Statistics and probability

📚The mathematics

4.1 Sampling and data

A population is everyone you want to draw a conclusion about; a sample is who you actually measured. Data is discrete if it counts and continuous if it measures. The five sampling techniques to know are simple random, convenience, systematic, quota and stratified, and you should be able to describe each and say what could go wrong with it.

Bias is the useful idea here. A convenience sample of your friends is quick and unrepresentative; a systematic sample can align with a hidden periodic pattern; a survey with a low response rate hears only from people who cared enough to reply. When a question asks you to comment on a sampling method, name the specific mechanism by which the sample stops resembling the population — not just “it might be biased”.

An outlier is more than \( 1.5 \times \mathrm{IQR} \) beyond the nearer quartile. Note that this is a definition, not a verdict: some outliers are genuine extreme values that belong in the data, and others are recording errors that do not. Say which you think it is and why.

4.2 & 4.3 Presenting and summarising

Cumulative frequency graphs are the tool for medians, quartiles and percentiles: read across from the relevant fraction of the total and down to the axis. Box-and-whisker diagrams compare two distributions at a glance — compare medians for centre, IQR or range for spread, and the relative 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; the answer is an estimate because you have thrown away the individual values. The modal class only means something when the class intervals are equal.

Calculate variance and standard deviation with technology — you are expected to use the statistics functions on your calculator, not the formula. What you must know is how they behave:

• Add a constant \(k\) to every value: the mean increases by \(k\); the standard deviation is unchanged. Sliding the whole data set along does not spread it out.
• Multiply every value by \(k\): the mean multiplies by \(k\), and so does the standard deviation. The variance multiplies by \( k^{2} \).

4.4 & 4.10 Correlation and regression

Pearson’s product-moment correlation coefficient \(r\) lies between \( -1 \) and \( 1 \) and measures linear association only. Describe it in two words — strength (strong, moderate, weak) and direction (positive, negative). A value near zero means no linear relationship, which is not the same as no relationship: a perfect parabola has \( r \approx 0 \).

The regression line of \(y\) on \(x\) always passes through the mean point \( (\bar{x}, \bar{y}) \). Use it to predict \(y\) from \(x\); to predict \(x\) from \(y\) you need the regression line of \(x\) on \(y\) instead, which is a different line.

4.5, 4.6 & 4.11 Probability

For equally likely outcomes, \( \mathrm{P}(A) = \dfrac{n(A)}{n(U)} \), and \( \mathrm{P}(A') = 1 - \mathrm{P}(A) \). The complement is often the fast route: “at least one” nearly always means \( 1 - \mathrm{P}(\text{none}) \).

Two events are mutually exclusive if \( \mathrm{P}(A \cap B) = 0 \) — they cannot both happen. They are independent if \( \mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) \), equivalently \( \mathrm{P}(A|B) = \mathrm{P}(A) \) — knowing one tells you nothing about the other. These are different ideas and, awkwardly for intuition, two events with non-zero probability cannot be both.

To test for independence, compute \( \mathrm{P}(A)\mathrm{P}(B) \) and compare it with \( \mathrm{P}(A \cap B) \). If they are equal the events are independent; if not, they are not. Show both numbers — the comparison is the answer.

Tree diagrams handle sequences of events. Multiply along the branches, add between distinct paths, and check that the probabilities at each set of branches sum to 1. Watch whether sampling is with or without replacement: without replacement, the second set of branches has different probabilities and a denominator one smaller.

4.7 & 4.8 Discrete random variables and the binomial distribution

A discrete probability distribution must have all its probabilities summing to 1 — which is usually how an unknown in the table is found. The expected value is

and it is a long-run average, not a value \(X\) has to be able to take. In a game where \(X\) is the player’s gain, \( \mathrm{E}(X) = 0 \) means the game is fair.

\( X \sim \mathrm{B}(n,p) \) applies when there is a fixed number \(n\) of independent trials, each with two outcomes and the same probability \(p\) of success. Then \( \mathrm{E}(X) = np \) and \( \mathrm{Var}(X) = np(1-p) \). Find the probabilities with technology, but be careful to distinguish \( \mathrm{P}(X = k) \) from the cumulative \( \mathrm{P}(X \le k) \), and remember that for discrete variables \( \mathrm{P}(X > 3) = 1 - \mathrm{P}(X \le 3) \) — where the boundary sits matters.

4.9 & 4.12 The normal distribution

\( X \sim \mathrm{N}(\mu, \sigma^{2}) \) is symmetric about \( \mu \), with about 68% of values within one standard deviation, 95% within two and 99.7% within three. Note the notation carries the variance, not the standard deviation.

Three question types, all done with technology:

• Forwards — given boundaries, find a probability.
• Inverse normal — given a probability, find the boundary value. Check whether the tail you are given is the left or the right one; calculators want the left by default, so a “top 10%” question needs 0.90 entered, not 0.10.
• Standardising — \( z = \dfrac{x - \mu}{\sigma} \) tells you how many standard deviations from the mean a value sits. This is how you find an unknown \( \mu \) or \( \sigma \): convert the given probability to a \(z\)-value, then solve the equation.

AHL 4.13 Bayes’ theorem

Bayes reverses a conditional probability: you know the chance of a positive test given the disease, and you want the chance of the disease given a positive test. Those two are wildly different when the disease is rare, which is the point of every question set on this. A tree diagram with the totals filled in usually answers a Bayes question faster than the formula — the denominator is just the total probability of the evidence, gathered from every branch that produces it.

AHL 4.14 Continuous random variables

A continuous random variable is described by a probability density function \( f(x) \ge 0 \) with \( \int_{-\infty}^{\infty} f(x)\,\mathrm{d}x = 1 \) — which is how an unknown constant in a pdf is found. Probability is area, so \( \mathrm{P}(X = a) = 0 \) for any single value, and this is why it makes no difference whether an inequality is strict.

The mode is where \( f \) is greatest, so it is a maximum problem — differentiate. The median \(m\) splits the area in half. And for linear transformations, \( \mathrm{E}(aX+b) = a\mathrm{E}(X) + b \) while \( \mathrm{Var}(aX+b) = a^{2}\mathrm{Var}(X) \): the \(b\) shifts the distribution without spreading it, exactly as adding a constant to a data set does.