AA Topic 4: statistics and probability

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4.1 to 4.9 are stated in both guides as content common to the two courses, so they are the same pages Applications students use rather than a second copy that could drift. The five below them exist only in this course.

Shared with Applications (4.1 to 4.9)

4.1 · Where the data came from SHARED Populations and samples, the five techniques, bias, and outliers. Push one student to 95 minutes and watch the median refuse to move. 4.2 · The picture is a choice SHARED Grouped frequency, histograms, cumulative frequency and box plots. Bin the same 60 values three ways and get three different stories. 4.3 · Where it sits, and how far it spreads SHARED Mean, median, mode, quartiles and standard deviation. The result that returns at Higher Level as Var(aX + b). 4.4 · Correlation and regression SHARED Pearson's r, the least squares line of y on x, and when prediction stops being allowed. Drag one point away and watch a convincing r collapse. 4.5 · Probability and sample spaces SHARED Counting outcomes, the complement, expected number of occurrences. Roll two dice four thousand times and watch the theory arrive slowly. 4.6 · Venn, tree, and the conditional bar SHARED Combined events, mutually exclusive against independent, with and without replacement. Drag the circles apart and P(A given B) drops to zero. 4.7 · Discrete random variables SHARED Distributions, the sum to 1, expected value, and a fair game. E(X) is a balance point, so it need not be a value X can take. 4.8 · The binomial distribution SHARED The four conditions, exactly and at least, mean and variance. Shade "more than 3" and find out whether the bar at 3 belongs. 4.9 · The normal distribution SHARED Areas and probabilities, the empirical rule, and the inverse direction. Drag the boundaries: the number in the panel is the area, not the z value.

Analysis and Approaches only

These five do not appear in Applications at all. If you are following an Applications course, you can stop at 4.9.

4.10 · The regression line of x on y Two least squares lines, which one predicts what, and when a prediction is not reliable. Both lines on one scatter. They cross at the mean and open out as r falls. 4.11 · Conditional probability, formally P(A∩B) = P(B)P(A|B), the independence test, and why P(A|B) and P(B|A) are different numbers. Slide the overlap. The two readouts agree at exactly one position. 4.12 · Standardising and z-values What z measures, comparing across different tests, and finding an unknown μ or σ. 70 on one test and 102.5 on another turn out to be the same standing. 4.13 · Bayes' theorem HL Total probability, up to three events, and the base rate fallacy. A 99% accurate test, and a positive result that means about 9%. 4.14 · Continuous random variables HL Density functions, E(X) and Var(X) by integration, and the variance of a discrete variable. Squeeze the interval onto a point: the probability goes to zero, the density does not.

Where the two courses part company

	Analysis and Approaches	Applications and Interpretation
x on y regression	yes, 4.10	not on the course
Bayes' theorem	HL, 4.13	not on the course
continuous random variables	HL, 4.14	not on the course
Spearman's rank	not on the course	yes, 4.10
chi-squared tests	not on the course	yes, 4.11

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