Histograms
Practise using frequency density to draw and interpret histograms with unequal class widths. Build complete and partly drawn histograms, find frequencies from bar areas, and estimate a grouped median and mean. Jump to the questions
Practise now
Worksheet preview and key skills
Worksheet preview
Eight GCSE Higher questions develop histogram skills from frequency density through to grouped-data estimates.
Interactive graphs let you set bar heights while the class boundaries fix each bar’s width.
What you’ll practise
- Calculate frequency density and complete grouped tables.
- Draw and complete histograms with unequal class widths.
- Use bar area to estimate frequencies and the median.
- Estimate a grouped mean using class midpoints.
Use the interactive worksheet below, or read the Topic guide for the method and worked examples.
Use frequency density and bar area to complete all eight questions. For drawing questions, drag or tap a bar and use the matching density field if needed.
Topic guide
A histogram represents grouped continuous data. Its bars touch because the horizontal scale is continuous. When class widths are unequal, the height is frequency density, not frequency.
Frequency density and area
Frequency density = frequency ÷ class width
Frequency = class width × frequency density
So the area of each bar represents its frequency. For example, suppose parcel masses are grouped as 100–120 g with frequency 30 and 120–150 g with frequency 24. The first density is 30 ÷ 20 = 1.5 and the second is 24 ÷ 30 = 0.8. Their bars have different widths and heights, but areas 30 and 24.
Drawing a histogram
- Calculate every class width from its boundaries.
- Divide each frequency by its class width.
- Put the continuous variable on the horizontal axis and frequency density on the vertical axis.
- Draw touching rectangles across the exact class boundaries at the calculated heights.
Reading all or part of a bar
A bar from 40 to 60 seconds at density 2.5 represents 20 × 2.5 = 50 observations. If estimating how many lie from 40 to 48 seconds, assume values are spread uniformly within that class: 8 × 2.5 = 20. This is an estimate because a histogram does not show where individual values lie inside a class.
Estimating the median
First recover the frequency in every bar from its area and add them to get the total N. Use the grouped median position N ÷ 2. Locate the class containing that position, then interpolate linearly within it.
For classes 0–10, 10–20 and 20–40 with frequencies 12, 18 and 40, the total is 70 and the median position is 35. There are 30 values before the 20–40 class, so the estimate is 20 + (5 ÷ 40) × 20 = 22.5.
Estimating the mean
Use each class midpoint as a representative value, multiply it by the class frequency, add these products, then divide by the total frequency:
Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency
For intervals 0–10 and 10–30 with frequencies 8 and 24, the midpoints are 5 and 20. The estimated mean is (5 × 8 + 20 × 24) ÷ 32 = 16.25. Use frequencies recovered from bar areas as the weights, never the density heights alone.
Common pitfalls
- Using frequency as the bar height instead of frequency density.
- Leaving gaps or drawing equal-width category bars.
- Reading height as frequency and forgetting to multiply by class width.
- Treating a partial-class answer as exact rather than an estimate based on a uniform spread.
Quartiles and the interquartile range can be estimated in the same way as the median, using positions N ÷ 4 and 3N ÷ 4.