Data Analysis, Statistics, and Probability
Core concepts
You need to read and interpret data displays (bar graphs, line plots, histograms, stem-and-leaf plots, box plots, circle graphs, and line graphs), compute and compare measures of center (mean, median, mode) and spread (range, and informally the interquartile range), recognize misleading graphs, and reason about sampling. Probability covers sample spaces, theoretical versus experimental probability, simple and compound events, and basic counting.
Common traps
- The median requires data sorted first; with an even number of values, average the two middle values.
- The mean is pulled toward outliers, while the median resists them.
- A data set can have no mode or several modes; range is a single number, not a pair.
- Probability is always between 0 and 1 inclusive. Drawing without replacement changes the denominator for the second event.
- For independent events, multiply probabilities; add only for mutually exclusive "or" cases.
- Circle graph percentages must total 100%, and broken axes or uneven intervals make a graph misleading.
Worked example
For the data set 6, 9, 9, 12, 19: Mean = (6 + 9 + 9 + 12 + 19) ÷ 5 = 55 ÷ 5 = 11. Median = 9 (the middle of the sorted list). Mode = 9. Range = 19 − 6 = 13. Note how the outlier 19 raises the mean above the median.
Probability example: a bag holds 4 red and 6 blue marbles. P(red) = 4/10 = 2/5. With replacement, P(red then red) = 2/5 × 2/5 = 4/25.
Rule of thumb: sort the data, state which measure the question asks for, and check that any probability you compute is between 0 and 1.