Histograms & Frequency Polygons (Edexcel GCSE Statistics)

Frequency Polygons

What is a frequency polygon?

• Frequency polygons are a very simple way of showing frequencies for continuous, grouped data

  • They give a quick guide to how frequencies change from one class interval to the next

What are the key features of a frequency polygon?

• Apart from plotting and joining up points with straight lines there are 3 rules for frequency polygons:

  • Plot points at the midpoint of class intervals

  • Unless one of the frequencies is 0 do not join the frequency polygon to the x-axis

  • Do not join the first point to the last one

• The result is not actually a polygon

  • It's more of an 'open' polygon that ‘floats’ in mid-air!

• You may be asked to draw a frequency polygon and/or use it to make comments and compare data

How do I draw a frequency polygon?

• This is most easily shown by an example

  • e.g.  The lengths of 59 songs, in seconds, are recorded in the table below

• Frequencies are plotted at the midpoints of the class intervals, so in this case we would plot the points (135, 4), (165, 10), (195, 24), (225, 18) and (255, 3).
  Join these up with straight lines (but do not join the last to the first!)

• If you have a histogram with equal class widths

  • then a frequency polygon can be drawn

    • by marking the points at the centres of the tops of all the histogram bars

    • and joining those points together to make the 'polygon'

How do I use and interpret a frequency polygon?

• Think about what you could you say about the data above, particularly by looking at the diagram only

  • The two things to look for are averages and spread

    • The modal class is 180 ≤ t < 210

      • Because the graph reaches its highest point there

      • It would be acceptable to say that 195 seconds is (an estimate of) the modal song length

    • The diagram (rather than the table) shows (an estimate of) the range of song lengths is 255 – 135 = 120 seconds

      • i.e. midpoint of highest class interval minus midpoint of lowest class interval

  • If 2 frequency polygons are drawn on the same graph comparisons between the 2 sets of data can be made

Histograms

What is a histogram?

• A histogram is similar to a bar chart, but there are important differences

• Bar charts are used for discrete (and possibly non-numerical) data

  • In a bar chart, the height (or length) of a bar determines the frequency

  • There are usually gaps between the bars

• Histograms are used with continuous data, grouped into class intervals

  • If the class intervals are of equal width then the height of a bar indicates the frequency

  • But there are no gaps between the bars

• Note that the situation is much more complex for histograms with unequal class widths

  • But these are not included on your Foundation course

How do I draw a histogram?

• If you know the frequencies and class boundaries, then the bars (rectangles) for each class interval can be drawn

  • The class boundaries are marked on the horizontal (x) axis

  • The height of each bar (the frequency) is measured on the vertical (y) axis

  • As the data is continuous, the bars will be touching (no gaps)

What else can I be asked to do with a histogram?

• You may be asked to complete an incomplete histogram using information from a grouped frequency table

  • Just draw in the missing bars based on the information in the table

• You may be asked to complete an incomplete grouped frequency table using information from a histogram

  • Use the information from the histogram bars to complete the table

    • The class boundaries can be read off the horizontal axis

    • The frequencies can be read off the vertical axis

• You may be asked to draw a frequency polygon on the histogram

  • Mark the points at the centres of the tops of all the histogram bars

  • Join those points together to make the 'polygon'

• You may be asked to comment on the skew shown on a histogram

  • See the 'Skewness' revision note

• You can also use histograms for two data sets to compare the data distributions

  • but only if they have the same class intervals and the same frequency scales