Histogram density and cumulative-frequency construction · Higher
| English | Português |
|---|---|
| frequency density/ˈfriːkwənsi ˈdensɪti/ | densidade de frequência |
A wide class can contain many observations even if its histogram bar is short. Frequency is represented by bar area, not height alone.
- A wide class can contain many observations even if its histogram bar is short. Frequency is represented by bar area, not height alone.
- This lesson studies frequency density · densidade de frequência 频率密度: Frequency divided by class width, used as histogram height.
Choose the mathematical structure
- Use continuous adjoining class boundaries on the horizontal axis. Density=frequency/class width, so bar area recovers frequency. Unequal widths require this adjustment. A cumulative-frequency graph plots each upper class boundary against the running total, beginning at the first lower boundary with zero. Read percentiles at fractions of the total; values within classes are estimates.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines frequency density?
Frequency divided by class width, used as histogram height.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For classes 0≤x<10,10≤x<20,20≤x<40 with frequencies 20,30,20, widths are 10,10,20 and densities 2,3,1. The last two bars have different heights but areas 30 and 20. Cumulative points are (0,0),(10,20),(20,50),(40,70). The median is at cumulative count 35; a straight-line estimate in the second class gives 10+(35-20)/30×10=15. Q1 is at 17.5 and Q3 at 52.5, giving corresponding interpolated estimates 8.75 and 22.5. Class midpoints 5,15,30 give estimated mean (100+450+600)/70≈16.43. A sketch that joins upper boundaries with a smooth curve can give slightly different readings, so use the stated precision and graph.
Histogram density and cumulative-frequency construction
Use continuous adjoining class boundaries on the horizontal axis
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find density for frequency 20 in width 20.
20/20=1.
Test a tempting shortcut
- A histogram uses density on its vertical axis when widths differ. Plot cumulative totals at boundaries, not midpoints. Grouped percentiles depend on interpolation assumptions and are not exact individual measurements.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Histogram bar height always equals the number of observations. This claim is false. Explain which definition or assumption it violates.
Find total frequency of 20,30,20.
Add all class frequencies.
Histogram bar height always equals the number of observations.
A histogram uses density on its vertical axis when widths differ. Plot cumulative totals at boundaries, not midpoints. Grouped percentiles depend on interpolation assumptions and are not exact individual measurements.
Interpret a new situation
- AQA S3 Higher includes equal/unequal-class histograms and cumulative-frequency graphs, with suitable interpretation. Explain why bar area represents frequency and label the graph axes/units.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the linear-interpolated median from the stated cumulative points.
Count 35 is halfway from 20 to 50, so halfway from 10 to 20.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.6. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Frequency divided by class width, used as histogram height. Choose the relationship, show the method, check its assumptions and interpret the result.