Statistics and Probability: reading data, summarising it and reasoning about chance
Statistics and probability in the numeracy CAT asks you to read real data critically: a news graph, a survey result, a table of test scores, a game of chance. You need to choose sensible summary statistics, describe spread and shape, spot misleading displays, and calculate straightforward probabilities and expected counts.
Core rules
- Numeric data is discrete (counts) or continuous (measurements); categorical data is grouped rather than measured. Match the display to the data: bar chart for categories, histogram for continuous data, dot plot for small numeric sets, scatter plot for two numeric variables.
- Mean = total ÷ count; median = middle value of ordered data (average the two middle values when n is even); mode = most frequent value; range = maximum − minimum; IQR = Q3 − Q1, the middle half of the data.
- Outliers and skew pull the mean but barely move the median, so the median often describes ‘typical’ better.
- P(event) = favourable outcomes ÷ total outcomes, always between 0 and 1. P(not A) = 1 − P(A). Expected count = P × number of trials.
- Larger, randomly chosen samples give more reliable estimates than small or self-selected ones.
Worked example
Weekly pay in a six-person team: $620, $640, $650, $680, $710 and $2400 (the owner). Mean = 5700 ÷ 6 = $950. Median = (650 + 680) ÷ 2 = $665. The mean is inflated by a single outlier, so the median is the fairer description of a typical worker’s pay in this team.
Traps that cost marks
- Finding the median without sorting the data first.
- Comparing counts rather than percentages when groups are different sizes (12 of 40 is 30%, but 15 of 60 is only 25%).
- Expecting experimental results to match theoretical probability in a small sample.
- Trusting a graph with a truncated y-axis, uneven gridlines or a 3-D effect.