Group some data into "0–5, 5–10, 10–30" and an ordinary bar chart lies to you — that last wide group looks huge just because it's wide. A histogram fixes this by using area, not height, to show frequency. And cumulative frequency gives us a running-total curve for pinning down the median and the spread of a big data set.
Histograms — frequency density
On a histogram the bars touch (the scale is continuous), and the area of each bar equals its frequency. Since area = height × width, the height must be frequency divided by width. That height has a name: the frequency density.
The analogy: think of butter spread on toast. The frequency is how much butter; the width is how far you spread it. Frequency density is how thickly it's spread — a wide slice with the same butter looks thinner. That's why we compare density, not raw height.
Cumulative frequency
Cumulative frequency is a running total — add each group's frequency to all the ones before it, so it only ever climbs.
Plot each cumulative total against the top of its class and join the points into a smooth S-shaped curve. Because the curve reaches every value up to a point, you can read estimates straight off it.
Median, quartiles and the IQR
From a data set of n values you read off key positions on the cumulative frequency axis:
- Median at the ½n position — the middle value.
- Lower quartile (Q1) at ¼n, and upper quartile (Q3) at ¾n.
- Interquartile range (IQR) = Q3 − Q1 — a measure of spread that ignores the extremes.
| Statistic | Position (n values) | Tells you |
|---|---|---|
| Lower quartile Q1 | ¼n | bottom-quarter mark |
| Median | ½n | the middle value |
| Upper quartile Q3 | ¾n | top-quarter mark |
| IQR | Q3 − Q1 | spread of the middle 50% |
In a nutshell: histograms use area (frequency density = frequency ÷ width); cumulative frequency is a running total whose curve gives the median at ½n, quartiles at ¼n and ¾n, and the IQR = Q3 − Q1.