AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-6)
Unit 1 · Topic 1.6
1.6 Descriptions for One Quantitative Variable Distributions
Describing a quantitative distribution means covering four things: shape, center, variability and unusual features, always in context. This checklist shows up on almost every free-response question that includes a graph of data.
Key terms
- shape, center and variability
- skewed right / skewed left
- symmetric
- unimodal / bimodal / uniform
- outlier
- gaps and clusters
The four parts of a description
Whenever you describe a distribution, cover all four of these, and tie each one to the variable and its units:
- Shape: skewed, symmetric, number of peaks.
- Center: a typical value, like the median or mean.
- Variability (spread): how spread out the values are, like the range or IQR.
- Unusual features: outliers, gaps and clusters (or say there are none).
Shape
A distribution is skewed right when the high values trail off farther than the low values do, so the tail on the right side is the long one. Most values are low and a few stretch far to the right. Incomes and house prices are classic examples. Skewed right is also called positively skewed.
It's skewed left when the long tail is on the low side instead. Scores on an easy test often look like this: most are high, with a few low ones trailing off. Skewed left is also called negatively skewed.
It's roughly symmetric when the left half looks like a mirror image of the right half.
Count the peaks too. One clear peak is unimodal. Two clear peaks is bimodal, which often means two different groups are mixed in the data. If every bar is about the same height with no real peak, the distribution is approximately uniform.
Name the skew after the tail, not the peak. A pile of values on the left with a long tail to the right is skewed right.
Center and variability
For now, you can estimate the center as the value with about half the data on each side, and describe variability with the smallest and largest values. Topic 1.7 gives you exact measures (mean, median, standard deviation, IQR).
A description like "the times range from about 12 to 31 minutes, with a center around 18 minutes" covers both.
Unusual features
An outlier is a value that's unusually far from the rest of the data. A gap is a stretch of the number line with no data. Clusters are groups of values bunched together, usually separated by gaps.
If there are none, say so: "There are no obvious outliers or gaps." That still counts as addressing unusual features.
Later you'll use a formal rule to decide whether a value is an outlier. For a description from a graph, "appears to be an outlier" is fine.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Describe a distribution from a dotplot
A dotplot shows the number of text messages 25 students sent yesterday. Most values pile up between 10 and 40, with the tallest stack at about 20. A few values trail off to the right: 55, 62 and 70. There's one more value at 140, with nothing between 70 and 140. Describe the distribution.
Show the solutionHide the solution
- Step 1: Shape: the long tail is on the right, so the distribution is skewed right, and it has one main peak (unimodal).
- Step 2: Center: the typical student sent about 20 to 25 texts.
- Step 3: Variability: the values run from about 10 to 140 texts.
- Step 4: Unusual features: 140 is far from the rest, with a gap from 70 to 140, so it appears to be an outlier.
Answer: The distribution of texts sent yesterday is unimodal and skewed right, with a center around 20 to 25 texts and values from about 10 to 140. The student who sent 140 texts appears to be an outlier, separated from the rest by a gap between 70 and 140.
- Example 2Calculator allowed
Trap: naming skew after the peak
A histogram of scores on an easy quiz has a tall peak at 9 and 10 points (out of 10), and the bars get shorter toward 2 and 3 points. A student calls it skewed right because the peak is on the right. Correct the description.
Show the solutionHide the solution
- Step 1: Skew is named for the longer tail, not where the peak is.
- Step 2: Here the tail stretches to the left, toward low scores.
- Step 3: So the distribution is skewed left.
Answer: The quiz scores are skewed left: most students scored high, and a tail of lower scores stretches to the left.
Common mistakes
- Naming the direction of skew after the peak instead of the tail.
- Leaving out one of the four parts, especially unusual features. If there are none, say so.
- Describing a graph without context: "it's skewed right" instead of "the distribution of commute times is skewed right."
- Calling every unequal distribution "skewed." A distribution can be roughly symmetric without being perfectly so.
On the exam
- Free-response questions often say "describe the distribution." Readers typically look for shape, center, variability and context. Use all four every time.
- When comparing two graphs, the same four parts apply, with comparison words (see topic 1.9).
Connected topics
Videos
Check yourself
4 questions on 1.6 Descriptions for One Quantitative Variable Distributions. Pick an answer to see if you got it, and why.
The annual incomes of all 300 employees at a company are recorded. Most employees earn between $40,000 and $70,000, but a few executives earn more than $500,000. Which is the most likely shape of the distribution of incomes?
A dotplot shows the heights of all players on a school's girls' basketball team and boys' basketball team combined. The dotplot has one cluster centered near 66 inches and another centered near 72 inches. Which is the most reasonable explanation?
A teacher records how many minutes each of 25 students spent on a homework assignment. Which of the following is the best description of the distribution?
| Stem | Leaves |
|---|---|
| 3 | 2 5 8 |
| 4 | 0 1 4 4 7 9 |
| 5 | 2 3 3 6 |
| 6 | 1 8 |
| 7 | (no leaves) |
| 8 | 4 |
Invented data: wait times, in minutes, for 16 patients at a clinic. Key: 3 | 2 means 32 minutes.
Which of the following best describes the distribution of wait times?
0 of 4 answered