Frequency Trees (OCR GCSE Maths)

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Frequency Trees

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

Examiner Tip

  • Work carefully when completing a frequency tree
    • a quick check is that your values in the bubbles at the end of each set of branches add up to the total frequency
  • Some of the frequencies may be given as fractions or percentages of others
    • e.g.  60% of the females passed their driving test
  • If there are errors in your tree, your probabilities will be incorrect and you will lose marks

Worked example

80 budding DJ's attend a club to practice their skills each week.
60% if those attending the club practice scratch mixing, the rest practice beat mixing.
Of those practising scratch mixing, 15 are female.
Of those practising beat mixing, 12 are male.

(a)
Use the information above to complete the frequency tree.

5-2-4-frequency-tree-we-question

Start with the total frequency bubble at the start of the frequency tree - 80.
Work out 60% of 80 to find the frequency for scratch mixing.

table row cell 10 percent sign space of space 80 space end cell equals cell 80 divided by 10 equals 8 end cell row cell 60 percent sign space of space 80 space end cell equals cell space 8 cross times 6 equals 48 end cell end table

Work your way through the rest of the tree.

beat: 80 space minus space 48 space equals 32

We are given that 15 of the scratch mixers are female and that 12 of the beat mixers are male.

scratch and female: 48 space minus space 15 space equals space 33
beat and female: 32 minus 12 equals 20

Now we have all the values, we can complete the frequency tree.


5-2-4-frequency-tree-we-solutionQuick check on bubble totals:  48 + 32 = 80, 33 + 15 + 12 + 20 = 80.

(b)
i)
A member of the club is chosen at random.  Find the probability that the member is a male practising beat mixing.

ii)
A scratch mixer is chosen at random.  Find the probability that the scratch mixer is female.

i)
Rephrasing this is P("male AND beat mixing").
The numerator will be the value in the bubble at the end of the branches "beat mixing" and "male" (12).
We are choosing from the whole club, so the denominator will be the total frequency (80).

bold P bold left parenthesis bold male bold space bold practising bold space bold beat bold space bold mixing bold right parenthesis bold equals bold 12 over bold 80 stretchy left parenthesis equals 3 over 20 stretchy right parenthesis
ii)
Rephrasing this is P("female AND scratch mixing").
The numerator will be the value in the bubble at the end of the branches "scratch mixing" and "female" (15).
This time though we are only choosing from scratch mixers, so the denominator will be at the end of the scratch mixing branch (48).

bold P stretchy left parenthesis " female " space GIVEN thin space " scratch " stretchy right parenthesis bold equals bold 15 over bold 48 stretchy left parenthesis equals 5 over 16 stretchy right parenthesis

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.