CIE A Level Maths: Probability & Statistics 1

Revision Notes

1.2.4 Histograms

Test Yourself

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

How do I draw a histogram?

  • 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:

frequency space density equals fraction numerator frequency over denominator class space width end fraction

  • 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

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 frequency of a class is found by
      • begin mathsize 16px style Frequency space equals space Frequency space Density space cross times space Class space Width end style
  • 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

Worked example

The table below and its corresponding histogram show the mass, in kg, of some new born bottlenose dolphins.

Mass, m kg

Frequency

 4 less or equal than m less than 8

4

 8 less or equal than m less than 10

15

 10 less or equal than m less than 12

19

 12 less or equal than m less than 15

9

 15 less or equal than m less than 30

6

 1-2-4---histogram-we-diagram

(a)
Complete the histogram.

 

(b)
Estimate the number of dolphins whose weight is greater than 13 kg.

1-2-4-histogram-we-solution-1

Exam Tip

  • Look carefully at the scales on the axes, it will rarely be a simple 1 unit to 1 square.

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.