Frequency Polygons

A frequency polygon is a graph that shows the information in a grouped frequency table. Each class is shown by one point, and the points are joined with straight lines.

Each point goes above the middle of its class, at the height of its frequency:

(\text{midpoint of the class},\ \text{frequency})

In the exam you are usually given a grouped frequency table and asked to draw a frequency polygon on a grid, for 2 marks. You may also be asked to read frequencies from a polygon, to find the total or the class that contains the median, to compare two polygons, or to write down two mistakes in a polygon that someone else has drawn.

The midpoint of a class is the number exactly halfway between its two ends. Add the two ends and divide by 2:

\text{midpoint} = \frac{\text{lower end} + \text{upper end}}{2}

For the class 30 < h \le 40, the midpoint is \frac{30+40}{2} = 35. For the class 35 < w \le 40, it is \frac{35+40}{2} = 37.5, which is not a whole number. That is fine: plot it halfway between 35 and 40

When the classes are all the same width, the midpoints go up by that width each time. With the classes 0 < t \le 10, 10 < t \le 20 and 20 < t \le 30, the midpoints are 5, 15 and 25

• Work out the midpoint of each class.
• Plot a point for each class at (midpoint, frequency), with a small cross or dot.
• Join the points in order, from left to right, with straight lines drawn with a ruler.

Start at the first point and stop at the last one. Don't join the last point back to the first, and don't draw lines down to the axis at the ends. Don't draw bars either: a frequency polygon is just the points and the lines between them.

Plot each point carefully. In the exam a point counts if it is within half a small square of the right place, so work out what one small square is worth on each axis first.

Work out the scale first. Count the small squares between two numbers on the frequency axis, and work out what one small square is worth.

Each point stands for one class. Read down from a point to find its midpoint, which tells you its class. Read across to the frequency axis to find its frequency.

• The total is all the frequencies added together.
• The modal class is the class with the highest point.
• For "more than 30 minutes", add the frequencies of the classes above 30, when 30 is the end of a class.
• If you know the total, a missing frequency is the total minus the frequencies you can read.

The median is the middle value. If there are n values, it is the value in position

\frac{n+1}{2}

Add up the frequencies class by class until you pass that position. The class you have reached is the class that contains the median.

Two frequency polygons can be drawn on the same grid, with a key to show which is which. Read each polygon on its own, using the key.

A polygon that is further to the right shows bigger values on the whole. Compare the modal classes as well. Then say what the comparison means in the question's own words, such as "Group B tended to take longer than Group A".

A higher polygon may just mean that group has more people. When the two groups are different sizes, compare the fraction of each group, not the number of people.

Some questions show a frequency polygon that someone else has drawn and ask you to write down two mistakes. Check the polygon against the table one point at a time.

• Is each point above the midpoint of its class?
• Is each point at the right height?
• Does the frequency scale go up in equal steps, with no number missed out?
• Are the points joined in order with straight lines?
• Is the last point left unjoined to the first?

Say exactly what is wrong and where, such as "the point for 10 < m \le 15 is plotted at 15 instead of 12.5". A vague answer, such as "the graph is wrong", does not get the mark.

Two answers that examiners often see are not mistakes. The horizontal axis does not have to start at 0, and the first and last points should not be joined.

• Plotting at the ends of the classes. Plot each point above the midpoint of its class.
• Joining the points with a curve. Join them with straight lines, drawn with a ruler.
• Joining the last point to the first point. Stop at the last point: a frequency polygon is not closed up.
• Drawing bars instead of points. Plot one point for each class and join the points with lines.
• Misreading the scale. Work out what one small square is worth before you plot or read a point.
• Giving a frequency instead of a class. For the modal class or the class with the median, write down the class interval.