What are frequency trees?

• Frequency trees allow us to consider the frequencies associated with  two characteristics of a set of data
• Frequency trees are usually used when each characteristic has only two possible outcomes of interest
  • For example, passing or failing a driving test
    • which would be one characteristic
    • male or female could be the outcomes of a second characteristic
  • A frequency tree would allow us to quickly see, say, how many females passed their driving test
• The total frequency appears in a 'bubble' at the start of a frequency tree
  • The first set of branches then break down the total frequency into the two frequencies for the two outcomes of the first characteristic
    • These are written in bubbles at the end of the first set of branches
  • The second set of branches would then further break down each of the frequencies from the first set of branches
• Strictly speaking it does not matter which set of branches has which characteristic
  • However the information given in problems usually lends itself to one way rather than the other
  • You will usually be given a frequency tree to complete in the exam
• It is possible to have three, or more, characteristics in a frequency tree by adding more sets of branches
  • However such diagrams would quickly become large and cumbersome
• For situations with more than two options of interest for a characteristic, two-way tables are more useful

How do I draw a frequency tree?

• If drawing a frequency tree from scratch
  
  • identify the two characteristics
  • decide which characteristic to put on the first set of branches and which to put on the second set of branches
  • remember to include a 'bubble' at the start for the total frequency and a 'bubble' at the end of each branch

• Notice how the outcomes for the second characteristic are repeated for each of the outcomes from the first characteristic

How do I complete a frequency tree?

• To complete an empty or partially complete frequency tree from information given in words
  • work your way through each sentence in the question
  • fill in any values you can directly from the question
    • look for information about the total frequency (a sentence containing this information may not mention either of the characteristics)
    • think about whether each statement is talking about a single characteristic, or both
  • the rest of the frequencies should be able to be filled in by adding or subtracting as appropriate
    • in general the values decrease from left to right as the total frequency is broken down

How do I find probabilities from a frequency tree?

• Very similar to two-way tables, this is a matter of picking the appropriate numbers from the diagram
• Start by converting the words used in the question to probability phrases
  • Aim to rephrase the question in your head using AND and/or OR statements
  • e.g.  The probability of selecting a male who failed the driving test is P("male AND failed")
• Use the branches on the tree to help you work out which values you need in order to write down the probability
  
  • for "male AND failed" this would be the along the branch saying 'male' on the first characteristic and 'failed' on the second characteristic
    • the value in the bubble at the end of the required branch(es) would be the numerator
  • the denominator will be the total of the group we are choosing from
    • this could be the whole group - the total frequency at the start of the diagram
    • if we are choosing from just females, say, it would be the frequency in the bubble at the end of the female branch