A *frequency polygon* is another way to show
the information in a frequency table. It looks a little bit like a line
graph. To make a frequency polygon, you just need to plot a few points and
then join the points by straight lines. So what points do you need to plot?
Well, first you have to find the *midpoints* of each class. The midpoint
of a class is the point *in the middle* of the class. So for instance, if
I have a class “10 – 19”, then the midpoint is 14.5. A class of
“0 – 5” has a midpoint of 2.5.

So say we have the frequency table from the earlier example:

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Mark Class |
Frequency |

1-5 |
1 |

6-10 |
5 |

11-15 |
8 |

16-20 |
6 |

What people usually do is add an extra column showing the midpoints of each class like this:

Mark Class |
Frequency |
Midpoint |

1-5 |
1 |
3 |

6-10 |
5 |
8 |

11-15 |
8 |
13 |

16-20 |
6 |
18 |

Now what we’re going to do is plot a graph showing the frequency for each midpoint. For instance, the frequency for the midpoint value 8 is 5. The midpoint values are shown along the horizontal axis, and the frequency values are shown along the vertical axis like for a histogram:

The reason it’s called a polygon is because the line sort of forms a plane shape with the horizontal axis as one side of the shape:

Now a frequency polygon and a histogram both show
the same information, but in a different way. Why would we want to draw a
frequency polygon instead of a histogram? Well, the most common reason is if
we want to *compare* two different sets of data. For instance, say we had
the exam marks for another class, also with 20 students, let’s call this other
group of students Class B. Class A will be the first class of students we
looked at.

Class B |
||

Mark Class |
Frequency |
Midpoint |

1-5 |
1 |
3 |

6-10 |
3 |
8 |

11-15 |
9 |
13 |

16-20 |
7 |
18 |

Don’t get confused by all the use of the word
‘class’. In this question, the word has two meanings. We are using it to
describe both the two groups of school kids – Class A and Class B, but also the
*mark classes *we’re sorting the data into: 1 – 5, 6 – 10, 11 – 15, 16 – 20.

We could plot this frequency table on the same set
of axes as we used for our first class. We’d have to make sure it was easy for
the reader to tell the difference between the two lines. You may want to use a
different colour pen for each line, one line could be blue and one line could
be red. Or you could do what we do here (since we haven’t got any colour), and
make one line a *solid* line, and the other a *dashed* line. It also
helps if you label the two lines as well. You can also use different symbols
for the points you plot – you could plot one line with crosses, and one line
with circles for instance.

Now in an exam you might get a question like this:

Comment on the marks obtained by Class A and Class B |

Solution |

That’s a pretty general question – what the heck
do they want us to do? Well, you’ve got to think about the So, the bigger the mark, the better a student has done right? A high mark out of 20 is good, and a low mark out of 20 is bad! Pretty simple stuff. Okay, so let’s look at the frequency polygon. Notice how Class A had more students with low marks – the dashed line (representing Class A) is higher than the solid line (representing Class B) in the left hand area of the graph. This tells us that Class A has more students who performed poorly than Class B. What about on the right hand side of the graph, in the higher marks section? Well, in this section the solid line (representing Class B) is higher than the dashed line (representing Class A). This tells us that Class B had more students with high marks on the exam. So all up this tells us what? Well, Class B has From looking at the frequency polygons, Class A has more students who scored a low mark on the exam than Class B. The graph also tells us that Class B has more students who scored a high mark on the exam than Class A. This tells us that Class B performed better on the exam than Class A. |

Handy Hint #1 - When can you compare frequency polygons

Notice how the two classes we compared
in the last section both had the *same number of students* in them – 20 in
each. This makes it very easy to compare how the two classes went on their
exam. It becomes a little more difficult to compare the two school classes if
they’re different sizes. For instance, if Class B had 30 students in it, the
overall graph might have looked like this:

Now, the entire line representing
Class B is above Class A. This tells us that Class B had more students who
bombed on the exam *and* more students who went well on the exam than
Class A. But this is partly due to Class B just having more students than Class
A. So which is the better class? Well, it’s not that easy to tell from the
graph. Another way to analyse and compare the two classes is to look at the mean,
median and mode of the data. The next section shows how to calculate these
from a frequency table.