Read a distribution, not just its average

Read frequencies from intervals, compare distributions fairly, and distinguish a descriptive pattern from a claim about the population. A histogram groups numerical measurements into adjacent intervals. A bar chart instead separates categories. The distinction matters because the horizontal distance has a numerical meaning only in the histogram.

Build the method

Check axis units, bin boundaries and whether the vertical scale shows counts, proportions or density. The examples here use equal-width bins and counts. Write a boundary convention such as including the left endpoint and excluding the right endpoint; otherwise a value at a boundary could be double counted.

Worked example

For bins [0,5), [5,10), [10,15), [15,20) with counts 3,5,2,2, the count from 5 through values below 15 is 5+2=7. Neither the first nor final bin belongs. A histogram cannot reveal the exact individual observations within a bin, so it usually cannot give an exact sample mean.

Compare the alternatives

Group A has 13 supporters out of 21 and B has 21 out of 51. B has more supporters, but A has the greater support rate. Counts answer how many; relative frequencies answer how prevalent. Use the same measurement scale when comparing shape, center, spread and unusual observations.

Avoid a specific trap

An outlier fence is a screening rule, not permission to delete a measurement. With Q1=10 and Q3=18, IQR=8 and the upper fence is 30. A value of 30 is on the fence; a value of 31 is beyond it. Investigate recording errors and substantive explanations before changing the dataset.

Transfer check with explanation

Transfer: if every bin count doubles, the total doubles while each relative frequency stays unchanged. If only the high-end bin doubles, both relative frequencies and potentially the apparent shape change. Explain these separately rather than assuming any larger dataset has a different distribution.

WORKED EXAMPLE

Use the histogram. How many delays are at least 5 but less than 15 minutes?

Equal-width histogram: [0,5):3; [5,10):5; [10,15):2; [15,20):2.

Translate both endpoint inequalities into the included bins before summing.
Include bins [5,10) and [10,15): 5 + 2 = 7.
Why an alternative fails: Count only the first of the two requested intervals.

Answer: 7

Compare the other answer choices

Count only the first of the two requested intervals.

Include the final interval starting at 15 even though 15 is excluded.

Include observations below 5 even though the lower bound excludes them.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

AP Statistics Course and Exam Description, effective Fall 2026