Lesson 10.1 · 20 min
Frequency Polygons
Plotting frequencies at the midpoints of each class and joining them with straight lines — and using two polygons to compare data sets.
In this lesson you will learn to…
- Draw a frequency polygon from a grouped frequency table
- Read frequencies and midpoints from a frequency polygon
- Draw two frequency polygons on one graph to compare distributions
- Know when a comparison may not be fair
Why this matters
Frequency polygons make it easy to compare distributions at a glance — for example, the heights of boys and girls, or waiting times at two hospitals — without two sets of bars getting in each other’s way.
Emma’s mistakes
Emma thinks you should plot each point at the start of the class interval and join them with a curve. Both are wrong:
- plot at the midpoint, because the midpoint represents the whole class, and
- join with straight lines, because you do not know what happens between midpoints.
Reading a polygon
To read the data back, look at each point: the -value is the midpoint and the -value is the frequency. Add the frequencies to get the total — for example, the number of cars for sale in a garage.
Fair comparisons. Two groups of very different sizes are hard to compare directly — 200 people will have taller “peaks” than 20, even if the pattern is the same. Compare the shape and position, not just the heights.
Key rules and formulas
Plot at the midpoints
Each point goes at the midpoint of its class interval, at the height of its frequency. For with frequency 8, plot .Join with straight lines
Join neighbouring points with straight lines using a ruler — not a curve. Do not join the last point back to the first.Why use one?
A frequency polygon shows the same shape as a histogram but with less clutter, so two or more can be drawn on the same axes and compared directly.
Histogram and frequency polygon
- Total frequency, Σf
- 29
- Σfx (midpoint × frequency)
- 527.5
- Estimated mean = Σfx ÷ Σf
- 18.19
- Modal class
- 15 ≤ x < 20
- Median class (value number 15)
- 15 ≤ x < 20
- Estimated range
- 25
Times taken by 29 children to travel to school. The red line joins the tops of the bars at their midpoints — that is the frequency polygon.
Worked examples
Drawing a frequency polygon
Try it yourself first, then reveal the solution one step at a time.
Comparing two groups
Try it yourself first, then reveal the solution one step at a time.
Practice: frequency polygons
8 questions
Question 1 of 8Choose all that applyFoundation Emma describes how to draw a frequency polygon. Select every correct statement.Select every correct answer — there may be more than one.
Question 2 of 8Complete the table Insect larvae lengths. Complete the points you would plot.Length (mm) Frequency Point to plot 20 ≤ l < 25 15 25 ≤ l < 30 33 30 ≤ l < 35 58 35 ≤ l < 40 50 Question 3 of 8Type the answer Larvae: –: 15, –: 33, –: 58, –: 50, –: 4. How many larvae were measured?Question 4 of 8Explore and match Horses sold at a market: – kg: 16, –: 130, –: 42, –: 12. Set the polygon to match. (The slider maximum is 140.)Move the controls until you match the question, then press Check.
- Total frequency, Σf
- 0
- Σfx (midpoint × frequency)
- 0
- Estimated mean = Σfx ÷ Σf
- —
- Modal class
- —
- Median class (value number 0.5)
- —
- Estimated range
- 200
Question 5 of 8Type the answer What is the mid-interval value of the class ?Question 6 of 8Multiple choice Students' travel times peak at – minutes; teachers' at – minutes. Which conclusion is best?Question 7 of 8True or false You should join the last point of a frequency polygon back to the first to close the shape.Question 8 of 8Type the answer A frequency polygon of 40 cars' prices has points at , , , and one more at . What is the frequency at ?