Statistics and probability

📚The mathematics

4.1 Sampling and data

A population is everyone you want a conclusion about; a sample is who you measured. Data is discrete if it counts and continuous if it measures. Know the five techniques — simple random, convenience, systematic, quota and stratified — and be able to describe each and say what could go wrong.

When a question asks you to comment on a method, name the mechanism by which the sample stops resembling the population, not just “it could be biased”. A convenience sample reaches only people who were easy to reach; a survey with a low response rate hears only from people who cared enough to reply; a systematic sample can lock onto a hidden periodic pattern.

An outlier is more than \( 1.5 \times \mathrm{IQR} \) beyond the nearer quartile. That is a definition, not a verdict: decide whether it is a genuine extreme value or a recording error, and say which and why.

4.2 & 4.3 Presenting and summarising

Cumulative frequency graphs give medians, quartiles and percentiles — read across from the right fraction of the total and down to the axis. Box-and-whisker diagrams compare two distributions: medians for centre, IQR or range for spread, whisker lengths for skew. Roughly symmetric box and whiskers suggest the data may be normally distributed.

For grouped data use mid-interval values to estimate the mean; it is an estimate because the individual values are gone. The modal class only means something for equal class intervals.

Compute variance and standard deviation with technology. What you must know is their behaviour: adding a constant \(k\) to every value raises the mean by \(k\) and leaves the standard deviation unchanged; multiplying every value by \(k\) multiplies both the mean and the standard deviation by \(k\), and the variance by \( k^{2} \).

4.4 & 4.10 Correlation

Pearson’s \(r\) measures linear association, between \( -1 \) and \( 1 \). Describe it in two words: strength and direction. A value near zero means no linear relationship — a perfect parabola has \( r \approx 0 \), which is why you look at the scatter diagram as well as the number.

Spearman’s rank correlation \( r_s \) is Pearson’s coefficient applied to the ranks rather than the values. It detects any monotonic relationship, not just a linear one, and it is far less sensitive to outliers because an extreme value only ever moves one place in the ranking. Use \(r\) when you are testing for linearity and \( r_s \) when the relationship may be curved but consistently increasing or decreasing, or when outliers are distorting \(r\). If values tie, average their ranks.

The regression line of \(y\) on \(x\) passes through the mean point \( (\bar{x},\bar{y}) \). The gradient and intercept both mean something in context, and interpreting them is regularly worth marks: the gradient is the change in \(y\) per unit change in \(x\), in the units of the problem.

4.5 & 4.6 Probability

\( \mathrm{P}(A') = 1 - \mathrm{P}(A) \), and the complement is usually the fast route: “at least one” almost always means \( 1 - \mathrm{P}(\text{none}) \).

Events are mutually exclusive when \( \mathrm{P}(A \cap B) = 0 \) and independent when \( \mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) \). These are different ideas, and testing for independence means computing both sides and comparing them — show both numbers.

On tree diagrams, multiply along branches and add between paths, and check each set of branches sums to 1. Watch whether sampling is with or without replacement.

The expected number of occurrences is \( n \times \mathrm{P} \): if 300 components are made and each has probability 0.04 of being faulty, you expect 12 faulty ones.

4.7–4.9 Distributions

For a discrete random variable, the probabilities must sum to 1 — that is how an unknown in the table is found — and \( \mathrm{E}(X) = \sum x\,\mathrm{P}(X = x) \). In a game where \(X\) is the player’s gain, \( \mathrm{E}(X) = 0 \) means the game is fair; a casino ensures it is negative.

\( X \sim \mathrm{B}(n,p) \) requires a fixed number of independent trials, two outcomes, and a constant \(p\). Then \( \mathrm{E}(X) = np \) and \( \mathrm{Var}(X) = np(1-p) \). Distinguish \( \mathrm{P}(X = k) \) from \( \mathrm{P}(X \le k) \), and remember \( \mathrm{P}(X > 3) = 1 - \mathrm{P}(X \le 3) \) for a discrete variable.

\( X \sim \mathrm{N}(\mu, \sigma^{2}) \) is symmetric about \( \mu \), with roughly 68% within one standard deviation, 95% within two and 99.7% within three. Both directions are examinable: forwards (boundaries to probability) and inverse (probability to boundary). For inverse normal, check which tail you have been given — “the top 10%” needs 0.90 entered, not 0.10.

4.11 Hypothesis testing

This is the distinctive AI SL material and it follows a fixed structure. State \( \mathrm{H}_0 \) and \( \mathrm{H}_1 \), find the \(p\)-value with technology, compare it with the significance level, and write a conclusion in context.

• \( \chi^{2} \) test for independence. Uses a contingency table. \( \mathrm{H}_0 \): the two variables are independent. Degrees of freedom \( = (\text{rows}-1)(\text{columns}-1) \). In examinations tables have at most 4 rows or columns, and expected frequencies will exceed 5.
• \( \chi^{2} \) goodness of fit. Tests whether data follows a stated distribution. \( \mathrm{H}_0 \): the data fits that distribution. At SL the degrees of freedom are \( n - 1 \).
• \(t\)-test. Compares the means of two populations. At SL samples are unpaired, the population variance is unknown and assumed equal, so use the pooled two-sample \(t\)-test. The underlying variable must be normally distributed. Choose one-tailed or two-tailed from the wording: “is there a difference” is two-tailed, “is A greater than B” is one-tailed.

AHL 4.12 Designing data collection

Questionnaire design: questions should be unbiased rather than leading, structured with consistent answer choices, and precise. Choosing which variables to collect, and how to categorise numerical data into a \( \chi^{2} \) table, are examinable judgements — and categories must be chosen so expected frequencies exceed 5. When parameters are estimated from the data itself, the degrees of freedom fall accordingly.

Reliability is consistency: would you get the same result again? Test it by test–retest or parallel forms. Validity is whether you are measuring what you intended at all: content validity and criterion-related validity. A bathroom scale reading 3 kg heavy every time is perfectly reliable and completely invalid, which is the example worth remembering.

AHL 4.13 Non-linear regression

Beyond linear, you may be asked for quadratic, cubic, exponential, power or sine regression, all with technology. Two measures of fit:

• \( SS_{res} \), the sum of the squared residuals — smaller is better, and the regression itself is chosen to minimise it.
• \( R^{2} \), the coefficient of determination — the proportion of the variability in the response variable accounted for by the model. An \( R^{2} \) of 0.87 means 87% of the variation is explained by the model, which is the sentence an interpretation question wants. For a linear model \( R^{2} = r^{2} \).

But \( R^{2} \) alone is a poor way to choose between models: adding parameters always raises it, and a model can fit the data beautifully while being nonsense in context. Look at the residuals, the shape and the plausibility too.

AHL 4.14–4.16 Estimation and confidence intervals

For a linear transformation, \( \mathrm{E}(aX+b) = a\mathrm{E}(X)+b \) and \( \mathrm{Var}(aX+b) = a^{2}\mathrm{Var}(X) \) — the \(b\) shifts without spreading. The sample mean \( \bar{x} \) is an unbiased estimate of \( \mu \), and \( s_{n-1}^{2} \) is an unbiased estimate of \( \sigma^{2} \). Note the \( n-1 \): the ordinary \( s_n^{2} \) systematically underestimates the population variance.

A linear combination of independent normal variables is normal, and in particular \( \bar{X} \sim \mathrm{N}\!\left(\mu, \dfrac{\sigma^{2}}{n}\right) \). The central limit theorem says that for large \(n\) — taken as \( n > 30 \) in examinations — the distribution of the sample mean approaches normality regardless of the distribution the sample was drawn from. That is what makes inference possible on data that is not itself normal.

A confidence interval for a mean uses the normal distribution when \( \sigma \) is known and the \(t\)-distribution when it is not, at any sample size. Interpret it carefully: a 95% confidence interval is a procedure that captures the true mean 95% of the time. When comparing two groups, heavily overlapping intervals mean you cannot claim a difference.

AHL 4.17 & 4.18 Poisson and formal testing

The Poisson distribution \( X \sim \mathrm{Po}(m) \) counts events in a fixed interval when they occur independently at a uniform average rate. Its mean and variance are both \(m\), which is a useful diagnostic: if sample data has mean and variance close together, Poisson is plausible. Sums of independent Poisson variables are Poisson, so a rate per hour scales to a rate per day by scaling \(m\). Contexts: calls to a helpline, emergency admissions, flaws per metre of cable, typos per page.

You should be able to choose between normal, binomial and Poisson from a context. Binomial has a fixed number of trials; Poisson counts occurrences with no fixed maximum; normal is continuous.

Formal testing at HL extends to critical values and critical regions, tests for a population mean, tests for a proportion using the binomial, tests for a Poisson mean, and testing whether a population correlation coefficient is zero. And the two errors:

• Type I — rejecting a true \( \mathrm{H}_0 \). A false positive. Its probability is the significance level.
• Type II — failing to reject a false \( \mathrm{H}_0 \). A false negative.

Lowering the significance level reduces Type I errors and increases Type II errors. Which matters more is a contextual judgement: for a smoke alarm you accept false alarms to avoid missing a fire; for a criminal conviction you accept the reverse.

AHL 4.19 Markov chains

A transition matrix \( T \) has \( T_{ij} \) equal to the probability of moving from state \(j\) to state \(i\), so its columns sum to 1. After \(n\) steps the state is

For a regular chain the state settles to a steady state independent of where it started. Find it either by raising \(T\) to a high power and reading off the columns, or exactly by solving \( T\mathbf{s} = \mathbf{s} \) together with the condition that the entries sum to 1 — and examination questions will say when an exact solution is required. Notice what that equation means: the steady state is the eigenvector of \(T\) with eigenvalue 1, which is why Topic 1 teaches eigenvectors.