Frequency Polygons
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Revision notes for Edexcel GCSE Maths Frequency Polygons. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Frequency Polygons

What you'll learn

  • How to read grouped data from a frequency table.
  • How to find the midpoint of each group.
  • How to plot points for a frequency polygon.
  • How to choose a sensible scale when the axes are blank.

1. Reading grouped frequency tables

In this topic, the data is usually already sorted into groups. For example, a table might show how many people took between 10 and 20 minutes, between 20 and 30 minutes, and so on.

A letter such as ttt, www or hhh is a variable. It stands for the measurement in the question, such as time, weight or height.

Definition

Frequency and class intervals

  • Frequency means how many values are in a group.
  • A class interval is one range of values, such as 20<t≤3020 < t \leq 3020<t≤30. This means “more than 20, up to and including 30”.
Example

Reading a grouped table

A table about 20 runners gives these groups:

Grouped frequency table showing each time interval and the number of runners in that interval.

  • 15<t≤2015 < t \leq 2015<t≤20: frequency 4
  • 20<t≤2520 < t \leq 2520<t≤25: frequency 5
  • 25<t≤3025 < t \leq 3025<t≤30: frequency 8
  • 30<t≤3530 < t \leq 3530<t≤35: frequency 3

  1. The variable ttt stands for time in minutes.

  2. The interval 15<t≤2015 < t \leq 2015<t≤20 means times greater than 15 minutes and up to 20 minutes.

  3. The frequency 4 means 4 runners were in that first time group.

  4. Check the total number of runners by adding the frequencies: 4 + 5 + 8 + 3 = 20.

2. Finding midpoints

A frequency polygon uses one point for each class interval. The horizontal position of that point is the middle of the interval.

The two numbers at the ends of a class interval are called the ends or boundaries.

Definition

Midpoint

The midpoint is the value halfway between the two ends of a class interval. Find it by adding the two ends and dividing by 2: midpoint=lower end+upper end2\text{midpoint}=\frac{\text{lower end}+\text{upper end}}{2}midpoint=2lower end+upper end​.

The midpoint of a class interval is halfway between the lower and upper ends.

Example

Finding midpoints for weights

A table about cat weights has these groups:

Each weight interval is represented by its midpoint before plotting a frequency polygon.

  • 0<w≤10 < w \leq 10<w≤1: frequency 6
  • 1<w≤21 < w \leq 21<w≤2: frequency 9
  • 2<w≤32 < w \leq 32<w≤3: frequency 18
  • 3<w≤43 < w \leq 43<w≤4: frequency 15
  • 4<w≤54 < w \leq 54<w≤5: frequency 10
  • 5<w≤65 < w \leq 65<w≤6: frequency 7

  1. For 0<w≤10 < w \leq 10<w≤1, the midpoint is 0+12=0.5\frac{0+1}{2}=0.520+1​=0.5.

  2. For 1<w≤21 < w \leq 21<w≤2, the midpoint is 1+22=1.5\frac{1+2}{2}=1.521+2​=1.5.

  3. Keep going in the same way to get the midpoints: 0.5, 1.5, 2.5, 3.5, 4.5 and 5.5.

  4. The frequencies stay the same. You only change each interval into its midpoint.

Common Mistake

Plotting the class ends

For the interval 20<t≤3020 < t \leq 3020<t≤30, the midpoint is 25. Do not plot at 20 or 30.

3. Plotting the points

A frequency polygon is drawn on a pair of axes.

Definition

Axes and coordinates

  • The x-axis is the horizontal axis.
  • The y-axis is the vertical axis.
  • A coordinate tells you where to plot a point. For a frequency polygon, use: midpoint first, frequency second.

So if a group has midpoint 25 and frequency 18, you plot the point (25, 18).

For a frequency polygon, the x-coordinate is the midpoint and the y-coordinate is the frequency.

Example

Plotting travel-time points

A table shows how long people took to travel to an event:

The travel-time data is plotted as midpoint-frequency points before the points are joined.

  • 0<t≤100 < t \leq 100<t≤10: frequency 12
  • 10<t≤2010 < t \leq 2010<t≤20: frequency 16
  • 20<t≤3020 < t \leq 3020<t≤30: frequency 23
  • 30<t≤4030 < t \leq 4030<t≤40: frequency 28
  • 40<t≤5040 < t \leq 5040<t≤50: frequency 13
  • 50<t≤6050 < t \leq 6050<t≤60: frequency 8

  1. Find the midpoints: 5, 15, 25, 35, 45 and 55.

  2. Pair each midpoint with its frequency: (5, 12), (15, 16), (25, 23), (35, 28), (45, 13), (55, 8).

  3. The x-axis should show time from 0 to 60 minutes.

  4. The y-axis should show frequency from 0 to at least 28.

  5. Plot each point carefully on the grid.

Tip

Use the grid lines

Check the small grid squares before plotting. For example, if the y-axis goes up in 2s, then frequency 14 is two small squares above 10.

4. Joining to make the frequency polygon

A polygon is a shape made from straight line pieces. A line segment is one straight piece of line between two points.

Definition

Frequency polygon

A frequency polygon is a graph made by plotting the midpoint-frequency points, then joining neighbouring points with straight line segments.

Key Idea

The whole method

Find the midpoint, plot the frequency, then join the points with straight lines.

Example

Drawing a frequency polygon for speeds

A table gives these car speed groups:

A frequency polygon is made by joining neighbouring midpoint-frequency points with straight line segments.

  • 0<s≤200 < s \leq 200<s≤20: frequency 5
  • 20<s≤4020 < s \leq 4020<s≤40: frequency 16
  • 40<s≤6040 < s \leq 6040<s≤60: frequency 30
  • 60<s≤8060 < s \leq 8060<s≤80: frequency 24
  • 80<s≤10080 < s \leq 10080<s≤100: frequency 18
  • 100<s≤120100 < s \leq 120100<s≤120: frequency 7

  1. Find the midpoints: 10, 30, 50, 70, 90 and 110.

  2. Make the coordinates: (10, 5), (30, 16), (50, 30), (70, 24), (90, 18), (110, 7).

  3. Plot all six points on the grid.

  4. Use a ruler to join the points from left to right.

  5. Do not draw bars. The straight joined lines are the frequency polygon.

Common Mistake

Joining to the origin

Do not draw separate lines from each point to (0, 0). Join each plotted point to the next plotted point.

The points should be joined to their neighbours, not separately back to the origin.

5. Choosing a scale when the axes are blank

Sometimes the grid is given, but the numbers are not written on the axes. You must choose a sensible scale.

Definition

Scale

A scale is the way numbers increase along an axis, such as going up in 5s, 10s or 20s.

The vertical axis for frequency should usually start at 0. The horizontal axis does not always need to start at 0 if all the data is much larger, such as heights from 140 cm to 190 cm.

Example

Choosing axes for heights

A table gives heights of plants:

When axes are blank, choose scales that fit all midpoints and frequencies clearly.

  • 140<h≤150140 < h \leq 150140<h≤150: frequency 6
  • 150<h≤160150 < h \leq 160150<h≤160: frequency 21
  • 160<h≤170160 < h \leq 170160<h≤170: frequency 32
  • 170<h≤180170 < h \leq 180170<h≤180: frequency 27
  • 180<h≤190180 < h \leq 190180<h≤190: frequency 14

  1. Label the x-axis Height (cm).

  2. Label the y-axis Frequency.

  3. Choose an x-axis scale from 140 to 190, going up in 10s.

  4. Choose a y-axis scale from 0 to 35, going up in 5s, because the largest frequency is 32.

  5. Find the midpoints: 145, 155, 165, 175 and 185.

  6. Plot the points: (145, 6), (155, 21), (165, 32), (175, 27), (185, 14).

  7. Join the points with straight line segments using a ruler.

Tip

Scale sanity check

Before plotting, check that your biggest midpoint and your biggest frequency both fit on the grid.

Exam technique

In the exam

  1. Find every midpoint before you start plotting.

  2. Check the axes: label the variable and units, and make sure the frequency axis starts at 0.

  3. Plot small, accurate points, then join neighbouring points with straight ruler lines.

Self review

Check yourself

  • What midpoint would you plot for 30<t≤4030 < t \leq 4030<t≤40?
  • In a frequency polygon, what goes on the x-axis and what goes on the y-axis?
  • If the largest frequency is 29, would a y-axis ending at 20 be suitable?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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