutes or less: Right: Route 1’s maximum is 39. Route 2 took longer than 40 minutes on 3 of its 15 trips.
  • Either route: Medians under 40 still allow trips over 40; Route 2 had three.

Table

Scores in two chemistry classes

Two chemistry classes took different tests. Ana, in class A, scored 86. Ben, in class B, scored 88 (invented data).
Test scores by class
ClassMeanStandard deviation
A748
B814

5. Type the expression for the z-score of Ana’s score of 86 in class A.

Type the expression as on the exam, with the values substituted.

Show the answer

zz=x−x¯s= \frac{x - \bar{x}}{s}=86−748= \frac{86 - 74}{8}=1.5= 1.5.

  • Answer: 86−748\frac{86 - 74}{8}

6. Type the expression for the z-score of Ben’s score of 88 in class B.

Type the expression as on the exam, with the values substituted.

Show the answer

z=88−814=1.75z = \frac{88 - 81}{4} = 1.75.

  • Answer: 88−814\frac{88 - 81}{4}

7. Which student did better compared with their own class?

  1. Ana: her score is 12 points above her class mean, and Ben’s is 7
  2. Ben, because his score of 88 is higher than Ana’s 86
  3. Ben: his z-score of 1.75 is higher than Ana’s 1.5
  4. Neither
Show the answer

Standardize: Ana z=1.5z = 1.5, Ben z=1.75z = 1.75. Ben is further above his class in standard deviation units.

  • Ana: her score is 12 points above her class mean, and Ben’s is 7: Raw distances ignore spread. Class B’s scores vary less, so 7 points there is a bigger achievement.
  • Ben, because his score of 88 is higher than Ana’s 86: The classes took different tests, so raw scores cannot be compared.
  • Correct: Ben: his z-score of 1.75 is higher than Ana’s 1.5: Right: Ben’s score is farther above his class mean, measured in standard deviations.
  • Neither: z-scores are made exactly for this: they put both scores on the same scale.

8. Interpret Ana’s z-score of 1.5 in context.

Write the conclusion in context.

Show the answer

A z-score tells how many standard deviations a value is above (positive) or below (negative) the mean, in context.

  • A full-credit answer: Ana’s score of 86 is 1.5 standard deviations above the mean score of class A.

Part 9 · Summary

Summary

To compare distributions, describe each feature (shape, center, spread, outliers) with a comparing word and numbers, in context. To compare single values from different distributions, use relative position: the z-score z=x−x¯sz = \frac{x - \bar{x}}{s} counts standard deviations from the mean, and a percentile gives the percent of values at or below.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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