Histograms (AQA A Level Maths): Revision Note

Histograms

What is a histogram?

• A histogram is similar to a bar chart but with some key differences

  • A histogram is for displaying grouped continuous data whereas a bar chart is for discrete or qualitative data

  • There will never be any gaps between the bars of adjacent groups in a histogram

  • Whilst in a bar chart the frequency is read from the height of the bar, in a histogram the height of the bar is the frequency density

• On a histogram frequency density is plotted on the y – axis

  • This allows a histogram to be plotted for unequal class intervals

  • It is particularly useful if data is spread out at either or both ends

• The area of each bar on a histogram will be proportional to the frequency in that class

What are the key features of a histogram?

• You will not be asked to draw a histogram but you may have to add information to one so you should make sure you are familiar with the process for drawing one

• Step 1.  Always check that there are no gaps between the upper boundary of a class and the lower boundary of the next class

  • If there are gaps you will need to close them by changing the boundaries before carrying out any calculations

    • Consider whether the values are rounded or truncated before closing the gaps

• Step 2.  Find the class width of each group by subtracting the lower boundary from the upper boundary

• Step 3.  Calculate the frequency density for each group using the formula:

• Step 4.  The histogram will be drawn with the data values on the x – axis and frequency density on the y – axis

  • Remember that the scale on both axes must be even, although the class widths may be uneven

    • Both axes should be clearly labelled and units included on the x – axis

  • Most often, the bars will have different widths

• Occasionally you will be asked to add a frequency polygon to the histogram

  • This is done by joining up the midpoints at the top of each bar

  • You should not join up the first or last midpoint to the x – axis (it is not really a polygon!)

How do I interpret a histogram?

• It is important to remember that the y – axis does not tell us the frequency of each bar in the histogram

• The area of the bar gives information about the frequency

  • Most of the time, the frequency will be the area of the bar and is found by multiplying the class width by the frequency density

  • Occasionally, the frequency will be proportional to the area of the bar

    • Frequency =

    • In these cases more information will be given to help you find the value of k

• You may be asked to find the frequency of part of a bar within a histogram

  • Find the area of that section of the bar using any information you have already found out

  • You will need to have found the value of k first

What are frequency polygons?

• Frequency polygons are a very simple way of showing frequencies/frequency densities for continuous, grouped data and give a quick guide to how frequencies change from one class to the next

What do I need to know about frequency polygons?

• Apart from plotting and joining up points with straight lines there are 2 rules for frequency polygons:

  • Plot points at the MIDPOINT of class intervals

  • Unless one of the frequencies/densities is 0 do not join the frequency polygon to the x-axis, and do not join the first point to the last one

• The result is not actually a polygon but more of an open one that ‘floats’ in mid-air!

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