Histograms.
Use frequency density to draw unequal-width bars whose areas represent the frequencies.
Calculate class widths and density · draw and interpret histograms · recover frequencies from bar areas
Let area represent frequency
A histogram uses numerical intervals as bar bases, so some bars can be wider than others. If frequency were represented by height alone, a wide bar would occupy extra area without necessarily containing more observations.
Frequency density corrects for that width. Divide each class frequency by its width to obtain the height. Then width multiplied by height recovers the frequency, so the bar area represents how many observations lie in the class.
A histogram displays grouped numerical data using bars over the class intervals. With unequal class widths, bar height alone cannot represent frequency fairly.
Frequency = class width × frequency density
Density = 3020 = 1.5
Frequency = width × density = 20 × 1.5 = 30
The 10–20 bar is tallest, but the wider 20–40 bar contains more observations: 30 compared with 20. Compare areas to compare frequencies.
Pause and explainCan a shorter histogram bar represent more observations than a taller one?
Yes, if it is sufficiently wider. Frequency is represented by area, not height alone.
Calculate widths and heights before drawing
Drawing a histogram requires a table of widths and densities, not just the original frequencies. A class from 20 to 40 has width 20 even if its neighbours each have width 10.
Keep the horizontal scale numerical so those widths appear in the correct proportions. Set each bar’s height to its calculated density and let adjacent class bars meet. Label the vertical axis “Frequency density” to explain what the heights measure.
| Time t (min) | Frequency | Width | Density |
|---|---|---|---|
| 0 ≤ t < 10 | 10 | 10 | 1 |
| 10 ≤ t < 20 | 20 | 10 | 2 |
| 20 ≤ t < 40 | 30 | 20 | 1.5 |
| 40 ≤ t < 60 | 10 | 20 | 0.5 |
Draw the bar for 20 ≤ t < 40, with frequency 30.
Frequency density = 3020 = 1.5
Draw the bar from 20 to 40 on the horizontal axis, with height 1.5. Do not use height 30.
Pause and explainA class of width 5 has frequency 20. What height belongs on a density axis?
4, since 20 ÷ 5 = 4. A height of 20 would incorrectly use frequency directly.
Recover frequencies from areas
Reading a histogram reverses its construction. Identify a bar’s horizontal width and vertical density, then multiply them to find the represented count.
For part of a class, only the full-class count is known. Estimating a subinterval count by its fraction of the bar width assumes observations are spread evenly within that class. State that assumption instead of treating the estimate as an exact count from raw data.
A histogram bar covers 40 ≤ t < 60 and has density 0.5. Find its frequency.
One bar has width 5 and density 4. Another has width 10 and density 2. Compare their frequencies.
The areas are 5 × 4 = 20 and 10 × 2 = 20. Both classes contain 20 observations, despite different heights.
To find the total frequency, add the frequencies represented by all the bar areas. For the explorer, the total is 10 + 20 + 30 + 10 = 70.
Pause and explainWhy is half a class width often taken to contain half the class frequency only as an estimate?
The original observations may not be evenly distributed inside the class; the histogram does not show their exact locations.
Distinguish a histogram from a bar chart
Bar charts and histograms can look similar, but their horizontal axes have different meanings. Category bar widths are a design choice; histogram widths are numerical intervals that carry information.
This is why touching bars and frequency density matter in a histogram. Comparing just the tallest bars can mislead when class widths differ. Name the graph type first, then use height or area according to what that graph actually represents.
| Bar chart | Histogram |
|---|---|
| Categories or discrete values | Grouped data on a continuous numerical scale |
| Usually equal-width bars with gaps | Widths match class widths; adjacent bars touch |
| Height represents frequency | Area represents frequency; height is frequency density |
Why would frequency heights be misleading for unequal widths?
A class twice as wide would receive twice the area for the same height. Using density compensates for width, so area remains proportional to frequency.
Pause and explainWhy do adjacent histogram bars touch?
Their bases are adjoining intervals on one continuous numerical scale, with no category gap between them.
Your turn. Use area, width and density.
Calculate every class width before its density, and check that the bar’s width times height returns its frequency.
When reading an unfamiliar histogram, inspect both axis scales. Add bar frequencies for totals, and clearly mark any estimate for only part of a grouped interval.
A class 10 ≤ x < 25 has frequency 45. Find its frequency density.
Reveal worked answer
A bar spans 30 ≤ x < 50 and has density 1.2. Find its frequency.
Reveal worked answer
A histogram bar has frequency 18 and density 3. Find its class width.
Reveal worked answer
Bars have (width, density) values (5, 4), (10, 3), (20, 0.5). Find the total frequency.
Reveal worked answer
In the explorer, estimate the frequency for 20 ≤ t < 30.