Math Core

Most interesting statistical questions are comparisons. Does a new fertilizer produce taller plants? Do students who sleep more get better grades? Are commute times longer on one route than another? This lesson shows how to display two or more distributions together and how to write a comparison that would earn full credit on the AP exam.

Displays for comparing groups

To compare distributions of a quantitative variable across groups, put the displays on the same scale so your eye can compare them directly.

Parallel box plots (side-by-side box plots) stack one box plot per group over a common number line. They are the quickest way to compare centers, spreads and outliers, even for many groups at once.
Back-to-back stemplots share one column of stems, with one group's leaves on the right and the other's on the left. They keep every data value and show shape well for small data sets.
Dot plots or histograms stacked one above the other on the same horizontal scale show shape in more detail. Use relative frequency histograms if the groups have different sizes.

Writing a comparison

Compare, don't just list

A complete comparison addresses shape, outliers, center and spread, in context, using comparative language: "greater than," "less than," "about the same as," "more variable than."

Listing two sets of numbers ("Group A's median is 42. Group B's median is 24.5.") is not a comparison. Say which is larger: "Group A's median is greater than Group B's."

Choose measures to match the shapes. If either distribution is skewed or has outliers, compare medians and IQRs. If both are roughly symmetric with no outliers, you may compare means and standard deviations. Use the same measures for both groups.

Worked example: Parallel box plots

A teacher timed students solving a logic puzzle. One group had practiced similar puzzles the day before; the other had not.

Time to solve a puzzle (seconds)

Compare the distributions of solving time for the two groups.

Shape. The practiced group's distribution is roughly symmetric. The no-practice group's distribution appears skewed right: its right whisker is longer and it has a high outlier.
Outliers. The no-practice group has one high outlier at about $118$ seconds. The practiced group has no outliers.
Center. The median time for the practiced group ( $50$ seconds) is less than the median for the no-practice group ( $63$ seconds).
Spread. The no-practice group's times are more variable: its IQR is $74 - 55 = 19$ seconds, compared with $56 - 45 = 11$ seconds for the practiced group. Its range is also larger.

Reading across the plots adds one more insight: $Q_3$ for the practiced group ( $56$ seconds) is less than the no-practice median ( $63$ seconds). So about $75\%$ of the practiced students finished faster than the median unpracticed student.

Back-to-back stemplots

Worked example: Exercise by age group

Twelve adults aged $20$ – $35$ (Group A) and twelve adults aged $55$ – $70$ (Group B) recorded their minutes of exercise on one day.

Group B leaves	Stem	Group A leaves
8	0
8 5 2	1
7 5 4 0	2	2 8
3 1	3	1 5 8
0	4	0 4 5 7
	5	2 8
2	6	5

Key: $2 \mid 3 \mid 1$ means $32$ minutes for Group B (left) and $31$ minutes for Group A (right).

Compare the two distributions.

Group B's leaves read outward from the stem, so its values are $8, 12, 15, 18, 20, 24, 25, 27, 31, 33, 40, 62$ . Group A's are $22, 28, 31, 35, 38, 40, 44, 45, 47, 52, 58, 65$ .

Center. The median for Group A is $\dfrac{40 + 44}{2} = 42$ minutes, greater than Group B's median of $\dfrac{24 + 25}{2} = 24.5$ minutes. The younger adults typically exercised about $17.5$ minutes longer.
Spread. The spreads are similar. Group A's IQR is $49.5 - 33 = 16.5$ minutes and Group B's is $32 - 16.5 = 15.5$ minutes.
Shape and outliers. Group A is roughly symmetric. Apart from one value, Group B is also fairly symmetric, but its $62$ minutes is an outlier: B's upper fence is $32 + 1.5(15.5) = 55.25$ .

Since Group B has an outlier, medians and IQRs are the right measures for this comparison.

Worked example: Is a difference meaningful?

Two classes took the same quiz. Class 1's median was $8$ points and Class 2's median was $7.5$ points, and each class's scores ranged over about $6$ points with similar IQRs of about $3$ points. A student claims Class 1 "did much better." Evaluate the claim.

The difference in medians ( $0.5$ point) is small compared with the variability within each class (IQRs of about $3$ points). The two distributions overlap almost completely, so the data don't show a meaningful difference. A difference in centers is only impressive when it is large compared with the spread.

Common mistake

Two common ways to lose credit on a comparison: (1) describing each group separately without ever comparing them, and (2) leaving out context. "The median of A is greater" earns less than "The median exercise time for younger adults (42 minutes) is greater than for older adults (24.5 minutes)."

Tip

Read across parallel box plots, not just within them. Where does one group's median fall compared to the other group's quartiles? If one group's median is beyond the other group's $Q_3$ , at least half of one group exceeds three quarters of the other.

Practice

The next three problems use these box plots of one-way commute times for a driver who takes Route X on some days and Route Y on others.

Commute time (minutes)

Practice 1

How many minutes greater is the median commute on Route Y than on Route X?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

About what percent of the Route Y commutes took longer than the median Route X commute?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which statement is best supported by the box plots?

Practice 4

The dot plots show quiz scores for two classes of $10$ students each. How many points greater is Class A's mean score than Class B's?

Class A quiz scores

Class B quiz scores

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

A researcher compares household incomes in two counties. Both distributions are strongly skewed right with several high outliers. Which pair of statistics is most appropriate for comparing them?

Practice 6

The back-to-back stemplot shows the number of books read in a year by $9$ members of Book Club P and $9$ members of Book Club Q. What is the median for Book Club Q?

Club P leaves	Stem	Club Q leaves
9 7 4	1
8 5 1 1	2	6
3	3	0 5 8
0	4	1 2 7
	5	3 4

Key: $1 \mid 2 \mid 6$ means $21$ books for Club P and $26$ books for Club Q.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which sentence is a proper comparison of center for two groups of test scores?