Statistics and Probability
Statistics and Probability
Statistics tests collecting, tabulating and displaying data and summarising it with averages and measures of spread, including from grouped frequency tables and cumulative frequency curves. Probability tests assigning likelihoods to equally likely outcomes and combining them using tables and tree diagrams.
Summarising data
- Mean = sum of values ÷ number of values; median = middle value of the ordered list; mode = most frequent value.
- Grouped data: use class midpoints for the mean. The median class is where cumulative frequency first reaches n/2; the modal class has the highest frequency.
- Range = highest − lowest; interquartile range = upper quartile − lower quartile.
Displaying data
- Pie chart: sector angle = frequency/total × 360°.
- On a cumulative frequency curve, read the quartiles at n/4, n/2 and 3n/4.
Probability
- P(event) = favourable outcomes ÷ total outcomes, and 0 ≤ P ≤ 1.
- P(not A) = 1 − P(A).
- Mutually exclusive events: P(A or B) = P(A) + P(B). Independent events: P(A and B) = P(A) × P(B).
Worked example. Marks are grouped as 0–9 (f = 4), 10–19 (f = 6), 20–29 (f = 10). Estimate the mean. Midpoints are 4.5, 14.5 and 24.5. Σfx = (4 × 4.5) + (6 × 14.5) + (10 × 24.5) = 18 + 87 + 245 = 350. Σf = 20. Mean = 350 ÷ 20 = 17.5 marks.
Traps that cost marks.
- Using 5 as the midpoint of the class 0–9 - for continuous data it is 4.5.
- Picking the class with the highest frequency as the median class.
- Drawing without replacement: the denominator must fall on the second draw.
- For "at least one", calculating directly instead of using 1 − P(none).