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This page sets out the method for frequency polygons in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Frequency polygons topic page.
When you see a grouped frequency table and 'draw a frequency polygon', add a midpoint column first.
When you see two polygons on one grid and 'compare', make one statement about the peak and one about the spread, in context.
For a class 0 to 10, the midpoint is 5.
Plot the frequency at that midpoint.
The lines between points are straight.
They are not a curve of best fit.
The height of a point is the frequency of that class.
Add the heights if you need the total frequency.
Two polygons on the same axes can be compared by eye.
The higher point is the higher frequency at that midpoint.
Sometimes the polygon is closed down to the axis at a midpoint before the first class and after the last.
That does not add a new frequency.
The polygon uses points and lines. A bar chart uses bars.
Do not read the line between points as extra data.
A class such as 10 to 20 contains many possible values. One point cannot sit on all of them, so you choose the middle. Add the lower end and the upper end, then divide by 2. (10 + 20) ÷ 2 = 15. The same rule works when the numbers are less friendly. A class from 14 to 22 has midpoint (14 + 22) ÷ 2 = 18.
Plotting at the end of the class shifts every point to the right. The shape of the polygon changes, and a comparison with a correct polygon becomes unfair. The end of the class is the boundary, not the typical value. Halving the sum of the ends is what puts the point in the middle. Check by seeing that the midpoint is the same distance from both ends.
Method: 1. Note both ends of the class interval. 2. Add those two ends. 3. Divide the sum by 2. That is the midpoint. 4. The plotted point should sit halfway along the interval.
The horizontal coordinate is the midpoint. The vertical coordinate is the frequency. A class from 20 to 30 with frequency 8 is the point (25, 8). You do not plot (20, 8) and you do not plot (30, 8). Once the points are plotted, join them in order of the midpoints with straight lines.
Sometimes the polygon is closed down to the axis at a midpoint before the first class and after the last class. Those extra points have frequency 0. They tidy the picture. They are not new classes full of people. Do not add them into a total frequency.
Method: 1. Work out the midpoint of each class. 2. Plot the frequency at that midpoint. 3. Join the points with straight lines, in order. 4. Leave a zero-frequency end point out of the total.
The total frequency is the sum of the frequencies, which are the heights. If the points have heights 8 and 13, the total of those two classes is 8 + 13 = 21. The midpoints 15 and 25 tell you where the points sit. Adding 15 and 25 gives 40, which is not a number of people.
Read the question carefully when several classes are listed. "These two classes" means add those two heights only. "The whole polygon" means every class. A point you skip is a group of people left out of the total.
Method: 1. List the frequency of each class you were asked about. 2. Add those frequencies. 3. Do not add the midpoints. 4. Do not read a value off the line between two points. Only the plotted points are data.
Two polygons on the same axes can be compared at the same midpoint. If group A has frequency 4 at midpoint 5, and group B has frequency 6 there, group B has 2 more in that class. Do this at each midpoint the question names, or add each group and subtract the totals.
A higher point means a higher frequency only when both polygons use the same vertical scale. This lesson prints the frequencies, so you can add them directly. The midpoints must match before you say one class is larger. Comparing midpoint 15 on one graph with midpoint 25 on the other is a comparison of different classes.
Method: 1. Read the frequency of each group at the same midpoint. 2. Add the frequencies inside each group. 3. Subtract the two totals for a difference. 4. Do not add the midpoints, and do not compare different midpoints as if they were the same class.
A grouped frequency table gives classes such as 20 < t ≤ 30 and a frequency for each. To plot a frequency polygon you need a coordinate for every class. The first number is the midpoint of the class and the second is the frequency. For 20 < t ≤ 30 with frequency 14 the point is (25, 14) because (20 + 30) ÷ 2 = 25.
Write all the coordinates in a list before you touch the graph. Check each midpoint is the same distance from both ends of its class. The classes do not need whole number midpoints: a class 2.5 < y ≤ 3.5 has midpoint 3 and a class 0.4 < m ≤ 0.8 has midpoint 0.6. A list of coordinates also stops you plotting the second class in the position of the third.
Method: 1. For each class, add the two ends and halve to get the midpoint. 2. Write the frequency beside it as a coordinate (midpoint, frequency). 3. Check that the midpoints go up in equal steps if the classes are equal. 4. Plot all the points before you join any of them.
Tip: The horizontal coordinate is a midpoint. The vertical coordinate is the frequency. Never swap them.
Mark each point with a small cross, then join neighbouring points with straight lines and a ruler. Do not draw a smooth curve, do not draw bars, and do not stop plotting early. You do not usually join the first or last point to the horizontal axis unless the question tells you to. Check the vertical scale before plotting: if the grid goes up in 5s, then a frequency of 12 is a little more than halfway between 10 and 15.
If you have to draw the polygon yourself, mark the axes with a title and use the same scale as the labels given. The graph is marked on accuracy of the points, then on joining them with straight lines. A point plotted at the end of a class, such as (30, 14) instead of (25, 14), shifts the whole polygon to the right and loses the accuracy marks.
Method: 1. Read the vertical scale and find the value of one small square. 2. Plot each point at (midpoint, frequency) with a small neat cross. 3. Join the points in order with a ruler. No curves and no bars. 4. Check each point against your list of coordinates.
A frequency polygon is a picture of a frequency table, so you can rebuild the table from it. Each point gives the frequency of one class. The total frequency is the sum of the heights. To find which class holds the median, halve the total frequency and add up the heights from the left until you pass that position. For 40 people the median is between the 20th and 21st values.
Work out each class from its midpoint and the class width. A midpoint of 35 with width 10 gives the class from 30 to 40. Fractions follow the usual rule: the frequency you want over the total frequency, in its simplest form. Never include the midpoints in a total. They tell you where a class is, not how many people are in it.
Method: 1. Read the frequency of every class from the vertical axis. 2. Add the frequencies to get the total. 3. For the median class, halve the total and add frequencies from the left until you reach it. 4. For a fraction, write the frequency you want over the total and simplify.
A common exam question shows a polygon someone else drew and asks for two mistakes. There are only a handful to look for. The points may be plotted at the ends of the classes instead of at the midpoints. The points may be joined with a curve, or drawn as bars. The points may be plotted at the wrong height because the scale was read wrongly, or the last point may be missing.
Compare the plotted points with the table. Work out what each coordinate should be, then see where the drawing differs. Describe the mistake in one clear sentence: "the points are plotted at the upper end of each class instead of at the midpoint". Two different mistakes earn two marks. Saying the same fault twice in different words only earns one.
Method: 1. Work out the correct coordinates from the table. 2. Compare each plotted point with the correct one and note any pattern. 3. Check how the points are joined: straight lines, not a curve or bars. 4. Write two different mistakes as short sentences.
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