Below the bars are numerical intervals, not category names
The first example asks what a histogram represents. Group numerical values into intervals such as “at least 10 and less than 20,” and count the observations in each. These are class intervals, and each count is a frequency. Numerical ranges run along the horizontal axis, showing where observations are concentrated.
Among the choices, comparisons of totals for named categories describe the organizing principle of a categorical bar chart. Changes over time and relationships between pairs of variables serve different purposes too. Look for frequencies grouped into numerical intervals, rather than choosing only because a display has bars.
Include 10, exclude 20: count a boundary only once
In the second example, “at least 10 and less than 20” means 10 ≤ x < 20. It includes 10 and excludes 20. Of the other choices, 9.9 falls below the lower limit and 21 exceeds the upper limit. Only 10 meets the condition.
Using the same convention for “0 up to 10,” “10 up to 20,” and “20 up to 30,” with each lower endpoint included and each upper endpoint excluded, prevents 10 or 20 from entering two intervals. The notation [10,20) also includes 10 and excludes 20. Read the stated endpoint rule rather than assuming every display uses the same one.
With equal frequencies, twice the width means half the height
The third example specifies that bar areas are proportional to frequencies. The intervals [0,10) and [10,30) each have frequency 10, but their widths are 10 and 20. Equal frequencies require equal areas, so the second bar must be half as tall because it is twice as wide. The first-to-second height ratio is 2:1.
If the vertical axis is frequency divided by interval width, the heights are 10 ÷ 10 = 1 and 10 ÷ 20 = 0.5. This height is called frequency density. Multiplying back gives areas 10 × 1 = 10 and 20 × 0.5 = 10, matching the equal frequencies. Scaling the vertical axis by the same factor leaves the height ratio at 2:1.
A 1:1 height ratio would give the wider second bar twice the area. A 1:2 ratio reverses the required height relationship, while 4:1 exaggerates it. If every interval has equal width, however, comparing heights also compares areas. Decide whether heights alone are enough only after checking widths and the vertical axis.
Reconstruct width, height and area in the same example
On paper, write the third example’s intervals and add “width 10, frequency 10, height 1” and “width 20, frequency 10, height 0.5.” Check that width × height gives 10 in each case. Then consider what would change if you made the second bar equally tall. Its area would change, failing to preserve the original frequency of 10.
The practice button selects standard middle- and high-school questions on probability and statistics. It includes other matching questions beyond these three. When revisiting each example, explain whether your answer rests on the interval, the boundary or the area, as well as naming the answer.
Continue with graph axes or counting probability outcomes
The related graph-and-coordinate guide extends the habit of reading axis meanings to slopes and intercepts. In the probability basics guide, check what counts as one outcome. Reading a frequency and establishing that outcomes are equally likely each require their own check.