Statistics and Probability: IB Maths AA SL Made Simple

A population is every member of the group you care about; a sample is the subset you actually measured, and almost every exam data set is a sample. Mean, median and mode each answer a different question: the mean uses every value and is sensitive to outliers, the median only cares about position and is robust to outliers, and the mode suits categorical or repeated discrete data. For an odd number of values the median is the middle sorted value; for an even number it's the average of the two middle values.

When data is grouped into a frequency table, the class midpoint stands in for every value in that class when calculating the mean or standard deviation — this makes the result an estimate, not the exact figure, because individual values are lost. In a histogram with unequal class widths, it's the area of each bar (frequency density × width) that represents frequency, not the bar's height alone. The modal class of grouped data is therefore the class with the highest frequency density, not simply the highest raw frequency, whenever widths differ.

If every value in a data set is transformed as $y=ax+b$, the new mean becomes $\bar{y}=a\bar{x}+b$ and the new variance becomes ${\sigma }_y^2=a^2{\sigma }_x^2$ — adding a constant shifts the mean but never changes the spread, while multiplying scales both mean and SD. When a single value is corrected (not a full transformation), adjust $\sum x$ by the difference between the correct and incorrect value and divide by $n$ again, rather than re-adding everything. Adding one new data point equal to the current mean always leaves the mean unchanged but slightly reduces the variance.

Range (max − min) is quick to calculate but wrecked by a single extreme value; the interquartile range $IQR=Q_3-Q_1$ describes the spread of the middle 50% and ignores extremes. A value is flagged as an outlier if it falls below $Q_1-1.5IQR$ or above $Q_3+1.5IQR$ — this labels it as unusual, it doesn't automatically get deleted. The five-number summary (min, Q1, median, Q3, max) builds a box plot, and comparing two box plots properly means using SOCS: shape, outliers, centre, and spread.

For equally likely outcomes, $P(A)=n(A)/n(U)$, and the complement rule $P(A^{\prime })=1-P(A)$ is the fastest route through any 'at least one' question. Mutually exclusive events satisfy $P(A\cap B)=0$, while the general addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ always holds and corrects for double-counting the overlap. Independent events satisfy $P(A\cap B)=P(A)\times P(B)$, and in tree diagrams you multiply along a branch for one path but add across branches when several paths lead to the same outcome.