Check the Structure Before Reading a Histogram
A histogram can make a quantitative distribution easy to see, but a small drawing error can change its meaning. In Constructing a Frequency Histogram, you learned to group quantitative values into intervals and represent their frequencies with touching bars. Here, the task is to inspect a histogram for common errors: unintended gaps between bars, misleading treatment of unequal-width bins, and axes that do not clearly describe the data.
A histogram’s horizontal axis represents a quantitative variable, such as time or distance, divided into intervals called bins. The vertical axis represents a quantity such as frequency, relative frequency, or density. The bars meet at shared bin boundaries because the intervals follow one another on a numerical scale. This differs from a bar chart, whose separated bars represent categories, as explained in Why Histograms and Bar Charts Are Different.
Gaps Between Bars: Spacing or Data?
For adjoining bins, the corresponding bars should touch at their shared boundary. A small blank space between every pair of bars suggests that the graph has been drawn like a bar chart, or that its layout has added unwanted spacing. Such spaces can make consecutive numerical intervals look like unrelated categories.
A blank region can also have a meaningful explanation: one or more bins may have frequency zero. That is a gap in the data, not an instruction to separate all the other bars. The empty interval still belongs on the horizontal number line. For example, if the bin \([20,30)\) has no observations, bars for \([10,20)\) and \([30,40)\) should stop at the edges of that empty interval. The blank area represents values from 20 up to, but not including, 30.
This distinction matters. An unwanted sliver between bars can falsely suggest that a range of values was excluded or empty. A full empty bin, by contrast, communicates that no observations fell within that interval. The earlier tutorial What a Boxplot Cannot Show noted that a histogram can reveal gaps; this tutorial adds a drawing check for making sure a gap shown in a histogram corresponds to the bins and data, rather than to decorative bar spacing.
Also inspect how the bins cover the values. Consecutive bins should meet at their boundaries, and each observation should be assigned to exactly one bin according to a consistent endpoint convention. If a bin is accidentally omitted, the graph may suggest a gap that the data do not contain. If bins overlap, an observation might be counted twice.
Worked Example: Separate an Empty Bin From a Drawing Error
A fictional field class records the lengths of 36 leaves, in millimeters. The class uses consecutive bins of width 5 millimeters, with frequencies shown here.
| Leaf length (mm) | Frequency |
|---|---|
| [10, 15) | 5 |
| [15, 20) | 9 |
| [20, 25) | 0 |
| [25, 30) | 14 |
| [30, 35) | 8 |
Plan. Check whether the bins are consecutive and locate the zero-frequency interval. Then decide which blank spaces in a sketch are justified by the data.
Do. The intervals meet in order: the first ends at 15, the next begins at 15, and so on through 35. Their frequencies sum to \(5+9+0+14+8=36\), matching the stated number of leaves. The interval \([20,25)\) is empty, so no bar should be drawn over that interval. The bars over \([10,15)\) and \([15,20)\) should touch, as should the bars over \([25,30)\) and \([30,35)\). The empty interval creates one wider blank span from 20 to 25 millimeters.
Conclude. A correct sketch has touching bars wherever consecutive bins both have observations, and a blank span only over \([20,25)\), where the frequency is zero. If every bar has a narrow space around it, those spaces are a drawing error, not evidence of additional gaps in leaf lengths.
Unequal Bin Widths Need Special Care
Equal-width bins make it straightforward to use frequency as the bar height: with all widths equal, a taller bar also has a larger area, so the visual comparison of areas agrees with the comparison of counts. With unequal widths, raw count heights can be misleading. A wide bin may contain more observations simply because it covers a wider interval.
When bin widths differ, use frequency density as the bar height if the histogram is meant to represent counts through bar areas. Frequency density is the bin frequency divided by its width. Then the area of a bar equals the frequency in that bin:
For example, if a bin has 12 observations and width 10 units, its frequency density is \(12/10=1.2\) observations per unit. A bin twice as wide should not automatically have twice the height. Its height is determined by its frequency per unit of horizontal scale.
The vertical axis must identify what the heights show. If the heights are frequency densities, label the axis accordingly and include the relevant units, such as “frequency density (observations per minute).” For a relative frequency density histogram, divide each bin’s relative frequency by its width; each bar’s area then represents the relative frequency in that bin. As described in Relative Frequency and Density Histograms, the bar area—not the height alone—represents the share when widths differ.
Worked Example: Correct the Heights for Unequal Bins
A fictional equipment shop records the battery life, in hours, of 40 sample devices. Its chosen bins have unequal widths.
| Battery life (hours) | Width (hours) | Frequency | Frequency density |
|---|---|---|---|
| [0, 10) | 10 | 12 | 1.2 |
| [10, 30) | 20 | 20 | 1.0 |
| [30, 40) | 10 | 8 | 0.8 |
Plan. Divide each bin’s frequency by its width. Use those densities as bar heights, then check that height multiplied by width returns the bin frequency.
Do. For \([0,10)\), the density is \(12/10=1.2\) devices per hour. For \([10,30)\), it is \(20/20=1.0\) device per hour. For \([30,40)\), it is \(8/10=0.8\) devices per hour. Check the areas: \(1.2(10)=12\), \(1.0(20)=20\), and \(0.8(10)=8\). The areas add to \(12+20+8=40\), the total number of devices.
If the three bars instead had heights 12, 20, and 8, the width-20 bar would have twice the width of either other bar, and its area would be \(20(20)=400\) in count-times-hours units. That display would not make its area represent its frequency. In the correctly scaled histogram, the middle bin has the greatest count, but its bar is not the tallest; the narrow first bin has the greatest density.
Conclude. Draw the bars over the stated intervals with heights 1.2, 1.0, and 0.8, and label the vertical axis “frequency density (devices per hour).” The areas then represent the three bin counts.
Axes and Labels Must Describe the Graph
A reader should be able to identify the variable, its units, the bin boundaries, and what the vertical scale measures without guessing. The horizontal axis needs a meaningful variable name and units when the variable has units. Its numerical scale should match the bin intervals. A vertical axis labeled only “frequency” is not appropriate if the heights are percentages or density.
Check the scale as well as the words on the axes. Tick marks should be in numerical order and evenly spaced when they represent equal numerical steps. The labels need to match the actual scale: an axis with values marked 0, 5, 10, and 15 should not be described as though each step were 10 units. The displayed vertical values should correspond to the plotted bar heights. For a frequency histogram, the count scale ordinarily starts at zero so that bar heights communicate frequencies from a common baseline.
A title can add useful context, but it does not replace axis labels. “Student data” does not identify which quantitative variable was measured or its units. Likewise, a horizontal axis labeled “value” and a vertical axis labeled “amount” leave the reader to infer what the graph shows. The earlier tutorial Identifying Graph Errors on Exam Questions emphasized checking titles, labels, and units in a graph; apply that same habit to the quantitative intervals and vertical measure in a histogram.
Worked Example: Repair Mismatched Histogram Labels
A fictional survey records weekly study time for 30 students. A histogram uses the bins \([0,2)\), \([2,4)\), \([4,6)\), and \([6,8)\) hours, with bar heights 4, 10, 12, and 4. The horizontal axis says “Time,” and the vertical axis says “Percent.”
Plan. Compare each axis label with the variable and the displayed numbers. Check whether the vertical heights are counts or percentages and determine what relabeling or recalculation is needed.
Do. The horizontal bins show that the variable is weekly study time, measured in hours, so “Time” is too vague. The counts add to \(4+10+12+4=30\). The heights shown are these counts, not percentages: for example, the first bin’s relative frequency is \(4/30\approx0.1333\), or about \(13.3\%\), not 4%. The last bin likewise contains 4 students, which is about \(13.3\%\), not 4%.
If the bar heights remain 4, 10, 12, and 4, relabel the vertical axis “Frequency (students).” Label the horizontal axis “Weekly study time (hours)” and use numerical ticks that mark 0, 2, 4, 6, and 8 hours. If the graph is intended to show relative frequency instead, recalculate the heights as \(4/30\approx0.133\), \(10/30\approx0.333\), \(12/30=0.400\), and \(4/30\approx0.133\), and label the vertical axis “Relative frequency.”
Conclude. The original vertical label does not match its bar heights. A correct frequency histogram identifies weekly study time in hours on the horizontal axis and frequency in students on the vertical axis—or uses calculated relative frequencies and labels those instead.
Common Mistakes and AP Exam Tips
- Adding spaces between all bars. Touching bars show adjoining numerical intervals. Leave a blank span only when the displayed bins have zero frequency or are otherwise explicitly omitted and explained.
- Confusing a zero-count bin with a decorative gap. Check the interval boundaries and frequencies. A real gap in the data must correspond to an interval containing no observations, not just a thin space between bars.
- Using counts as heights for unequal-width bins. If area is supposed to represent frequency, divide each count by its bin width and use frequency density. Then check that each bar’s area matches its count.
- Labeling density as frequency. Frequency density is a rate per unit of the horizontal variable, not a count. State the density and its units on the vertical axis.
- Using vague or incorrect labels. “Time” is less informative than “Weekly study time (hours),” and “Percent” is incorrect when the heights are raw counts. Label the quantity actually shown.
- Ignoring the bin boundaries or scale. Verify that the horizontal axis numbers agree with the intervals, and that the bins have no accidental overlap or omission.
For a full-credit explanation, name the error and state how to correct it. For example: “The bars have gaps even though the intervals are consecutive; draw adjoining nonempty-bin bars so they touch.” For unequal widths, identify the mismatch and give the needed adjustment: “Use frequency density, frequency divided by bin width, as the height so bar area represents the count.” For a label error, state both what the axis currently implies and what it should say.
Check Your Understanding
Use the histogram checks in this tutorial to answer each question.
- Two consecutive bins have positive frequencies, but the sketch leaves a narrow white space between their bars. What should be changed, and why?
- A bin from 12 to 18 units has no observations. What should its histogram region show? How is that different from adding small spaces between every pair of bars?
- A bin has frequency 15 and width 5 units. What is its frequency density, including units? What is the area of its bar when density is the height?
- Two bins have widths 4 and 12 units. Why could using their raw counts as bar heights mislead a reader when comparing area?
- A histogram’s vertical axis says “Percent,” but the bar heights are counts of people. Give one correction that would make the graph consistent.