Goal and prerequisites
Use frequency density when class widths differ and distinguish exact counts from grouped estimates. Review statistics before starting.
Method and its conditions
Frequency density equals frequency divided by class width. Bar area, not height alone, represents frequency. Recover a count by multiplying density and width. A grouped mean uses class midpoints as representative values, so it is an estimate.
Worked example: show the decisions
A class 0≤x<10 has frequency 20, giving density 2. A class 10≤x<30 has frequency 30, giving density 1.5. The second bar is lower but has larger area and more observations. The estimated mean is(20×5+30×20)/50=14.
Contrast: diagnose an incorrect approach
Comparing bar heights as if they were counts fails when widths differ. A split within a class requires an assumption about how observations are distributed; proportional allocation is an estimate, not an exact reconstruction. Always label the vertical axis frequency density.
Transfer: try a changed situation
Classes 0–4 and 4–10 have densities 3 and 2. Find their frequencies and estimate the mean using midpoints.
Transfer solution and interpretation
The counts are 12 and 12. Midpoints are 2 and 7, so the estimated mean is(12×2+12×7)/24=4.5. Equal bar areas represent equal frequencies even though the heights and widths differ.
A histogram class 3≤x<8 has frequency 30. Find its frequency density.
- Divide frequency by class width; here the width is five.
- Using the stated quantities: 30/5 = 6.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Compare the other answer choices
Use frequency as the height despite a non-unit class width.
Multiply frequency by class width instead of dividing.
Reverse the frequency-to-width ratio.
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.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027