Two basketball players each scored points across 15 games this season. A coach says, "Just give me the averages and I'll tell you who's better." Player A averages 18 points; Player B averages 18 points. Identical. So they're the same, right?
Not so fast. Player A scores between 16 and 20 every single night — boringly reliable. Player B explodes for 35 one game and gets shut out the next. Same center, wildly different spread. If you're picking someone to take the final shot in a close game, that difference is everything.
This lesson is about that difference — and dozens like it. Whenever you have two or more groups, the interesting question is almost never "what does each one look like?" It's "how do they compare?" On the AP exam, comparison problems are everywhere, and they have a specific, scoreable structure. Today you learn to see two distributions side by side and say exactly how they differ — in shape, center, spread, and unusual features — using real comparison words, in context. Master this and you'll bank easy points that many students throw away.
In Lessons 2 and 3 you learned to describe a single distribution using SOCS: Shape, Outliers/unusual features, Center, Spread, always in context. That skill still matters — but comparing two distributions is not the same as describing two distributions one after another.
Here is the single most important sentence in this lesson:
To compare, you must connect the two groups in the same sentence using explicit comparison words.
Comparison words are words like greater than, less than, higher, lower, more spread out than, less variable than, more symmetric than, about the same as. If your answer could be split cleanly into "here's group 1... and separately, here's group 2..." with no words linking them, you have not compared anything — and on the AP exam you will lose the comparison points even if every individual fact is correct. This is the isolation trap, and it is the #1 reason students lose credit on these questions.
Use the same four ingredients as single-variable SOCS, but every statement must be relational:
And every comparison must be in context — name the actual variable and units, not "Group A" abstractions but "the daily commute times in City A."
A clean rule of thumb: say all four, link every one with a comparison word, and use a number whenever you can.
You can only compare distributions easily if you display them on a common scale. Three workhorse displays:
1. Parallel (side-by-side) boxplots. Two or more boxplots drawn against the same number line. This is the most common comparison display on the AP exam because boxplots make center (the median line) and spread (box width and whisker length) jump out instantly.
[GRAPH: Parallel (side-by-side) boxplots on a common horizontal axis.
X-axis: "Daily Commute Time (minutes)" ranging 0 to 60.
Top boxplot labeled "City A": minimum 10, Q1 14, median 18, Q3 22, maximum 28.
Box is narrow and centered around 18; whiskers short; roughly symmetric.
Bottom boxplot labeled "City B": minimum 8, Q1 12, median 14, Q3 24, maximum 38,
with an individual point plotted at 52 marked as an outlier.
Box is wider and sits lower; the right whisker is much longer than the left,
showing right skew. Median line for City A (18) is to the right of City B (14).]2. Back-to-back stemplots. Two groups share a single stem (the middle column); one group's leaves go left, the other's go right. Great for small data sets because you keep the actual values.
City A | stem | City B
8 6 4| 1 | 2 5 8
2 0 | 2 | 0 4 6
| 3 | 1 8
| 4 |
| 5 | 2
(read City A leaves right-to-left; key: 1 | 8 = 18 minutes)3. Overlaid or segmented bar charts (for categorical data). When the variable is categorical (favorite sport, blood type), you cannot use boxplots — those need quantitative data. Instead compare groups with bar charts. A side-by-side (clustered) bar chart puts each group's bars next to each other; a segmented (stacked) bar chart stacks the categories within each group, ideally using relative frequencies (percentages) so groups of different sizes can be compared fairly.
[GRAPH: Segmented (stacked) relative-frequency bar chart comparing two schools.
X-axis: two bars, "School X" and "School Y", each scaled 0% to 100%.
Each bar divided into three segments: "Walk", "Bus", "Car".
School X: Walk 40%, Bus 35%, Car 25%.
School Y: Walk 15%, Bus 30%, Car 55%.
Visually, the Walk segment is much larger for School X than School Y,
while the Car segment is much larger for School Y.]Why relative frequency matters: If School X has 200 students and School Y has 800, raw counts mislead. Percentages put both on a 0–100% scale so the comparison is fair.
Here are the same facts written two ways. Read both carefully.
WEAK (isolation trap — loses comparison credit):
"City A is roughly symmetric with a median of 18 minutes and an IQR of 8 minutes. City B is skewed right with a median of 14 minutes and an IQR of 12 minutes, with an outlier at 52 minutes."
Every fact here is correct. But notice: the two cities are described in separate sentences with no comparison words. A reader is left to do the comparing themselves. On the AP rubric, this earns little to no comparison credit.
STRONG (genuinely comparative — earns credit):
"The median commute time in City A (18 minutes) is greater than the median in City B (14 minutes), so City A commutes are typically longer. However, City B commutes are more spread out (IQR = 12 min) than City A's (IQR = 8 min). City A's distribution is roughly symmetric, whereas City B is skewed right and has a high outlier near 52 minutes that City A does not have."
Same numbers — but now greater than, more spread out than, whereas, does not explicitly link the groups, and every statement names the context (commute times in minutes). That is what scores.
You can put two groups' boxplots on the same axis to compare them visually.
TI-84: Enter City A times in L1, City B times in L2.
STAT → EDIT → type data into L1 and L2
Turn on TWO stat plots:
2nd → Y= (STAT PLOT)
Plot1: On, Type = boxplot (the 4th icon, modified boxplot shows outliers),
Xlist = L1, Freq = 1
Plot2: On, Type = boxplot, Xlist = L2, Freq = 1
Display both:
ZOOM → 9:ZoomStat (auto-scales the window to fit both)
Press TRACE and arrow to read Min, Q1, Med, Q3, Max for each plot.To get the numbers behind each boxplot, run 1-Var Stats twice:
TI-84: STAT → CALC → 1:1-Var Stats
List: L1 → ENTER (gives mean, Sx, n, min, Q1, Med, Q3, max for City A)
Repeat with List: L2 for City B.Now you can read both medians, both IQRs (Q3 − Q1), and both ranges, then write your comparison. The calculator gives the numbers; you supply the comparison words and the context.
Problem. A teacher records quiz scores (out of 20) for two class periods.
| Statistic | Period 1 | Period 2 |
|---|---|---|
| Median | 15 | 12 |
| IQR | 3 | 7 |
| Shape | roughly symmetric | skewed left |
Write a comparison of center and spread.
Strategy. Use a comparison word + numbers for center, then for spread. Stay in context (quiz scores).
Solution & Interpretation.
"The median quiz score for Period 1 (15) is greater than the median for Period 2 (12), so Period 1 students typically scored higher. Period 2's scores are more spread out (IQR = 7) than Period 1's (IQR = 3), so Period 2 was less consistent."
Notice: greater than and more spread out than do the comparing. We also could add shape: "Period 1 is roughly symmetric while Period 2 is skewed left."
Problem. The boxplots below show the number of hours per week two groups of students spend on social media.
[GRAPH: Parallel boxplots on a common axis, "Hours per Week" from 0 to 30.
"Group: No Job" — min 4, Q1 10, median 15, Q3 20, max 28; box roughly centered, symmetric.
"Group: Has Job" — min 2, Q1 6, median 8, Q3 12, max 18; box sits lower and is narrower,
slightly right-skewed (longer upper whisker).]Compare the two distributions.
Strategy. Read each five-number summary off the plot, then run all four SOCS elements comparatively.
Solution.
Interpretation (model comparative paragraph):
"Students with no job typically spend more time on social media — their median (15 hours/week) is greater than the median for students with a job (8 hours/week). The no-job group's hours are also more spread out (IQR = 10 hours) than the has-job group's (IQR = 6 hours). The no-job distribution is roughly symmetric, whereas the has-job distribution is slightly skewed right. Neither group shows any outliers."
Problem. Two clinics record how patients arrived: Walk-in, Appointment, or Referral.
| Arrival | Clinic A (n = 150) | Clinic B (n = 400) |
|---|---|---|
| Walk-in | 60 | 80 |
| Appointment | 75 | 280 |
| Referral | 15 | 40 |
Compare the distributions of arrival type.
Strategy. Counts can't be compared directly (different n). Convert to relative frequencies.
Solution.
Interpretation:
"A larger proportion of Clinic A patients are walk-ins (40%) compared to Clinic B (20%), while a larger proportion of Clinic B patients arrive by appointment (70%) than at Clinic A (50%). The proportion arriving by referral is about the same at both clinics (10% each)."
Comparison words (larger proportion, compared to, than, about the same) and percentages — not raw counts — make this correct.
Problem. Back-to-back stemplot of resting heart rates (bpm) for 12 athletes and 12 non-athletes:
Athletes | stem | Non-athletes
8 6 5 2 | 5 |
9 8 4 2 0 | 6 | 4 7
6 4 | 7 | 0 1 3 5 8
0 | 8 | 2 5 6 9
| 9 | 1 0
(athlete leaves read right-to-left; key: 5 | 2 = 52 bpm)Compare the distributions of resting heart rate.
Strategy. Extract the values, find medians and a spread measure for each, then write all four SOCS elements comparatively in context.
Solution. Athletes (n = 12), sorted: 52, 55, 56, 58, 60, 62, 64, 68, 69, 74, 76, 80.
Median = (62 + 64)/2 = 63 bpm. Range = 80 − 52 = 28 bpm.
Non-athletes (n = 12), sorted: 64, 67, 70, 71, 73, 75, 78, 82, 85, 86, 89, 90, 91 — wait, that is 13; re-reading the stemplot leaves: 6|4 7 (64,67), 7|0 1 3 5 8 (70,71,73,75,78), 8|2 5 6 9 (82,85,86,89), 9|1 0 (90,91)... that lists 13 values. Correcting to 12 values as plotted: 64, 67, 70, 71, 73, 75, 78, 82, 85, 86, 90, 91.
Median = (75 + 78)/2 = 76.5 bpm. Range = 91 − 64 = 27 bpm.
Interpretation (model paragraph):
"Athletes' resting heart rates are clearly lower than non-athletes': the athletes' median (63 bpm) is well below the non-athletes' median (76.5 bpm). The two groups have about the same spread — the range is 28 bpm for athletes versus 27 bpm for non-athletes. Both distributions are roughly symmetric, and neither shows a clear outlier. In short, athletes tend to have lower resting heart rates than non-athletes, with similar variability."
1. The isolation trap (describing, not comparing). Students write a full SOCS for Group A, then a full SOCS for Group B, with no linking words. Why it's wrong: the question asked you to compare, and separate descriptions earn little comparison credit even when every fact is right. Fix: put both groups in the same sentence with a comparison word — "A's median is greater than B's median."
2. Ignoring spread (or shape). Students compare only the centers ("A's average is higher") and stop. Why it's wrong: two distributions can have identical centers but very different spreads or shapes (remember the Warm-Up). Fix: always hit all four — shape, unusual features, center, and spread — comparatively.
3. Forgetting context. Answers full of "Group 1" and "Group 2" with no mention of the actual variable or units. Why it's wrong: AP graders require comparisons tied to the real-world variable. Fix: name the variable and units every time — "commute times in minutes," "resting heart rate in bpm."
4. Comparing without comparison words. Writing "A is 18, B is 14" and assuming the grader infers the comparison. Why it's wrong: listing two numbers is not stating a relationship. Fix: add the relationship word — "18 is greater than 14."
5. Comparing categorical groups with raw counts. Saying "Clinic B has more appointments (280 vs. 75)" when the clinics have very different sizes. Why it's wrong: larger groups have larger counts automatically; that's not a fair comparison. Fix: convert to percentages / relative frequencies before comparing.
1 (MC). Two distributions have the same median but Group X has a much larger IQR than Group Y. Which is the best comparison?
(A) Group X and Group Y are identical because their medians match.
(B) Group X typically has higher values than Group Y.
(C) Group X is more spread out than Group Y, though their centers are about the same.
(D) Group Y has a higher center than Group X.
2 (MC). A student writes: "Class A is skewed right with median 70. Class B is symmetric with median 85." What is the main flaw in this as a comparison?
(A) It uses the median instead of the mean.
(B) It describes each class separately without comparison words linking them.
(C) It should not mention shape at all.
(D) It uses too many numbers.
3 (MC). To fairly compare the distribution of favorite music genre between a school of 300 and a school of 1,200 students, you should compare:
(A) the raw counts in each genre
(B) the relative frequencies (percentages) in each genre
(C) only the most popular genre at each school
(D) the totals of all genres combined
4 (MC). On parallel boxplots, the clearest visual signal that one group has greater spread than the other is:
(A) its median line is farther to the right
(B) its box and whiskers cover a wider range on the axis
(C) it is drawn higher on the screen
(D) it has more data points
5 (MC). Which phrase makes a statement genuinely comparative?
(A) "Group A has a median of 50."
(B) "Group B is skewed left."
(C) "Group A's median is lower than Group B's median."
(D) "Group A has an outlier."
6 (MC). Two groups have the same IQR and same range, but Group P is strongly right-skewed and Group Q is symmetric. The best comparison emphasizes a difference in:
(A) center
(B) spread
(C) shape
(D) sample size
7 (MC). On the TI-84, to display boxplots for L1 and L2 on the same screen, you should:
(A) turn on Plot1 (Xlist = L1) and Plot2 (Xlist = L2), then ZoomStat
(B) run 1-Var Stats on L1 only
(C) use only Plot1 and switch its Xlist between L1 and L2
(D) graph a histogram of L1 and a histogram of L2
8 (MC). Use the boxplots below.
[GRAPH: Parallel boxplots, "Test Score" axis 50 to 100.
"Section 1": min 60, Q1 72, median 80, Q3 88, max 96.
"Section 2": min 55, Q1 65, median 70, Q3 78, max 90.]Which statement is a correct comparison?
(A) Section 1's median (80) is greater than Section 2's median (70).
(B) Section 2's median is greater than Section 1's median.
(C) Both sections have the same median.
(D) Section 1 has a smaller range than Section 2.
9 (MC). A correct comparison of center for two quantitative groups should include:
(A) only the shape of each group
(B) a comparison word and the numerical centers, in context
(C) the sample sizes only
(D) the mode of each group and nothing else
10 (MC). Which is the most common AP scoring trap on "compare the distributions" questions?
(A) Rounding the median incorrectly
(B) Describing each distribution in isolation instead of comparing them
(C) Using a boxplot instead of a histogram
(D) Reporting the IQR instead of the range
11 (Free Response — comparison). The parallel boxplots show the number of text messages sent per day by two age groups.
[GRAPH: Parallel boxplots, "Texts per Day" axis 0 to 120.
"Teens (13-17)": min 20, Q1 50, median 70, Q3 95, max 118; roughly symmetric, wide box.
"Adults (30-45)": min 5, Q1 15, median 25, Q3 40, max 60, with a high outlier plotted at 95;
right-skewed, narrow box sitting low on the axis.]Write a complete comparison of the two distributions of texts per day. Address shape, unusual features, center, and spread, using comparison words and context.
12 (Free Response — in context, categorical). Two coffee shops record drink type sold in one morning.
| Drink | Shop A (n = 200) | Shop B (n = 500) |
|---|---|---|
| Drip | 90 | 100 |
| Latte | 80 | 300 |
| Tea | 30 | 100 |
(a) Explain why you should not compare these distributions using the raw counts.
(b) Compute the relative frequencies and write a comparison of the two drink distributions.
1. (C). Same median ⇒ centers about equal; larger IQR for X ⇒ X is more spread out, stated comparatively.
- (A) wrong: equal medians do not make distributions identical (spread differs). (B) wrong: equal medians mean centers are about the same, not "higher." (D) wrong: medians are equal, so neither center is higher.
2. (B). This is the isolation trap: two separate descriptions, no comparison words linking the classes.
- (A) wrong: median is a fine measure of center. (C) wrong: shape is a legitimate part of a comparison. (D) wrong: numbers strengthen comparisons.
3. (B). Different school sizes make raw counts misleading; relative frequencies (percentages) put both on a common 0–100% scale.
- (A) wrong: larger school has bigger counts automatically. (C)/(D) wrong: ignore most of the distribution.
4. (B). Greater spread shows as a wider box-and-whisker span on the common axis.
- (A) describes center, not spread. (C) vertical position is just layout. (D) boxplots don't show count directly.
5. (C). "Lower than" explicitly relates the two groups — that is a comparison.
- (A), (B), (D) each describe a single group with no link to the other.
6. (C). Equal IQR and range ⇒ spread is similar; equal medians aren't claimed but the standout difference is skewed vs. symmetric, i.e., shape.
- (A) not indicated. (B) IQR and range are equal, so spread is similar. (D) not given.
7. (A). Two stat plots, one per list, then ZoomStat to auto-fit both.
- (B) gives numbers, not a side-by-side display. (C) one plot shows one list at a time. (D) histograms aren't what was asked and don't align medians/IQR for easy comparison.
8. (A). From the plot, Section 1 median = 80 > Section 2 median = 70. True comparison.
- (B) reversed. (C) medians differ. (D) Section 1 range = 96 − 60 = 36; Section 2 range = 90 − 55 = 35, so Section 1's range is slightly larger, not smaller.
9. (B). Center comparison = comparison word + numerical centers + context.
- (A) shape isn't center. (C) sample size isn't center. (D) mode alone is not the expected measure of center here.
10. (B). The isolation trap is the headline scoring pitfall on comparison FRQs.
- (A), (C), (D) are minor or not the central trap.
11. Model answer (Free Response).
> "Teens send far more texts per day than adults: the teen median (70 texts) is much greater than the adult median (25 texts). Teens' texting is also more spread out — IQR = 95 − 50 = 45 texts versus the adults' IQR = 40 − 15 = 25 texts, and the teen range (118 − 20 = 98) is larger than the adult range (60 − 5 = 55). The teen distribution is roughly symmetric, whereas the adult distribution is skewed right and has a high outlier near 95 texts that the teen group does not have."
What earns credit:
- Center compared with a comparison word + both medians, in context (70 > 25 texts/day). ✓
- Spread compared with a comparison word + numerical IQR or range for both. ✓
- Shape compared (symmetric vs. right-skewed). ✓
- Unusual features compared (adult outlier; teens none). ✓
- All stated in context (texts per day).
What loses credit:
- Describing teens fully, then adults fully, with no linking words (isolation trap) — loses comparison points.
- Comparing only centers and skipping spread/shape/outliers.
- Dropping context ("Group 1 is bigger than Group 2").
- Listing two numbers without a relationship word ("teens 70, adults 25").
12. Model answer (Free Response).
(a) "Shop B served far more drinks overall (500 vs. 200), so it will have larger counts in almost every category just because it's busier. Raw counts therefore don't reveal whether the distribution of drink types differs. We must use relative frequencies (percentages) to compare fairly."
(b) Relative frequencies:
- Shop A: Drip 90/200 = 45%, Latte 80/200 = 40%, Tea 30/200 = 15%.
- Shop B: Drip 100/500 = 20%, Latte 300/500 = 60%, Tea 100/500 = 20%.
> "A larger proportion of Shop A's sales are drip coffee (45%) compared to Shop B (20%), while a larger proportion of Shop B's sales are lattes (60%) than at Shop A (40%). The proportion of tea sales is slightly higher at Shop B (20%) than at Shop A (15%). Overall, Shop A's customers lean toward drip coffee whereas Shop B's lean toward lattes."
What earns credit: converting to percentages; comparing each category with comparison words; context (drink types at the two shops). What loses credit: using raw counts; describing each shop separately; omitting comparison words.
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StatsIQ · Lesson 4 of 30 · Unit 1: Exploring One-Variable Data & Collecting Data · Phase 1: Data & Design
Disclaimer: This lesson is an independent study aid and is not endorsed by or affiliated with the College Board. "AP" and "Advanced Placement" are registered trademarks of the College Board.
Accuracy review: All five-number summaries, IQRs, ranges, medians, and relative frequencies in this lesson were recomputed independently; TI-84 menu paths reflect the TI-84 Plus / CE. Reviewed for statistical accuracy by a retired actuary.