From Bar Heights to Counts in a Range
In Choosing Class Width and Number of Bins, you saw how a histogram groups quantitative data into intervals, or bins. Now we will use the bar heights to answer a practical question: how many observations, or what percentage of the observations, fall in a stated range?
For a frequency histogram with equal-width bins, each bar’s height represents the number of observations in that bin. To find a count for a range that consists of whole bins, add their heights. To convert that count to a percentage of all observations, divide by the total number of observations represented in the histogram and multiply by \(100\%\).
The notation for bin boundaries matters. In this tutorial, an interval written \([10,20)\) includes 10 but excludes 20. Thus, adjacent bins \([10,20)\) and \([20,30)\) include every value from 10 up to, but not including, 30. The value 20 belongs to the second bin, not both. This convention, introduced in Constructing a Frequency Histogram, prevents double-counting at shared boundaries.
A histogram summarizes data, so it may not show individual observations. It can give an exact count for a range made up of complete bins when the bar heights can be read exactly. If the requested range cuts through a bin, any count for that partial bin is an estimate unless additional information is available. Be explicit about which kind of answer you are giving.
Reading Bar Heights and Calculating a Percentage
First read the vertical scale carefully. If the axis is marked every 5 observations and a bar reaches the tick mark at 15, its frequency is 15. If its top falls between labeled marks, estimate the height using the spacing of the scale and report the result as an estimate. Do not treat the bar’s width as a count; in the equal-width frequency histograms used here, it is the height that represents frequency.
Once the relevant count is found, use the total number of observations as the denominator. This is the whole data set represented by the histogram, not just the observations in one of the bins being discussed.
If the range includes complete bins only, the count is the sum of those bin frequencies. If part of a bin is included, a common graphical estimate assumes observations are spread roughly evenly across that bin. Under that assumption, the estimated count from part of a bin is the same fraction of the bin’s frequency as the included width is of the full bin width. The assumption is a convenience for estimation, not something the histogram confirms.
For example, if a bin is 10 units wide and has frequency 20, a range covering half of that bin would be estimated to contain \(20 \times \frac{1}{2}=10\) observations. The actual count in that half could be different. If a question asks for an exact count and the range cuts through a bin, explain that the histogram alone does not provide enough detail for an exact answer.
Worked Example: Add Complete Bins
Worked Example: Add Complete Bins
A fictional recreation center records the time, in minutes, that 60 visitors spend using an indoor climbing wall. A frequency histogram has equal-width 5-minute bins. Its bar heights are 8 for \([0,5)\), 17 for \([5,10)\), 21 for \([10,15)\), and 14 for \([15,20)\). Estimate the number and percentage of visitors whose time was at least 5 but less than 15 minutes.
Identify the bins. The requested interval is \([5,15)\). It consists of the complete bins \([5,10)\) and \([10,15)\). Because the upper endpoint is excluded, values of exactly 15 minutes are not included.
Add the frequencies. The first bin contains 17 visitors and the second contains 21, so the count in the range is \(17+21=38\) visitors. The four bar heights total \(8+17+21+14=60\), which agrees with the stated sample size.
Find the percentage. Use all 60 visitors as the denominator:
About 38 of the 60 visitors, or 63.3%, spent at least 5 but less than 15 minutes on the climbing wall. The count is exact if the displayed bar heights are exact; the percentage is rounded to one decimal place.
Worked Example: Estimate a Count From Part of a Bin
Worked Example: Estimate a Count From Part of a Bin
A fictional trail group records how long 80 hikers take to complete a short route, in minutes. The equal-width 10-minute bins and frequencies are shown below. Estimate the number and percentage of hikers whose times were at least 15 but less than 35 minutes.
| Time (minutes) | Frequency |
|---|---|
| [0, 10) | 12 |
| [10, 20) | 20 |
| [20, 30) | 28 |
| [30, 40) | 16 |
| [40, 50) | 4 |
| Total | 80 |
Separate complete bins from partial bins. The requested interval is \([15,35)\). It includes the upper half of \([10,20)\), all of \([20,30)\), and the lower half of \([30,40)\). The two partial pieces are each 5 minutes wide, while their bins are 10 minutes wide.
Estimate the partial-bin counts. Assuming, just for this estimate, that hikers are spread roughly evenly within each partial bin, the upper half of the 10-to-less-than-20 bin contributes approximately \(20 \times \frac{5}{10}=10\) hikers. The lower half of the 30-to-less-than-40 bin contributes approximately \(16 \times \frac{5}{10}=8\) hikers. The complete 20-to-less-than-30 bin contributes 28.
Combine the estimates and calculate the percentage. The estimated count is \(10+28+8=46\) hikers. Therefore:
An estimated 46 hikers, or 57.5%, took at least 15 but less than 35 minutes. The partial-bin assumption means the count is an estimate, not an exact tally. The histogram does not reveal how many of the 20 hikers in the 10-to-less-than-20 bin had times from 15 up to 20 minutes.
Worked Example: Read Approximate Heights From a Graph
Worked Example: Read Approximate Heights From a Graph
A fictional school garden tracks the amount of water, in liters, used by 60 student teams during a project. Its frequency histogram has 10-liter bins from 0 to 50 liters. The vertical axis is marked in increments of 5 observations. The bar tops appear to be about 8, 13, 19, 11, and 9 observations for the five bins, respectively. Estimate the number and percentage of teams that used at least 10 but less than 40 liters.
Read the scale and select the bins. Because the bars for \([10,20)\), \([20,30)\), and \([30,40)\) lie entirely in the requested interval, add their estimated heights. The approximate frequencies are 13, 19, and 11.
Estimate the count. The sum is \(13+19+11=43\) teams. The other two bars are about 8 and 9, so all five estimated bar heights sum to \(8+13+19+11+9=60\), consistent with the stated total.
Convert to a percentage. The requested range contains an estimated 43 of the 60 teams:
About 43 teams, or 71.7%, used at least 10 but less than 40 liters. Since the bar tops were read approximately from a graph with a scale marked every 5 observations, both the count and percentage should be described as estimates. The arithmetic does not make approximate bar readings exact.
A Reliable Process for Range Questions
A careful answer starts by translating the wording into an interval and comparing it with the bin boundaries. Words such as “at least,” “less than,” “from,” and “through” determine which endpoint values are included. Then decide whether the range consists of whole bins or cuts through one or more bins.
Identify the lower and upper endpoints and whether each endpoint is included. Match this wording to the histogram’s bin convention.
Use the labeled vertical scale. Record approximate heights as estimates if the bar tops fall between tick marks.
Add frequencies for complete bins. For a partial bin, either explain that an exact count is unavailable or make a clearly labeled proportional estimate.
Divide the range count or estimated count by the total number represented, then multiply by \(100\%\). State the result in context and preserve the word “about” when readings or partial-bin assumptions are involved.
A useful check is to add all the bin frequencies. Their sum should equal the total number of observations. For approximate bar readings, the sum may be approximate too, so compare it with any stated sample size and revisit readings that seem inconsistent. When the question asks only for a count, do not add an unnecessary percentage; when it asks for a percentage, include the denominator or make clear that the denominator is the whole data set.
Common Mistakes and AP Exam Tips
- Including an endpoint that the question excludes. The bins \([10,20)\) and \([20,30)\) cover 10 up to, but not including, 30. They do not give the count “through 30,” because observations exactly equal to 30 belong to the next bin. Match the interval wording to the histogram boundaries.
- Calling a partial-bin estimate exact. A bar gives the total for its full bin, not the location of observations inside it. Say “approximately” and identify the even-spread assumption, or state that an exact count cannot be determined from the histogram alone.
- Using the wrong denominator. For the percentage of all observations in a range, divide by the total sample size represented by the histogram, not by the count in one selected bin. Earlier tutorials on choosing percentage denominators emphasize identifying the group named in the question; here, “of all observations” means the whole data set.
- Reading bar width as frequency. In the frequency histograms here, frequency is represented by bar height. Read the vertical scale rather than estimating from how wide the bar looks.
- Reporting unjustified precision. A graph with coarse tick marks may support an estimate of about 43, not a claim of exactly 43. A percentage calculated from an estimated count should also be reported as approximate.
- Forgetting to check the total. The sum of all bin frequencies should agree with the sample size. This check can reveal a missed bin, a misread height, or a boundary error.
For a full-credit response, name the interval in context, show which bin heights are added, and show the percentage calculation using the correct total. If you estimate a partial bin or read approximate bar heights, label the result as an estimate and explain why the histogram cannot support an exact count.
Check Your Understanding
Use the bin boundaries and bar heights carefully. Treat a count as an estimate whenever the information shown does not support an exact value.
- A histogram has frequencies 9, 14, and 17 for \([0,5)\), \([5,10)\), and \([10,15)\). How many observations are in \([5,15)\)?
- In a data set of 50 observations, 20 fall in a specified range. What percentage of the observations is that?
- A bin covers \([20,30)\) and has frequency 18. Under an even-spread assumption, estimate the number in \([20,25)\).
- Why can’t a histogram with bins \([10,20)\) and \([20,30)\) give the exact number of observations from 10 through 30?
- A bar top appears between the 10 and 15 marks on the vertical axis. How should its frequency be reported, and what should you check against the histogram’s total?