Unit 1 · Topic 1.6 · about 20 minutes
Descriptions for One Quantitative Variable Distributions
Write a complete description of a distribution from its graph, in context, and use that description to support or reject a claim.
Minutes spent on homework last night, 60 students
Predict first
In the histogram above, most students are piled up on the left and a few stretch far out to the right. How would you describe its shape?
What to say about a distribution
The distribution of a variable tells you what values it takes and how often it takes each one. Every description of a quantitative distribution covers the same ground:
- Shape: skewed or symmetric, and how many peaks.
- Center: a typical value.
- Variability, also called spread: how far apart the values are.
- Unusual features: outliers, gaps and clusters, or a plain statement that there are none.
All of it goes in context. "Skewed right with a center around 40" means nothing until you add minutes of homework and 60 students.
Shape
A distribution is skewed right, or positively skewed, when its right tail, toward the larger values, is longer than its left. It is skewed left, or negatively skewed, when the left tail is longer. It is approximately symmetric when the left half is close to a mirror image of the right half. Real data are never perfectly symmetric, which is why approximately earns its place.
Count the peaks as well. A distribution with one main peak is unimodal. One with two prominent peaks is bimodal. If every bar is about the same height, with no prominent peak at all, the distribution is approximately uniform.
Scores on an easy quiz are a classic left skew: most students score near the top, and a few trail off toward the bottom.
Scores on an easy 20-point quiz, 30 students
Skewed left and unimodal. Most scores are 15 or higher, and the tail stretches down to one score of 8.
Minute past the hour when each of 180 customers entered a coffee shop
Approximately uniform. Customers arrive at a steady rate, so every 10-minute bin holds about 30 of them and there is no peak.
Unusual features
An outlier is a value that is unusually small or unusually large compared with the rest of the data. A gap is a stretch of the number line, between two values that do occur, where no data were observed. Clusters are concentrations of values, usually separated from each other by gaps.
The ages of the 40 people at a seven-year-old's birthday party show clusters and a gap at a glance: a group of children, a group of adults, and nobody in between.
Ages of the 40 people at a birthday party
Bimodal, with two clusters: children from 5 to 9 and adults from 29 to 52. Nobody at the party is between 10 and 28, a gap of almost 20 years.
Center and variability, read from a graph
From a graph you can estimate the center as the value that splits the data in half, with about half the observations below it and half above. For the 60 homework times, the 30th and 31st values both fall in the 30 to 45 minute bar, so the center is somewhere in that bar.
Variability is how spread out the values are. From a graph, describe it with the smallest and largest values, or say where most of the data lie: most of the 60 students did between 0 and 90 minutes of homework. 1.7 Summary Statistics for One Quantitative Variable turns center and variability into exact numbers.
Days absent last semester, 28 students in one class
Worked exampleDescribing a distribution, start to finish
A teacher recorded how many days each of the 28 students in her class was absent last semester. The dotplot above shows the results. Describe the distribution.
Shape. Most students missed only a few days, and a thin tail stretches out to the right. The distribution is skewed right and unimodal, with its peak at 2 days.
Center. With 28 students, the middle falls between the 14th and 15th values when they are sorted, and both are 3. A typical student missed about 3 days.
Variability. Absences range from 0 to 24 days, but 25 of the 28 students missed 9 days or fewer.
Unusual features. There is a gap from 13 to 23 days, and one student missed 24 days, far more than anyone else. That value is an apparent outlier.
The distribution of absences for these 28 students is skewed right, with a center of about 3 days. Most students missed between 0 and 9 days. One student missed 24 days, an apparent outlier separated from the rest of the class by a gap from 13 to 23 days.
Using a graph to back up a claim
A description is also evidence. Suppose the attendance office claims that a typical student in this class misses at least a week of school, 5 days, each semester. The dotplot says otherwise: 18 of the 28 students missed 4 days or fewer, so more than half of the class missed less than a week. A few students with long absences stretch the tail to the right, but a claim about a typical student has to rest on the center of the distribution, not on its tail.
Check your understanding
A histogram of the prices of 150 used cars listed for sale in a city has a tall peak around 8,000 dollars and a long tail that stretches out toward 40,000 dollars. Which describes its shape?
A city records the minute after the hour, from 0 to 59, when each of 600 parking tickets was written. A histogram with bins 10 minutes wide has bars of heights 98, 103, 101, 97, 100 and 101. Which word best describes the shape?
A dotplot of the weights of the 40 dogs boarded at a kennel shows 22 dogs between 8 and 25 pounds, 17 dogs between 45 and 70 pounds, no dogs between 25 and 45 pounds, and one dog at 175 pounds. Which description is accurate?
Use the histogram of homework times for 60 students at the top of this lesson. A student claims that most students spent more than an hour on homework. Which response correctly evaluates the claim?
Which of these is the most complete description of a distribution?
Course alignment, for teachers
AP Statistics topic 1.6, Unit 1: Exploring One-Variable Data and Collecting Data.
- Skill 4.A: Describe and compare tabular and graphical representations of data, as well as summary statistics.
- Skill 4.B: Justify a claim based on statistical calculations and results.