Frequency Trees and Tables
This topic involves organising data from experiments into frequency tables or frequency trees to analyse outcomes. These tools allow you to calculate unknown quantities and estimate probabilities based on observed results.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Frequency tables list outcomes alongside the number of times they occurred.
- A frequency tree is a diagram that splits a total population into smaller, distinct sub-groups based on different criteria.
- In a frequency tree, the sum of the frequencies on the branches originating from a point must equal the frequency at that point.
- Missing values in both tables and trees can be found by subtracting known values from totals.
- The relative frequency of an outcome is used as an experimental estimate of its probability.
- To judge if an experiment is biased, compare the relative frequencies of the outcomes; similar values suggest it's fair, while widely different values suggest bias.
Formulae
Relative Frequency = (Number of times an outcome occurs) / (Total number of trials) When estimating the probability of an event based on the results of an experiment.
Definitions
- Frequency
- The count of how many times a particular event or outcome is observed in a set of trials.
- Relative Frequency
- An estimate of probability calculated from experimental data. It is the frequency of an outcome divided by the total number of trials.
- Frequency Tree
- A diagram that shows how a set of items is categorised and sub-categorised. Branches show the number of items in each category.
Worked example
A group of 100 students were surveyed about whether they play football or tennis. 60 of the students are boys. 25 of the boys play tennis. In total, 40 students play football. How many girls play football?
- 1
Start a frequency tree with the total, 100 students.
The first branches are 'Boys' and 'Girls'.
- 2
Fill in the 'Boys' branch with 60.
Calculate the number of girls:
100 - 60 = 40Fill this into the 'Girls' branch.
- 3
From the 'Boys' branch, create sub-branches for 'Tennis' and 'Football'.
Fill in 'Tennis' with 25.
- 4
Calculate the number of boys who play football:
60 (total boys) - 25 (boys who play tennis) = 35.
- 5
The question states 40 students play football in total.
We know 35 of these are boys.
- 6
Therefore, the number of girls who play football is the total football players minus the boys who play football:
40 - 35 = 5
Answer: 5
Common mistakes
- ×Answering with a probability or fraction when the question asks for a frequency (a whole number count).
- ×In a frequency tree, subtracting from the wrong total. For example, subtracting a sub-category from the overall total instead of its parent category total.
- ×Making a simple arithmetic error early on, which then makes all subsequent calculations incorrect.
No-calculator tips
- ✓When filling in a tree or table, perform subtractions carefully. It's often helpful to check your work by adding the parts back together to see if you get the total.
- ✓When comparing relative frequencies like 19/50 and 16/50, you only need to compare the numerators (19 and 16) as the denominators are identical.
- ✓Before performing calculations with fractions, always check if they can be simplified to make mental arithmetic easier (e.g., 40/100 becomes 4/10 or 2/5).