Lesson 1.5 · Unit 1 · CED topic 1.6
Describing the distribution of a quantitative variable
A graph is only useful if you can say what it shows. On the AP Statistics exam, "describe the distribution" is one of the most common tasks, and it is also one of the easiest to do incompletely. A complete description covers four things, every time: the shape of the distribution, its center, its variability (spread), and any unusual features such as outliers, gaps or clusters, all in the context of the problem, with units. This lesson defines the vocabulary and shows what a complete description sounds like.
Learning objectives.
- 1.6.A. Describe distributions of quantitative one-variable graphical representations.
- 1.6.B. Justify a claim using distributions of quantitative one-variable graphical representations.
Essential knowledge.
- 1.6.A.1. Descriptions of the distribution of one quantitative variable include shape, center, and variability (spread) as well as any unusual features such as outliers, gaps, or clusters in context.
- 1.6.A.2. The shape of the distribution of one quantitative variable is skewed to the right (positively skewed) if the right tail (toward larger values) is longer than the left. The shape of the distribution is skewed to the left (negatively skewed) if the left tail (toward smaller values) is longer than the right. The shape of the distribution is approximately symmetric if the left half is approximately the mirror image of the right half.
- 1.6.A.3. Distributions of one quantitative variable with one main peak are called unimodal. Distributions with two prominent peaks are called bimodal. A distribution in which each frequency or each relative frequency is approximately the same with no prominent peaks is approximately uniform.
- 1.6.A.4. Outliers for one quantitative variable are data points that are unusually small or large relative to the rest of the data.
- 1.6.A.5. A gap is a region in a distribution between two values in which there are no observed data.
- 1.6.A.6. Clusters are concentrations of values usually separated by gaps.
- 1.6.B.1. Graphical representations of a quantitative variable may reveal information that can be used to justify claims about the variable in context.
Shape
The shape of a distribution describes its overall form. Three ideas are used together.
Skewness. A distribution is skewed to the right (positively skewed) if the right tail, the part toward larger values, is longer than the left. It is skewed to the left (negatively skewed) if the left tail, toward smaller values, is longer than the right. It is approximately symmetric if the left half is approximately the mirror image of the right half. The name tells you where the tail is, not where the peak is: a right-skewed distribution has its peak on the left and a long tail trailing off to the right.
Peaks. A distribution with one main peak is unimodal. A distribution with two prominent peaks is bimodal. A distribution in which each frequency or relative frequency is about the same, with no prominent peaks, is approximately uniform.
Skewness is common in real data. Incomes, house prices and waiting times are usually skewed right, because a few very large values stretch the right tail. Scores on an easy exam are often skewed left, because most students score near the top and a few score much lower. Bimodal distributions often signal that the data mix two different groups, such as the heights of a group containing both children and adults.
Center and variability
The center is a typical or middle value of the distribution, and the variability (or spread) is how widely the values range around it. At this stage you can estimate both from a graph: the center is roughly where the values balance or where about half lie on either side, and the spread can be described by the range from the smallest to the largest value or by how far the bulk of the data extends. The next lesson gives exact measures (the mean and median for center; the range, interquartile range and standard deviation for spread). Whatever the measure, write the center and spread with units and in context: "most commutes are between about 10 and 30 minutes."
Unusual features: outliers, gaps and clusters
- An outlier is a data point that is unusually small or large relative to the rest of the data.
- A gap is a region in a distribution between two values in which there are no observed data.
- Clusters are concentrations of values, usually separated by gaps.
Never leave these out. A response that talks about the shape and center of a distribution but ignores an obvious outlier is incomplete, and graders look for it. Outliers are also worth a sentence of explanation when the context suggests one (a data-entry error, or a genuinely unusual case).
Putting it together
A good description has four parts, often remembered as shape, outliers (and gaps, clusters), center and spread, in any order that reads naturally, each tied to the context. A weak description is only a label ("it is skewed right"). A strong one reads like a sentence a reporter could use.
The histogram of the 30 student commute times has bin counts 2, 11, 8, 4, 2, 1, 1, 0, 0, 1 for bins of width 10 minutes starting at 0. Describe the distribution.
Shape: unimodal and skewed to the right, with the peak in the 10 to 20 minute bin and a long tail toward larger commute times. Center: a typical commute is a little under 30 minutes (the middle of the data falls around the 20 minute mark). Spread: the commute times range from about 5 to 90 minutes, but most students (\(2+11+8=21\) of 30, or 70%) commute less than 30 minutes. Unusual features: one student commutes about 90 minutes, far longer than the rest and a likely outlier; there is a gap from about 70 to 90 minutes. A complete summary in context: "The distribution of commute times for these 30 students is unimodal and skewed right, centered near 20 minutes, with most between 5 and 30 minutes and one outlier near 90 minutes."
A museum records the ages of 200 randomly selected visitors one Saturday. The histogram has a tall bar from 5 to 15 years and another tall bar from 35 to 45 years, with few visitors between 20 and 30 and few over 60. Describe the shape and suggest an explanation.
The distribution is bimodal, with one peak around 10 years and one around 40 years, and it has a dip (nearly a gap) between about 20 and 30 years. The two peaks suggest two different groups of visitors, children and their parents or other adults, mixed in one sample. A single center would not describe such data well, so the right description mentions both peaks and the two groups.
In the commute histogram, a student claims that most students have a commute under 30 minutes. Is the claim supported?
The bins [0,10), [10,20) and [20,30) hold \(2+11+8=21\) of the 30 students, which is \(21/30=70\%\). A majority is more than half, so the claim is supported: "70% of the sampled students, 21 of 30, have a commute under 30 minutes." A claim is justified by citing the part of the graph that shows it, with counts or proportions.
For each variable, say whether the distribution is more likely to be skewed right, skewed left, approximately symmetric or approximately uniform, and explain: (a) the birth month of 1,000 randomly chosen people; (b) the times, in minutes, that customers wait for a table at a busy restaurant; (c) the scores on a very easy quiz (maximum 20) taken by 80 students.
(a) Approximately uniform: each month has about the same share of births, so the bars would be nearly equal. (b) Skewed right: most waits are short, but a few customers wait much longer, stretching the right tail. (c) Skewed left: most scores are near 20, with a few much lower scores forming a tail on the left. In each case the reason refers to the context.
- Describe every quantitative distribution with shape, center, variability, and unusual features (outliers, gaps, clusters), in context with units.
- Skew is named for the longer tail. Unimodal, bimodal and uniform describe the number of peaks.
- Never ignore an outlier, a gap or a cluster. A single label such as "skewed right" is an incomplete answer.
- Justify a claim with numbers read from the graph (counts or proportions), not just a general impression.
- A histogram of hours of sleep for 25 students has bars: 5 to under 6 h: 1 student; 6 to under 7: 2; 7 to under 8: 6; 8 to under 9: 9; 9 to under 10: 7. Describe the shape.
Show answer
The peak is in the 8 to 9 hour bin and the tail extends toward the smaller values, so the distribution is unimodal and skewed to the left. (1 + 2 + 6 + 9 + 7 = 25 students.)
- The salaries at a small company are mostly between \$40,000 and \$70,000, but the owner earns \$450,000. Describe the likely shape and identify the unusual feature.
Show answer
The distribution is skewed to the right, and the owner's salary of \$450,000 is an outlier far above the rest.
- A histogram of the daily number of customers at a shop has two clear peaks, one near 30 and one near 120. What does this suggest, and why would a single "typical value" be misleading?
Show answer
Bimodal data usually mix two different kinds of days (for example weekdays and weekends). A single typical value near the middle (about 75 customers) would describe almost no actual day, because the data cluster around 30 and 120.
- A student describes a distribution of test scores as "skewed right." List what is missing from this description.
Show answer
A complete description also gives the center and the variability (with units and context), and mentions any outliers, gaps or clusters. The answer should also be tied to the context, such as naming the test and the group of students.
- The ages of the 11 people in a hiking group are 15, 15, 16, 16, 16, 17, 17, 17, 18, 18 and 45. Describe the unusual feature and give a possible explanation in context.
Show answer
The age 45 is an outlier, much larger than the rest, which are all between 15 and 18. A plausible explanation is that 45 is an adult leader of a group of teenagers.
- A graph of exam scores shows no scores between 60 and 70, with many scores between 70 and 100 and a few between 30 and 60. Name the feature between 60 and 70.
Show answer
It is a gap: a region between two values with no observations. The scores above 70 form a cluster, separated from the lower scores by the gap.