AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-9)
Unit 1 · Topic 1.9
1.9 Comparisons of the Distributions for One Quantitative Variable
Comparing distributions means putting two or more groups side by side and saying how they differ in shape, center, variability and unusual features, using comparison words. You'll also use z-scores to compare individual values that come from different distributions.
Key terms
- comparing distributions
- side-by-side boxplots
- back-to-back stemplot
- z-score (standardized score)
Graphs for comparing
Use the same scale for every group, or the comparison is meaningless. Good choices are side-by-side (parallel) boxplots, back-to-back stemplots, and dotplots or histograms stacked over a common axis.
Dotplots, stemplots and histograms let you compare shape, center, variability, outliers, gaps and clusters. Boxplots let you compare center, variability, outliers and skew, but not gaps or clusters.
Write comparisons, not lists
A comparison uses words like "greater than," "less than," "similar to" or "about the same as." Two separate descriptions aren't a comparison.
Not a comparison: "Group A has median 12. Group B has median 16." A comparison: "The median growth for Fertilizer B (16 cm) is greater than for Fertilizer A (12 cm)."
Cover all four features, and give context. Use the same type of measure for both groups (compare medians with medians, IQRs with IQRs).
z-scores
A z-score (standardized score) tells how many standard deviations a value is above or below the mean: z = (x − μ)/σ. When you only have sample values, use z = (x − x̄)/s.
Positive z means above the mean; negative means below. z = 0 is exactly at the mean. A z-score has no units, which is what makes it useful: you can compare a test score to a race time, or a score on one test to a score on a harder test.
You can also solve backward: x = μ + zσ gives the value that sits z standard deviations from the mean.
How unusual is a value?
A z-score also tells you how unusual a value is within its own distribution. A value with z beyond 2 or −2 is more than 2 standard deviations from the mean, which matches the 2-standard-deviation outlier rule from topic 1.7. In a roughly bell-shaped distribution, values that far out are uncommon (you'll put a number on this with the normal distribution in topic 2.11).
A percentile is another measure of relative position: it tells you what percent of the values are at or below a given value. A z-score tells you distance from the mean in standard deviations; a percentile tells you the share of values below. Both let you compare positions across different distributions.
One caution: z-scores describe position, not shape. Standardizing every value doesn't make a skewed distribution symmetric.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Compare two distributions
Plant growth (cm) after 4 weeks. Fertilizer A five-number summary: 4, 9, 12, 14, 18. Fertilizer B: 6, 13, 16, 19, 31; by the 1.5 × IQR rule, 31 is an outlier and B's whisker ends at 24. Compare the distributions.
Show the solutionHide the solution
- Step 1: Check outliers. A: IQR = 14 − 9 = 5; fences 1.5 and 21.5; no outliers. B: IQR = 19 − 13 = 6; fences 4 and 28; 31 is a high outlier.
- Step 2: Center: B's median (16 cm) is greater than A's (12 cm).
- Step 3: Variability: the IQRs are similar (6 cm for B vs. 5 cm for A), but B's range (25 cm) is much larger than A's (14 cm), mostly because of the outlier.
- Step 4: Shape: A is roughly symmetric. B's upper whisker and outlier stretch farther to the right, so B looks skewed right.
- Step 5: Outliers: B has a high outlier at 31 cm; A has none.
Answer: Plants with Fertilizer B grew more in the typical case (median 16 cm vs. 12 cm). The middle halves are about equally spread (IQR 6 vs. 5 cm), but B has a larger range because of a high outlier at 31 cm. A is roughly symmetric, while B is skewed right.
- Example 2Calculator allowed
Compare with z-scores
Maya scored 88 on a history test with mean 76 and standard deviation 8. Jon scored 82 on a biology test with mean 70 and standard deviation 6. Who did better relative to their class?
Show the solutionHide the solution
- Step 1: Maya: z = (88 − 76)/8 = 12/8 = 1.5.
- Step 2: Jon: z = (82 − 70)/6 = 12/6 = 2.0.
- Step 3: Both are 12 points above the mean, but Jon is 2 standard deviations above his class mean, while Maya is 1.5.
Answer: Jon did better relative to his class (z = 2.0 vs. Maya's z = 1.5), even though Maya's raw score is higher.
- Example 3Calculator allowed
Working backward from a z-score
Battery lives have mean 50 hours and standard deviation 4 hours. A battery has z = −1.25. How long did it last?
Show the solutionHide the solution
- Step 1: x = μ + zσ = 50 + (−1.25)(4) = 50 − 5 = 45.
- Step 2: Negative z means the battery lasted less than average.
Answer: 45 hours.
Common mistakes
- Listing summaries for each group without comparison words.
- Comparing a median for one group with a mean for the other.
- Comparing graphs drawn on different scales.
- Concluding the higher raw score is always relatively better, without checking z-scores.
On the exam
- "Compare the distributions" on free response usually means shape, center and variability (and outliers if any), with comparison words and context. Missing context or comparison words is a common way to lose credit.
- z-score questions often hide a trap where the higher raw score has the lower z-score.
Connected topics
Videos
Check yourself
4 questions on 1.9 Comparisons of the Distributions for One Quantitative Variable. Pick an answer to see if you got it, and why.
Maya scored 88 on a biology test where the class mean was 76 and the standard deviation was 6. Jordan scored 92 on a history test where the class mean was 80 and the standard deviation was 8. Who did better relative to their own class?
The weights of the apples in a shipment have a mean of 180 grams and a standard deviation of 15 grams. An apple has a z-score of −1.4. What is its weight?
| Statistic | Section 1 | Section 2 |
|---|---|---|
| Number of students | 28 | 28 |
| Minimum | 52 | 61 |
| Q1 | 70 | 80 |
| Median | 78 | 85 |
| Q3 | 86 | 90 |
| Maximum | 98 | 99 |
| Mean | 77.6 | 84.1 |
| Standard deviation | 10.9 | 8.2 |
Invented data: scores on the same test for two sections of a statistics course
Which statement must be true?
Which is the best comparison of the two distributions?
0 of 4 answered