Count the Main Concentrations
In Describing Shape: Symmetric, Skewed, Uniform, you learned to describe the overall pattern of a histogram, including whether it is balanced, has a tail, or is roughly even. Another feature of shape is modality: how many distinct peaks or concentrations the distribution appears to have. Recognizing peaks can help you notice when a single summary of the data may hide an important pattern.
A histogram groups quantitative values into consecutive bins. A peak is a region where the bars are relatively high compared with bars on either side. A peak may span several neighboring bins; it does not have to be one unusually tall bar. Look across the full pattern from smaller values to larger values, rather than counting every small up-and-down change as a new peak.
The number of peaks is a description of the graph’s overall pattern. A distribution may have one broad peak, two separated peaks with a clear trough, or several concentrations. Some histograms do not show a clear number of peaks. In that case, describe the pattern cautiously rather than forcing it into a category.
How to Decide Whether a Bump Is a Peak
A histogram with one very high bar and small neighboring differences may still have one main peak. By contrast, two clusters of higher bars with a noticeable lower region between them are stronger evidence of two distinct peaks. The trough need not be empty: a bimodal distribution can have observations between its two concentrations.
Bin choices affect the appearance of a histogram. As you learned in Choosing Class Width and Number of Bins, narrow bins can show local detail while wide bins can smooth it away. A small bump that appears only with one narrow bin choice may not be a dependable second peak. When possible, check whether the overall pattern is still visible with another reasonable bin width. Do not change bin choices just to create or erase a feature; use them to judge how stable the feature is.
A histogram displays grouped values, so its bars do not reveal every individual observation. You can say that the histogram appears to have two peaks, but you generally cannot identify the exact most frequent value from a histogram alone. That would require individual values or a display that preserves them, such as a dotplot.
Worked Example: Identify One Main Peak
A fictional school club records how many minutes 50 students spend on a weekly puzzle. The constructed histogram frequencies are:
| Puzzle time (minutes) | Frequency |
|---|---|
| 0 to less than 10 | 2 |
| 10 to less than 20 | 6 |
| 20 to less than 30 | 13 |
| 30 to less than 40 | 15 |
| 40 to less than 50 | 9 |
| 50 to less than 60 | 4 |
| 60 to less than 70 | 1 |
Plan. Read the bar heights from shorter to longer puzzle times. Look for a main concentration and check whether the pattern falls into a separate second concentration after a lower region.
Do. The frequencies total \(2+6+13+15+9+4+1=50\) students. They rise toward the 30-to-less-than-40-minute interval, which has the highest bar, and then generally fall as times increase. The neighboring intervals also have fairly high frequencies, so the concentration is a region around 20 to less than 50 minutes, not just a single bar.
Conclude. The histogram of puzzle times appears unimodal, with one main concentration around 20 to less than 50 minutes. The frequencies taper toward both shorter and longer times; there is no distinct second concentration separated by a trough.
Why Two Peaks Can Suggest Subgroups
A bimodal histogram can suggest that the overall data combine two groups with different typical values. For example, a group of people with shorter travel times and another group with longer travel times could produce two concentrations in the combined distribution. The pattern is a useful clue to investigate, especially if there is a reasonable contextual variable that could distinguish the groups.
However, two peaks do not prove that there are two real subgroups, identify who belongs to each group, or establish why the values differ. Natural variation, an unusual sample, or bin choices can also affect the shape. A careful description separates what the histogram shows from what it might suggest: state the two concentrations and the trough, then frame a possible subgroup explanation as a possibility to check.
If relevant group information is available, examine the distributions separately or compare them with an appropriate display. For example, a histogram of commute times for drivers and another for transit riders could show whether those groups have different patterns. If the proposed grouping does not match the observed data, do not keep the explanation merely because it seems plausible.
Worked Example: Interpret Two Peaks as a Clue
A fictional workplace records commute times, in minutes, for 50 employees. The constructed histogram frequencies are:
| Commute time (minutes) | Frequency |
|---|---|
| 0 to less than 10 | 1 |
| 10 to less than 20 | 5 |
| 20 to less than 30 | 11 |
| 30 to less than 40 | 7 |
| 40 to less than 50 | 3 |
| 50 to less than 60 | 2 |
| 60 to less than 70 | 7 |
| 70 to less than 80 | 10 |
| 80 to less than 90 | 4 |
Plan. Check that the two higher regions are separated by a lower region. Then describe the pattern in commute-time units and consider whether a subgroup explanation is plausible without treating it as established.
Do. The frequencies sum to \(1+5+11+7+3+2+7+10+4=50\) employees. The first concentration is around 20 to less than 40 minutes, with a frequency of 11 in the 20-to-less-than-30 bin. Frequencies fall to 3 and then 2 in the next two bins. They rise again to 7 and 10 in the 60-to-less-than-80-minute region, before falling to 4 in the final bin. This gives two concentrations separated by a trough, rather than a steady taper from one peak.
Conclude. The distribution of employee commute times appears bimodal, with one concentration around 20 to less than 40 minutes and another around 60 to less than 80 minutes. The pattern could reflect subgroups with different commute arrangements, such as different transportation modes, but the histogram alone does not show that explanation is correct. If transportation mode were recorded, comparing the groups’ commute-time distributions would be a reasonable next step.
Multimodal Patterns and Uncertain Cases
A distribution with three or more distinct concentrations appears multimodal. As with bimodality, the peaks should be separated by noticeable lower regions; a series of slightly uneven bars is not automatically a set of distinct peaks. If several apparent peaks are close together, consider whether they form one broad, irregular concentration instead.
Sometimes the pattern is not clear enough for a firm label. A histogram may have a broad high region with several small bumps, or the apparent peaks may shift when bins change. It is acceptable to say that the histogram has no clearly distinct peaks or that it appears to have one broad concentration with minor irregularities. A precise description is better than an unsupported label.
Worked Example: Check Whether Several Bumps Are Distinct Peaks
A fictional community group summarizes the number of minutes 60 residents spend walking each day. With narrow, equal-width bins, the constructed frequencies are:
| Walking time (minutes) | Frequency | Combined wider interval | Combined frequency |
|---|---|---|---|
| 0 to less than 10 | 4 | 0 to less than 20 | 14 |
| 10 to less than 20 | 10 | ||
| 20 to less than 30 | 6 | 20 to less than 40 | 15 |
| 30 to less than 40 | 9 | ||
| 40 to less than 50 | 5 | 40 to less than 60 | 16 |
| 50 to less than 60 | 11 | ||
| 60 to less than 70 | 7 | 60 to less than 80 | 15 |
| 70 to less than 80 | 8 |
Plan. First check whether the narrow-bin bumps are separated by clear troughs. Then compare the broader pattern formed by combining adjacent bins. The total count should remain unchanged.
Do. The narrow-bin frequencies sum to \(4+10+6+9+5+11+7+8=60\). The bars rise and fall several times, which could tempt someone to call the distribution multimodal. But the changes are modest and do not make a clear sequence of distinct peaks and deep troughs. Combining adjacent bins gives frequencies \(4+10=14\), \(6+9=15\), \(5+11=16\), and \(7+8=15\), which also sum to 60. These wider-bin frequencies are fairly similar across the range.
Conclude. The narrow-bin bumps do not provide strong evidence of several distinct peaks. The broader display suggests a fairly even spread rather than clear multimodality. I would describe the distribution cautiously and avoid claiming that the small changes in the narrow-bin bars show separate groups. A dotplot or another reasonable histogram could provide additional detail if the individual observations were available.
Describe the Pattern, Then Consider the Explanation
When applying the SOCS framework from The SOCS Framework for Describing Distributions, treat modality as part of shape. Name the variable and group, state the number of main concentrations that the display supports, and use values or units to locate them. For a possible bimodal distribution, also mention the lower region between the peaks. This gives the reader evidence for the description rather than just a label.
Keep two claims separate. “The histogram of commute times appears bimodal, with concentrations around 20–40 and 60–80 minutes” describes the display. “The peaks show that employees use two different transportation modes” is a claim about the people and a possible explanation; the histogram does not establish it. You can instead say that the pattern could suggest subgroups and identify what information would help check that idea.
Modality does not replace the other parts of a distribution description. A bimodal histogram can also be skewed, have an outlier, or have a particular center and spread. Use the graph to describe each feature that is visible, and do not assume that the peaks alone tell the full story.
Common Mistakes and AP Exam Tips
- Counting every local high bar as a peak. A peak is a region of concentration, not any bar taller than its immediate neighbor. Look for a distinct high region separated from another by a lower region.
- Calling a histogram bimodal because two bars are tallest. Two tall bars without a noticeable separation may be part of one broad concentration. Describe the overall pattern, not just the two largest counts.
- Claiming that peaks prove subgroups. A second peak can suggest a mixture of groups, but it does not identify a cause. Use “could suggest” and name what additional group information could help investigate the idea.
- Ignoring bin width. Small fluctuations in narrow bins may disappear in a reasonable wider-bin view. Check the stability of the pattern before making a strong claim.
- Claiming an exact mode from a histogram. A bar covers a range of values. Describe the peak region or interval; do not claim that the histogram identifies the exact most common observed value.
- Using a label without context. For full-credit communication, identify the variable and group, name the apparent number of peaks, and locate the concentrations using the scale and units.
A clear AP-style statement might say: “The distribution of employee commute times appears bimodal, with concentrations around 20–40 and 60–80 minutes and fewer employees between them. This could suggest subgroups, but the histogram alone does not identify their cause.” The first sentence describes the evidence; the second gives a cautious interpretation.
Check Your Understanding
Use the full pattern of bars and describe what the display supports, not what it cannot prove.
- A histogram has one broad high region and then tapers toward both ends. What modality description fits? Does the peak have to be one bar?
- A histogram has two concentrations separated by a low region that still contains some observations. Can it appear bimodal? Explain.
- Why might a small bump in a narrow-bin histogram not be convincing evidence of another peak?
- A bimodal histogram of daily screen time shows concentrations near 2 hours and 6 hours. What can you say about the shape, and what can you not conclude about the people from the histogram alone?
- A histogram has several alternating bar heights, but combining adjacent bins gives similar frequencies across the range. What caution should you use when describing modality?