Why Describe Shape?
A histogram turns a set of quantitative measurements into a visible pattern. A useful description goes beyond saying that the bars are tall or short: it explains where observations concentrate, whether the distribution is balanced or has a long tail, and whether there are empty intervals or potentially unusual values. Those features help a reader understand the data without seeing every measurement.
Shape descriptions are summaries, not explanations of why the data look as they do. For example, a right-skewed histogram of household electricity use shows that higher-use households extend farther from the main concentration. It does not, by itself, explain what caused those values.
Terms for Describing Histogram Shape
A peak is often a region rather than one uniquely tallest bar. Two neighboring bars of similar height can form a broad peak, and a small fluctuation in bar heights does not necessarily create another meaningful mode. Look for clear concentrations separated by a lower region, rather than counting every small rise and fall.
The direction of skew names the direction of the tail, not the side where the highest bar appears. A histogram may have most of its observations near the lower end and a thin tail reaching toward high values; that is right-skewed. The same principle applies in reverse for left-skewness. Symmetry is usually approximate: real data rarely make perfectly matching sides.
A gap is an interval with no observations, shown by an empty region between bars. A gap can separate clusters or mark a break in the range of observed values. A potentially unusual value is one that appears separated from the main concentration or lies in a sparse tail. These are clues to investigate, not automatic proof that a value is erroneous or statistically exceptional.
Worked Example: Household Electricity Use
Worked Example: Household Electricity Use
Suppose a hypothetical utility summarizes the monthly electricity use of 50 households in equal-width bins. Use the table to describe the shape. Electricity use is measured in kilowatt-hours (kWh).
| Monthly use (kWh) | Number of households |
|---|---|
| 0 to under 200 | 2 |
| 200 to under 400 | 7 |
| 400 to under 600 | 14 |
| 600 to under 800 | 13 |
| 800 to under 1,000 | 8 |
| 1,000 to under 1,200 | 4 |
| 1,200 to under 1,400 | 2 |
Check the total. Adding the bin frequencies gives \(2+7+14+13+8+4+2=50\), matching the stated number of households.
Modality. The largest frequencies are 14 and 13 in neighboring bins, so the main concentration is around 400 to under 800 kWh. These adjacent high bars make one broad peak, not two separate peaks. The distribution is unimodal.
Symmetry and skewness. Frequencies decline over several bins to the right of the main peak: 8, then 4, then 2. The tail therefore extends toward larger monthly electricity use. The histogram is right-skewed, rather than symmetric.
Gaps and unusual values. None of the listed bins is empty, so this histogram shows no gap in the displayed range. The few households in the highest-use bins may be worth examining, but the histogram alone does not show whether any one household’s use is an error or unusually high for a relevant reason.
A contextual summary is: Monthly electricity use is unimodal, with a broad concentration from about 400 to 800 kWh, and is right-skewed, with a thin tail extending to higher use. There are no empty bins, and the sparsely populated high-use tail may merit further inspection.
Symmetry and Skewness in More Detail
A useful way to check approximate symmetry is to compare bars at similar distances on opposite sides of the central region. With equal-width bins, bars of similar height have similar frequency, so matching heights in corresponding positions suggest a balanced shape. For example, if a bar one bin to the left of the center has about the same height as the bar one bin to the right, and this pattern continues farther out, the histogram may be approximately symmetric.
By contrast, when the bars taper away more gradually on one side, that side forms the longer tail and determines the skew direction. Do not decide the direction from where most values lie, from the side of the tallest bar, or from whether a few high bars are present. Follow the overall tail.
Worked Example: A Roughly Symmetric Test-Score Distribution
A hypothetical class has 56 test scores, measured in points. The equal-width bins have the following frequencies. Describe the shape, including whether it is perfectly symmetric.
| Score (points) | Number of students |
|---|---|
| 30 to under 40 | 2 |
| 40 to under 50 | 6 |
| 50 to under 60 | 12 |
| 60 to under 70 | 16 |
| 70 to under 80 | 12 |
| 80 to under 90 | 6 |
| 90 to 100 | 2 |
Check the total. The frequencies add to \(2+6+12+16+12+6+2=56\), as stated.
Modality. The tallest bar is the 60-to-under-70 bin, and frequencies fall away on both sides. This is one central peak, so the distribution is unimodal.
Symmetry. Moving outward from the central bin, the frequencies match in pairs: 12 and 12, then 6 and 6, then 2 and 2. This supports describing the histogram as symmetric around the central region near 65 points. Its symmetry is a property of the grouped pattern; it does not mean the individual scores are paired or that every score has a matching score.
Gaps and unusual values. Every listed interval has observations, and the frequencies decrease gradually on both sides. There is no visible gap or isolated tail bin. A contextual summary is: Test scores are unimodal and approximately symmetric, with the greatest concentration from 60 to under 70 points and progressively fewer students in bins farther from that center.
Gaps and Potentially Unusual Values
A zero-frequency bin is a gap in the displayed histogram: no recorded observations fall in that interval. A gap can make two concentrations appear separated, but it does not establish that the data come from different groups or reveal why the interval is empty. A histogram also groups observations into bins, so it cannot show exactly where within a nonempty bin an observation falls.
Sparse bins far from the main concentration can flag possible unusual values. Use cautious language such as “there is a sparse upper tail” or “one observation appears separated from the main cluster.” Avoid calling a value an error simply because it is distant. Check the original records and the measurement context before deciding how to handle a potentially unusual observation.
Worked Example: A Gap and Two High Test Scores
In a second hypothetical class, 30 test scores are grouped into equal-width bins. The raw records confirm that the single scores in the final two nonempty bins are 84 and 96 points. Describe what the histogram indicates.
| Score (points) | Number of students |
|---|---|
| 30 to under 40 | 2 |
| 40 to under 50 | 7 |
| 50 to under 60 | 12 |
| 60 to under 70 | 7 |
| 70 to under 80 | 0 |
| 80 to under 90 | 1 |
| 90 to 100 | 1 |
Check the total. The counts give \(2+7+12+7+0+1+1=30\) students. The 84-point score falls in the 80-to-under-90 bin, and the 96-point score falls in the 90-to-100 bin.
Modality and gap. The main concentration has a peak in the 50-to-under-60 bin. The 70-to-under-80 bin is empty, so there is a visible gap between the main group and the two higher scores. The two high scores do not form a substantial second cluster; each occupies a separate bin with frequency one.
Potentially unusual values. The scores of 84 and 96 are separated from the main concentration by an empty interval, so they appear potentially unusual in this class. That description is not a judgment that either score is incorrect. A teacher would need to check the records and context before drawing that conclusion. A contextual summary is: Scores are concentrated from 30 to under 70 points, with a peak from 50 to under 60; there is a gap from 70 to under 80 and two higher scores, 84 and 96, that appear separated from the main group.
How to Write a Complete Shape Description
A concise description can follow a consistent order: name the variable and units, identify the number and location of peaks, describe symmetry or the direction of any tail, then note visible gaps or potentially unusual values. Mention only features supported by the graph. If a histogram does not show a clear pattern, say so rather than forcing a label.
For example, describe “monthly electricity use in kWh” or “test scores in points,” not just “the data.”
State whether there is one peak, more than one separated peak, or no clear peak, and give its approximate location.
Say whether the shape is approximately symmetric or has a longer tail toward smaller or larger values.
Give the relevant interval or range, and use cautious language rather than declaring an observation an error.
Common Mistakes
- Calling skew by the peak’s location. The correction is to trace the longer tail: a tail toward larger values means right-skewed, even when the peak is near the lower end.
- Calling every small rise a new mode. A mode is a meaningful concentration, not simply any bar taller than the bar immediately before it. Look for distinct peaks and the lower regions between them.
- Treating a gap as proof of separate populations. A gap means no observations appear in that interval in this histogram. It does not, by itself, explain why.
- Calling a distant value an error or outlier with certainty. A histogram can flag a value for review, but the data source and context must be checked before making a judgment.
- Leaving out context. “Right-skewed” is less informative than “monthly household electricity use in kWh is right-skewed, with a tail toward higher use.”
Shape descriptions help readers compare distributions and identify patterns worth investigating. They are strongest when they distinguish the visible evidence from conclusions the histogram cannot establish.
Check Your Understanding
For each question, use the shape vocabulary carefully and distinguish what the histogram shows from what it cannot establish.
- A histogram of daily water use has a main peak at low values and a long tail toward high values. Which direction is its skew, and what feature determines that direction?
- What is the difference between a broad peak made of neighboring tall bars and two modes?
- A histogram has no observations in the interval from 70 to under 80 points. What does this gap establish, and what does it not establish?
- A single bar in the highest bin contains one household. Give a cautious description of this feature and one reason not to label the observation an error immediately.
- What information should be included in a contextual description of a histogram’s shape?