Topic 4: Statistics and probability

Statistics and probability is the most data-driven part of the AA course, and it rewards careful, organised work more than clever tricks. The topic splits into two halves that meet in the middle. The statistics half is about describing real data: how you gather it, how you summarise its centre and spread, and how you measure the relationship between two variables. The probability half is about modelling uncertainty: assigning numbers to events, combining those numbers with the rules of probability, and using named distributions (binomial and normal) to predict outcomes. Both SL and HL students cover the same core, with HL adding a few refinements such as Bayes-style reasoning through conditional probability and more careful treatment of expectation. Throughout, your graphical display calculator (GDC) is essential: it computes standard deviations, regression lines, binomial and normal probabilities, and inverse-normal values. The skill being tested is choosing the right tool, entering data correctly, and interpreting the output in context. This guide walks through each strand with worked reasoning so you understand not just which button to press but why the answer means what it does.

Sampling and types of data

Before any analysis, you must know how data was collected, because a biased sample makes every later calculation misleading. The population is the entire group you want to study; a sample is the subset you actually measure. Good samples aim to be representative. The AA syllabus expects you to recognise several sampling techniques. In simple random sampling every member of the population has an equal chance of selection, often via random number generation. In systematic sampling you order the population and pick every $k$th member after a random start. In stratified sampling you split the population into groups (strata) such as year levels, then sample from each stratum in proportion to its size; for example, if a school is $60\%$ junior and $40\%$ senior, a stratified sample of $50$ takes $30$ juniors and $20$ seniors. In quota sampling you fill fixed counts per group but choose members non-randomly, and in convenience sampling you simply take whoever is easiest to reach. The last two are prone to bias. You should also distinguish discrete data, which takes separate countable values such as number of siblings, from continuous data, which can take any value in a range such as height or time. Reliability matters too: outliers (unusually extreme values) can distort summaries, and you should be alert to sampling bias, measurement error, and non-response. A clear comment on whether a sample is likely to be representative is often worth a mark.

Measures of central tendency and spread

Once you have data you summarise its centre. The mean is $\bar{x}=\frac{\sum x}{n}$, the sum of all values divided by how many there are; it uses every value but is pulled toward outliers. The median is the middle value when data is ordered (the average of the two middle values if $n$ is even); it resists outliers. The mode is the most frequent value. For spread, the range is simply largest minus smallest. The interquartile range is $\text{IQR}=Q_3-Q_1$, the spread of the middle half of the data, where $Q_1$ and $Q_3$ are the lower and upper quartiles. The most important measure in AA is the standard deviation, which measures the typical distance of values from the mean. The population standard deviation is $\sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}}$, and its square $\sigma^2$ is the variance. In practice you find $\sigma$ and $\bar{x}$ from the GDC's one-variable statistics, never by hand on long lists. A key exam fact: AA uses the standard deviation labelled $\sigma_x$ (the population formula dividing by $n$), not the sample standard deviation $s_x$ that divides by $n-1$. A useful transformation rule: if every data value is changed by $y=ax+b$, then the new mean is $a\bar{x}+b$ and the new standard deviation is $|a|\sigma$, because adding a constant shifts the data without changing spread while multiplying scales both.

Grouped data, cumulative frequency and box plots

Real data often arrives grouped into class intervals, for example masses recorded as $50\lt m\le 60$. To estimate the mean of grouped data you use the midpoint of each class as the representative value and weight by frequency: $\bar{x}=\frac{\sum f x}{\sum f}$ where $x$ is each midpoint. This is an estimate because the exact values within each class are unknown. For grouped data the modal class is the interval with the highest frequency. Cumulative frequency is the running total of frequencies up to the upper boundary of each class; plotting cumulative frequency against the upper class boundary and joining the points with a smooth curve gives a cumulative frequency graph (an ogive). From this curve you read off the median (at the $\frac{n}{2}$ level on the vertical axis), the quartiles (at $\frac{n}{4}$ and $\frac{3n}{4}$), percentiles, and you can estimate how many data points fall below or above a chosen value. A box-and-whisker plot is a compact five-number summary showing the minimum, $Q_1$, median, $Q_3$, and maximum; the box spans the IQR and the whiskers reach the extremes. Box plots make it easy to compare two distributions and to spot skew: if the median sits left of centre in the box and the right whisker is long, the data is positively skewed. A standard outlier test flags any value below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$.

Bivariate data, correlation and regression

When each observation has two measurements, such as hours studied and exam score, you analyse the relationship. A scatter diagram is the first step, revealing whether the association is positive, negative, or non-existent, and whether it is roughly linear. The Pearson product-moment correlation coefficient $r$ quantifies linear association on a scale from $-1$ to $1$: values near $\pm 1$ indicate a strong linear relationship, values near $0$ indicate a weak one, and the sign matches the direction of the trend. You obtain $r$ directly from the GDC. Interpret it carefully: a strong correlation does not prove that one variable causes the other, since a lurking third variable may drive both. The least-squares regression line of $y$ on $x$, written $y=ax+b$, is the straight line minimising the sum of squared vertical distances from the points; the GDC gives $a$ and $b$. This line passes through the mean point $(\bar{x},\bar{y})$. You use it to predict $y$ from a value of $x$. Prediction is reliable only by interpolation (within the range of the data) and when $r$ shows a reasonably strong relationship; extrapolation beyond the data range is unsafe. Note that the regression of $y$ on $x$ is designed to predict $y$ from $x$, so you should not rearrange it to predict $x$ from $y$.

Probability foundations: events, conditional probability and independence

Probability assigns a number between $0$ and $1$ to how likely an event is. For equally likely outcomes, $P(A)=\frac{n(A)}{n(U)}$, the number of favourable outcomes over the total in the sample space $U$. The complement satisfies $P(A')=1-P(A)$. The general addition rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, where you subtract the overlap so it is not double-counted; for mutually exclusive events $P(A\cap B)=0$ and the rule simplifies. Conditional probability, the chance of $A$ given that $B$ has occurred, is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Rearranging gives the multiplication rule $P(A\cap B)=P(A|B)\,P(B)$. Two events are independent when one occurring does not change the probability of the other, formally when $P(A\cap B)=P(A)\,P(B)$, equivalently $P(A|B)=P(A)$. Do not confuse independent with mutually exclusive: mutually exclusive events cannot happen together and are therefore strongly dependent. HL students lean on the conditional formula more heavily, often combining it with tree diagrams to handle reversed conditioning, effectively a Bayes calculation where you find $P(B|A)$ from $P(A|B)$ by working back through the branches.

Representing probability: tree and Venn diagrams

Two visual tools organise probability problems and prevent errors. A Venn diagram shows events as overlapping regions inside a rectangle representing the sample space. You place counts or probabilities into the regions, working from the intersection outward so each region is counted once. Venn diagrams make set operations concrete: the union is everything inside either circle, the intersection is the overlap, and the complement is everything outside. They are ideal for problems phrased with at least, neither, or only. A tree diagram suits sequential or multi-stage experiments, such as drawing two counters without replacement. Each branch carries a probability, probabilities on branches from a single node sum to $1$, and you multiply along a path to get the probability of that combined outcome, then add the probabilities of all paths that satisfy the condition. With replacement the branch probabilities stay constant between stages; without replacement they change because the population shrinks, which also makes the stages dependent. Tree diagrams are the natural setting for conditional probability: the second-stage branches are already conditional on the first stage, so reading $P(A\cap B)$ off a path is exactly applying $P(A\cap B)=P(A)\,P(B|A)$.

Discrete random variables and expected value

A discrete random variable $X$ takes separate numerical values, each with a probability given by its probability distribution. A valid distribution must have all probabilities between $0$ and $1$ and summing to one: $\sum P(X=x)=1$. This summation condition is frequently used to solve for an unknown probability or an unknown constant in a formula-defined distribution. The expected value, or mean, of $X$ is $E(X)=\sum x\,P(X=x)$, a weighted average of the possible values using their probabilities as weights. It represents the long-run average outcome if the experiment were repeated many times, and need not be a value $X$ can actually take. A game or process is called fair when its expected gain is zero, that is $E(X)=0$ once stakes are included; setting $E(X)=0$ and solving for a price or payout is a classic exam task. For example, if a game charges a fee and pays prizes, you compute the expected net gain and a positive value favours the player. HL extends expectation reasoning but the SL core, computing $E(X)$ from a table and interpreting fairness, underpins everything that follows, including the mean of the binomial distribution.

The binomial distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. The conditions are: a fixed number of trials $n$, two outcomes per trial (success or failure), constant success probability $p$, and independent trials. We write $X\sim B(n,p)$. The probability of exactly $r$ successes is $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$, where $\binom{n}{r}$ counts the ways to arrange $r$ successes among $n$ trials. In practice you use the GDC's binomial probability density function for exactly $r$, and the cumulative function for at most $r$; for at least, use the complement, since $P(X\ge r)=1-P(X\le r-1)$. Translating worded inequalities into the right GDC input is the main source of errors, so write out the boundary carefully, remembering that more than $r$ means $X\ge r+1$. The mean of a binomial variable is $E(X)=np$ and its variance is $\text{Var}(X)=np(1-p)$, so the standard deviation is $\sqrt{np(1-p)}$. These formulas let you find an expected count quickly, for instance the expected number of defective items in a batch.

The normal distribution

The normal distribution models continuous data that clusters symmetrically around a mean, producing the familiar bell curve. It is described by two parameters, the mean $\mu$ which locates the centre and the standard deviation $\sigma$ which sets the width, written $X\sim N(\mu,\sigma^2)$. The curve is symmetric about $\mu$, and the total area beneath it equals $1$, so probabilities correspond to areas. A useful benchmark is that about $68\%$ of values lie within one standard deviation of the mean, about $95\%$ within two, and about $99.7\%$ within three. To find a probability such as $P(X\lt a)$ you use the GDC's normal cumulative function with $\mu$ and $\sigma$; for a probability between two values you enter both bounds. The reverse problem, finding the value $a$ for which a given proportion lies below it, uses the inverse normal function: you supply the area to the left and the GDC returns $a$. This is how you find boundary marks for grade cut-offs or percentiles. Sketching the bell curve and shading the region you want before computing is strongly recommended; it catches errors such as confusing the area to the left with the area to the right. Standardisation converts any normal value to a $z$-score via $z=\frac{x-\mu}{\sigma}$, measuring how many standard deviations a value lies from the mean, which is occasionally needed when $\mu$ or $\sigma$ is the unknown to be found.