Frequency Polygons
A histogram
draws a solid bar over each class of grouped data. A frequency
polygon tells the same story with a single wiggly line instead. The recipe is
simple:
- For each class, find its midpoint — the value halfway between the lower
and upper boundary of the interval.
- Plot a point at that midpoint, at a height equal to the class's
frequency.
- Join the points, in order, with straight lines.
That joined-up line is the frequency polygon. Because it is just a line, it shows the overall
shape of a distribution at a glance — where the data pile up, and how the
tails trail off — and, best of all, you can lay two polygons on the same axes
to compare distributions, something a pair of histograms does clumsily.
Picture a histogram of fish lengths. Put a dot at the middle of the top of
each bar — right above the class midpoint, at the bar's height — and connect the dots. You
have drawn a frequency polygon straight from the histogram. The polygon is the histogram's
"skyline", so a tall bar becomes a high point on the line and a short bar a low one.
Building one from a table
A grower weighs the apples off a tree and groups the weights into five
10-gram classes. To draw the polygon we only need two columns: the
midpoint of each class, and its frequency.
| Class (g) |
Midpoint x |
Frequency f |
| 0–10 | 5 | 2 |
| 10–20 | 15 | 5 |
| 20–30 | 25 | 9 |
| 30–40 | 35 | 6 |
| 40–50 | 45 | 3 |
The midpoint of 0–10 is
\frac{0 + 10}{2} = 5, of
10–20 is 15,
and so on — the very same class midpoints used to find the
estimated mean.
That gives five points to plot,
(5, 2), (15, 5),
(25, 9), (35, 6) and
(45, 3) — then we join them up. Press Play to add
the points one class at a time:
The line climbs to a peak over the 20–30
class (midpoint 25) and then falls away — most apples weigh about
25 grams, with fewer very light or very heavy ones.
Comparing two distributions
The real power of a frequency polygon shows when you draw two on the same axes.
Here are the apples from Tree A (the table above) and a second
Tree B, whose apples run lighter:
Both trees produced 25 apples, so the polygons are directly
comparable. Tree B's line peaks further left, over the lighter
10–20 class, while Tree A peaks over
20–30. In one glance you can see that
Tree A grows heavier apples — a comparison that two separate histograms would
make you work for.
The traps that catch people out with frequency polygons:
- Plot each point at the class midpoint — not at the lower
boundary, the upper boundary, or the end of the interval. The midpoint of
20–30 is
25, so the point sits at 25.
- Frequency goes up the vertical axis; the measured
quantity (weight, height, time…) goes along the horizontal axis.
- Join the points with straight lines, not a smooth curve, and only when
the data are grouped and continuous.
Two schools measure their pupils' heights and group them into
10-centimetre bands. Overlay the two frequency polygons and the
answer to "which school is taller on average?" jumps out: whichever line's peak sits further
to the right along the height axis. The polygon turns a comparison of two
long tables into a single picture your eye can read in a second.
Read a frequency polygon
Below is a frequency polygon for five equal classes, drawn with its histogram bars faint behind
it so you can see the points sit on the midpoint of each bar's top. Press
Refresh for fresh data, then practise:
- The highest point marks the most common class — read its midpoint off the
bottom axis.
- To read the frequency of a class, find its midpoint, go up to the line,
and read across to the frequency axis.
- The heights of the points are just the class frequencies, so they add up to the total.