Describing Distributions of One Quantitative Variable
Turn a graph into a clear explanation. Describe shape, center, spread and unusual features, then use visible evidence to support a claim about the recorded data.
By the end of this lesson, you should be able to:
- Recognize symmetry, skew, peaks and approximately uniform distributions.
- Describe a distribution’s center, spread, gaps, clusters and possible outliers in context.
- Distinguish an estimate from a graph from an exact value supported by recorded data.
- Justify a claim with a graph feature, count or proportion and state the group it describes.
Before you start: Read dotplots and histograms and check their scales. Review Topic 1.5 when you need help connecting dots or bins to observations.
First time learning this? Follow the Cedar High graph through the four-part description, then study the model answers.
Here to revise? Use the checklist, then try the practice questions before opening the solutions.
The concept in 60 seconds
A distribution shows which values occur and how often they occur. Describing it means explaining the overall pattern, not reading every dot or bar aloud.
Ask four questions: What shape does it have? Where is its center? How spread out are the values? Are there unusual features? Give your answers using the variable’s name and units.
A useful memory aid: SOCS. Shape, Outliers and other unusual features, Center, Spread. Remember gaps and clusters too. Add context throughout. The order can change to fit the question; this is a writing aid, not an official scoring rubric.
Then go one step further when a question asks for a justification. A statement such as “most travel times were short” needs evidence. “ of the recorded travel times were below minutes” gives a count and a clear threshold.
Read the task first. If it asks only about shape, explain shape. If it asks for a full description, include the other relevant features. If it asks whether a claim is supported, connect the evidence directly to that claim.
What does the Cedar High graph tell us?
Return to the same fictional -student sample from Topic 1.5. The variable is each sampled student’s one-way home-to-school travel time on the Tuesday being studied, rounded to the nearest minute.
Recorded travel times, minutes:
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dot represents observation ·
On a small screen, scroll the plot sideways to read the full scale.
Start with the overall pattern. Most observations are in the lower and middle parts of the scale, with a longer stretch toward larger travel times. The middle of the ordered observations is near minutes. The recorded values extend from to minutes.
Predict before reading on: Is “a majority of these sampled students took less than minutes” supported? Count the dots below . A dot at exactly does not belong in that count.
Check your prediction
Exactly of the observations are below minutes, so the proportion is . A majority means more than half. The claim is not supported. “Half of the recorded travel times were below minutes” is accurate.
These values are invented teaching data. They describe this sample on the specified Tuesday; the graph alone does not establish the pattern for every Cedar High student or every school day.
Recognize shape, tails and peaks
Describe the broad pattern before focusing on individual observations. Two useful questions are “Do the sides look balanced?” and “How many prominent peaks are there?”
Separate distributions · equal-width bins · common frequency scale
On a small screen, scroll the plot sideways to read the full scale.
Approximately symmetric
The two sides are roughly mirror images around the center. Small differences are normal in recorded samples; the halves do not need to match perfectly.
Skewed right
The longer tail extends toward larger values. Many observations may be at lower values, with fewer stretching far to the right. “Positively skewed” means the same direction.
Skewed left
The longer tail extends toward smaller values. Many observations may be at higher values, with fewer stretching far to the left. “Negatively skewed” means the same direction.
Name the skew by the tail. The tallest stack or bar tells you where observations are concentrated. It does not determine the direction of skew.
Separate distributions · equal-width bins · common frequency scale
On a small screen, scroll the plot sideways to read the full scale.
Peaks describe concentration
- Unimodal: one main peak or broad high region.
- Bimodal: two prominent peaks, often with a lower region between them.
- Approximately uniform: frequencies or relative frequencies are fairly similar across the intervals, with no prominent peak.
Shape words can be combined. For example, a distribution can be approximately symmetric and bimodal. Symmetric does not automatically mean bell-shaped, and uniform does not mean all the recorded values are equal.
Look for prominent features. A few small bumps do not always justify calling a small sample multimodal. Histogram bin widths can hide or reveal peaks, so avoid a stronger claim than the display supports. Review how bin width changes a histogram.
The shape examples are separate invented distributions. All intervals within each panel have equal width. Their frequency axes use the same scale, and their sample sizes are labeled.
Describe center and spread with units
Center: where is the middle or a typical value?
A graph-based description should give a useful numerical location: “travel times are centered around minutes,” rather than “the center is in the middle.” A center summarizes where the distribution sits on the numerical scale.
The tallest bar is not automatically the center. It identifies a high-frequency interval. To locate the middle observation, consider how many observations lie on either side, including those in other bins.
Use about, around or approximately for estimates from a display. A histogram groups values into intervals, so it usually does not reveal an exact mean or median. If an exact summary is provided, or can be determined from fully readable raw values, identify that summary accurately.
For the Cedar High data: the two middle ordered travel times are and minutes, giving a median of minutes. The dotplot and the listed values support that exact result. A grouped histogram of these observations would support a less precise graph-based description. Topic 1.7 develops the calculation and interpretation of summary statistics.
Spread: how far do the values extend?
Report the observed low and high values with units and describe concentration when it helps. Cedar High’s recorded times extend from to minutes, with of the observations between minutes inclusive and minutes exclusive.
“From to minutes” names the endpoints. Their difference, minutes, is the numerical range. Do not call the largest value alone the range. A good description can also say where the bulk of the observations lies; endpoints alone can hide a tight main cluster and one distant value.
Match your precision to the graph
| Display or information | A defensible description | A limit to remember |
|---|---|---|
| Clearly scaled dotplot or keyed stemplot | Individual recorded values, repeated values and exact endpoints may be readable. | Respect the recording precision and the graph’s scale. |
| Histogram with bins from to minutes | Identify occupied intervals and estimate the center or overall extent. | The endpoints of the outer bins are not necessarily observed values. |
| A supplied numerical summary | Use the named statistic with its value and units. | Do not replace a mean with a median or invent a statistic. |
Two-cluster caution: a single center can fall between clusters where few or no observations occur. Describe the clusters too, so “typical” does not hide an important feature.
Look for gaps, clusters and possible outliers
A fictional school help desk records the wait times of visitors, in whole minutes: , , , , , , , , , , , and .
dot represents observation ·