MISSION · Data and probability
Histograms and Frequency Tables
The chart for measurements, and why bar charts aren't it
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A histogram groups numerical data into intervals and shows how many fall in each, which is how you see the shape of a set of measurements.
Not a bar chart
They look similar and answer different questions.
A bar chart shows separate categories — favourite colours, transport to school. The order of the bars is arbitrary and there are gaps between them.
A histogram shows a continuous number line cut into intervals — heights, times, masses. The order is fixed, and the bars touch, because there is no gap between “160–165 cm” and “165–170 cm”.
That touching is the visual signal. Gaps mean categories; no gaps mean a measured quantity.
Frequency tables
The table comes first. Choose intervals, tally how many values land in each, and you have the frequencies the histogram draws.
| Height (cm) | Frequency |
|---|---|
| 150–160 | 4 |
| 160–170 | 11 |
| 170–180 | 8 |
| 180–190 | 2 |
Intervals must not overlap and must not leave gaps, which is why the boundaries need a rule. “\(150 \le h < 160\)” puts a value of exactly 160 in the next group, and every real data set needs that decision made before counting starts.
Grouping loses information — once it’s a table you no longer know the individual heights — but it buys you the shape, which raw numbers hide.
Interval width changes the picture
This is the thing to play with. The same data, grouped differently, can look completely different.
Too wide and everything collapses into two or three bars that show nothing. Too narrow and each bar holds one or two values, so the chart is spiky noise. Somewhere in between, the real shape appears.
There is no formula for the right width — it’s a judgement, and it’s why looking at a histogram with one grouping isn’t enough to trust its shape. Five to fifteen intervals is a reasonable starting range.
Reading the shape
- Symmetric — the bell shape, values clustering in the middle.
- Right-skewed — a long tail of high values. Incomes, house prices, waiting times.
- Left-skewed — a long tail of low values.
- Bimodal — two peaks, which usually means two groups have been mixed together. The heights of a mixed-age class, or two machines with different settings.
Spotting bimodality is a genuine reason to draw the histogram rather than just calculate a mean, because a mean sits in the empty valley between the peaks and describes nobody.
Common mix-ups
- Leaving gaps between bars. That makes it a bar chart, claiming the data is categorical.
- Overlapping intervals, so some values could be counted twice.
- Trusting one grouping. Change the width and look again.
- Reading a bar’s height as a value rather than as a count of values.
Check yourself
Why do histogram bars touch when bar chart bars don’t? And what does a two-peaked histogram usually suggest?
Because a histogram’s horizontal axis is a continuous number line with no gaps between intervals, while a bar chart’s categories are separate. Two peaks usually mean two different groups have been combined into one data set.
LEVEL 3 · REFLECT
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