Free graphing resources · Original practice
Histogram worksheets.
Count every boundary once.
Sort observations into nonoverlapping intervals, then build four frequency histograms. Repeated values exactly on a boundary make the bin rule matter.
4 equal-bin datasets.
Lower included; upper excluded.
More ways to practice
Why endpoints decide the shape
A histogram represents continuous data grouped into intervals. Each bar spans the full width of its bin, and adjacent bars touch because the intervals are contiguous. A bar chart represents categories, so bars are usually separated and their widths carry no numeric meaning. Before drawing, write each bin as a half-open interval such as 10 to under 15, meaning 10 is included and 15 is not. State this rule once and apply it to every value.
The endpoint rule changes counts when values land exactly on a boundary. If the rule is left-inclusive, a reading of 15 belongs to the 15 to under 20 bin, not the 10 to under 15 bin. If you switch rules midway, totals drift and bars mislead. Choose one convention, record it beside the graph, and keep it for the whole dataset.
- List the bins in order and write each as left-inclusive, right-exclusive.
- Sort the raw values and place each one in exactly one bin using the stated rule.
- Count frequencies and confirm they sum to the number of observations.
- Draw touching bars whose widths equal the bin widths on the same scale.
- Label both axes with units and note the endpoint convention.
A fresh boundary case
Consider waiting times in whole minutes: 4, 7, 9, 10, 10, 12, 15, 15, 18, 20. Use bins 0 to under 5, 5 to under 10, 10 to under 15, 15 to under 20, and 20 to under 25. With left-inclusive bins, the counts are 1, 2, 3, 2, and 2. The two values of 10 enter the third bin, and the two values of 15 enter the fourth. The total is 10, matching the ten values; check that sum before drawing.
On graph paper, give each five-minute bin the same horizontal width, for example five small squares. Bar heights follow frequency, so the third bin is three squares tall. Because the bins touch, the bars touch. A line graph would instead connect points across time and imply interpolation between readings, which this grouped count does not support.
- Write the ten values in order and mark each boundary value once.
- Apply the left-inclusive rule and record five counts.
- Verify the counts sum to ten.
- Set a horizontal scale where one bin equals a fixed number of squares.
- Draw touching bars and label the axis with minute units.
Diagnostic checks before submitting
Run three quick checks. First, confirm the frequency total equals the observation count. Second, confirm every bar width matches its bin width on the same axis scale. Third, confirm no gap appears between adjacent bars unless a bin is genuinely empty. An empty bin still occupies its width with zero height, so the axis stays continuous.
Also check the axis labels. A histogram axis should show numeric interval endpoints, not category names. If your labels read as labels rather than numbers, you have likely drawn a bar chart. Keep the endpoint note visible so a reader can reproduce your counts.
- Add the frequencies and compare with the raw count.
- Measure one bar width against the bin width on the axis.
- Check for unintended gaps and for empty bins drawn at zero height.
- Confirm axis labels are numeric endpoints with units.
Try this example
Group these scores into bins 20 to under 30, 30 to under 40, and 40 to under 50 using left-inclusive bins: 22, 29, 30, 30, 41, 47, 50.
Show the worked explanation
The 20 to under 30 bin holds 22 and 29, count 2. The 30 to under 40 bin holds 30 and 30, count 2. The 40 to under 50 bin holds 41 and 47, count 2. The value 50 falls outside all three bins, so either extend the bins or note it as excluded. Counts sum to 6, matching the six included values; the seventh value is outside the stated range.
Common mistakes
- Treating a value on a bin boundary as belonging to both adjacent bins. State a left-inclusive, right-exclusive rule and assign each boundary value to the higher bin only.
- Leaving gaps between histogram bars or varying bar widths. Make bars touch and give every bar the same width as its bin on the shared horizontal scale.
- Labeling the horizontal axis with category names instead of numeric intervals. Write numeric endpoints such as 10 to under 15 and include the unit.
Check your work
- Bins are written as explicit intervals with a stated endpoint rule.
- Every observation is assigned to exactly one bin.
- Frequencies sum to the number of included observations.
- Bar widths equal bin widths and adjacent bars touch.
- Axis labels show numeric endpoints and units.
- The graph is identified as a histogram, not a bar chart or line graph.
Questions before you start
How is a histogram different from a bar chart?
A histogram groups continuous numeric data into contiguous intervals, so bars touch and widths match bin widths. A bar chart compares separate categories, so bars are usually separated and widths are not numeric.
What happens to a value exactly on a boundary?
It goes into one bin only, decided by the stated rule. With left-inclusive bins, a value equal to the right endpoint belongs to the next bin.
Can a histogram have an empty bin?
Yes. An empty bin has frequency zero and is drawn at zero height while keeping its full width, so the axis remains continuous.
