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

Frequency Density

What is frequency density?

• Frequency density is given by the formula

  • 

    • This formula is not on the exam formula sheet, so you need to remember it

• Frequency density is used with grouped data (i.e. data grouped by class intervals)

  • It is useful when the class intervals are of unequal width

  • It provides a measure of how dense data is within its class interval

    • relative to the width of the interval

  • For example,

    • 10 data values spread over a class interval of width 20 would have a frequency density of 

    • 20 data values spread over a class interval of width 100 would have a frequency density of 

    • As  

      • the data in the first interval is more dense (closer together) than in the second interval

      • even though the second interval has twice as many data values

How do I calculate frequency density?

• In questions it is usual to be presented with grouped data in a table

• So add two extra columns to the table

  • one to work out and write down the class width of each interval

  • the second to then work out the frequency density for each group (row)

Histograms

What is a histogram?

• A histogram looks 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 (usually of unequal width)

  • In a histogram, the area of a bar determines the frequency

    • This means it is difficult to tell anything simply from looking at a histogram

    • Some basic calculations will be needed for conclusions and comparisons to be made

  • There are no gaps between the bars

How do I draw a histogram?

• Drawing a histogram first requires the calculation of the frequency densities for each class interval (group)

  • Use  

  • Exam questions often ask you to finish an incomplete histogram, rather than start with a blank graph

• Once the frequency densities are known, the bars (rectangles) for each class interval can be drawn

  • with widths being measured on the horizontal (x) axis

  • and the height of each bar (the frequency density) being measured on the vertical (y) axis

  • As the data is continuous, the bars will be touching

How do I interpret a histogram?

• It is important to remember that the frequency density (y-) axis does not tell us frequency

  • The area of the bar is equal to the frequency

    • 

• Note that a very simple histogram may have equal class widths

  • In this case the y-axis may be labelled 'frequency' instead of 'frequency density'

  • If 'frequency' is on the y-axis, then you can use the heights of the bars to directly determine the frequencies

• You may be asked to estimate the frequency of part of a bar/class interval within a histogram

  • Find the area of the bar for the part of the interval required

  • Once area is known, frequency can be found as above

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

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