Histograms (AQA AS Maths: Statistics)

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 we 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?

• 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