Histograms
Suppose you measure how many minutes each child in a class spent reading: a long list of numbers
like 12, 7, 23, 18, 9, 31, 16, \dots A bare list is hard to make sense
of. A histogram turns it into a picture by sorting the numbers into
equal groups and showing how big each group is. The recipe is three steps:
- Bin the range of the
data into
equal-width intervals (called bins) — for example
0–10 minutes,
10–20 minutes, and so on.
- Count how many values fall in each bin.
- Draw a bar over each bin whose height is that count.
A tall bar means lots of values landed in that interval; a short bar means only a few did. The
bars sit flush against each other with no gaps, because the numbers run smoothly
from one bin straight into the next — there is no empty space between "0
to 10" and "10 to 20".
Histogram or bar chart?
A histogram looks a lot like a bar chart, but they answer different questions, and
there is one quick way to tell them apart: do the bars touch?
- A histogram shows ranges of numbers — continuous data like heights,
ages, weights or times. The bars touch, because the number line has no gaps.
Each bar covers an interval, such as "10 to
20".
- A bar chart shows separate categories — things you count but cannot
measure on a number line, like favourite fruit (apple, banana, pear). The bars have
gaps between them, because the categories don't flow into one another.
So when you read a histogram you always read an interval off the bottom axis (like
"the 10–20 band"), never a single value.
The two traps that trip people up:
- Histogram bars touch (they show number ranges). Bar-chart bars have
gaps (they show separate categories). If you see gaps, it is a bar chart, not a
histogram.
- Each bar stands for a whole interval, not one number. Read the band along the
bottom — "20 to 30" — not a single point
on it.
A fisher catches a bucket of fish and measures each one. Lengths like
9 cm, 14 cm, 11 cm
don't fall into neat groups by themselves — a fish can be any length. So we make bands:
0–10 cm,
10–20 cm,
20–30 cm. Drop each fish into its band, count
the piles, and the tallest bar shows the most common size of fish. Because length is continuous,
the bars touch.
The bin width changes the picture
Here is one fixed data set of 30 values, drawn twice. The first histogram uses a
narrow bin width of 2; the second uses a wide bin width of
5 — exactly the same numbers, only the binning differs.
The narrow bins reveal the real shape: a hump around 5–7
with a tail trailing off to the right. The wide bins flatten that story into three blunt bars —
the peak and the tail are smeared together. Too wide hides the detail;
too narrow (a width of, say, 0.5) would splinter the
data into a spiky, near-empty mess. Choosing a sensible bin width is half the craft of a good
histogram.
- A histogram bins the range, counts per bin, and draws a bar whose height is the count.
- The bars touch (no gaps) because the underlying data are continuous.
- The bin width is a choice: too wide hides detail, too narrow makes the picture spiky and noisy.
- The silhouette shows where the data pile up and how the tails stretch.
Read a histogram
Below is a histogram of how long the children in a class read for, sorted into five
10-minute bands. The bars touch and the bottom axis is labelled in
minutes. Press Refresh for a fresh class, then practise reading it:
- The tallest bar is the most common band — the time most children spent reading.
- To find how many children are in any band, read the height of that bar.
- To find the total number of children, add up the heights of all the bars.
Worked examples
Take a histogram with these four bands:
- 0–10: 3 children
- 10–20: 8 children
- 20–30: 5 children
- 30–40: 2 children
- Which band is most common? The tallest bar is over
10–20, with 8
children — so most children read for between 10 and
20 minutes.
- How many read for less than 10 minutes? That is the
first band only: 3 children.
- How many children altogether? Add every bar:
3 + 8 + 5 + 2 = 18 \text{ children.}
A grower weighs every apple from a tree and groups the weights into bands:
100–120 g,
120–140 g, and so on. The histogram's tallest
bar might sit over 120–140 g — that is the
most common apple size. Add up all the bars and you get the total number of apples picked. One
picture answers "what is a typical apple?" and "how many were there?" at the same time.