Statistics

Statistics is the science of collecting, analysing, and drawing conclusions from data under uncertainty. A-Level Statistics covers data representation, probability theory, statistical distributions, and hypothesis testing — the foundations of data-driven decision making.

Topics Covered

Data Representation

• Types of data — qualitative vs. quantitative, discrete vs. continuous, primary vs. secondary
• Measures of central tendency — mean $\bar{x} = \frac{\sum x}{n}$, median, mode; when each is appropriate
• Measures of spread — range, interquartile range (IQR), variance $\sigma^2 = \frac{\sum(x-\bar{x})^2}{n}$, standard deviation
• Visual representations — histograms (with varying class widths), cumulative frequency curves, box plots, stem-and-leaf diagrams
• Outliers — identification using $1.5 \times \text{IQR}$ or mean $\pm 2\sigma$; deciding whether to exclude

Correlation and Regression

• Scatter diagrams — visual assessment of correlation (positive, negative, none)
• Pearson’s product-moment correlation coefficient — $r$ measures linear correlation; $-1 \leq r \leq 1$
• Regression line — $y = a + bx$; least squares; interpreting $a$ (intercept) and $b$ (gradient) in context
• Interpolation vs. extrapolation — reliability of predictions within and beyond the data range

Probability

• Axioms — $0 \leq P(A) \leq 1$, $P(\Omega) = 1$, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Conditional probability — $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Independence — $P(A \cap B) = P(A) \times P(B)$
• Tree diagrams and Venn diagrams — systematic approaches to multi-stage probability problems
• Mutually exclusive vs. independent — these are different concepts; mutually exclusive events cannot be independent (unless one has probability 0)

Statistical Distributions

• Binomial distribution — $X \sim \text{Bin}(n, p)$; $P(X = x) = \binom{n}{x}p^x(1-p)^{n-x}$; conditions (fixed trials, two outcomes, constant probability, independence)
• Normal distribution — $X \sim N(\mu, \sigma^2)$; bell curve, symmetry, the empirical rule ($68$-$95$-$99.7\%$)
• Standard normal — $Z = \frac{X - \mu}{\sigma}$; using tables for $P(Z < z)$
• Approximations — normal approximation to binomial when $n$ is large and $p \approx 0.5$

Hypothesis Testing

• Null and alternative hypotheses — $H_0$ (no effect) vs. $H_1$ (effect exists); one-tailed vs. two-tailed
• Test statistics — calculating from sample data and comparing to critical values
• Significance level — $\alpha = 0.05$ or $0.01$; the probability of a Type I error
• $p$-values — $P(\text{observing this or more extreme} \mid H_0)$; reject $H_0$ if $p$-value $< \alpha$
• Critical regions — the set of values that lead to rejecting $H_0$
• Binomial hypothesis tests — testing a population proportion

Study Tips

1. Draw diagrams — Venn diagrams for probability, scatter plots for correlation, normal distribution sketches for every $Z$-score question.
2. Show full working in hypothesis tests — state $H_0$, $H_1$, calculate the test statistic, find the $p$-value or critical value, compare, state conclusion in context.
3. Know when to use Binomial vs. Normal — Binomial for counting successes in fixed trials; Normal for continuous measurements with a bell-shaped distribution.
4. Practise reading tables — normal distribution tables and binomial tables require careful reading. Check whether the table gives $P(Z < z)$ or $P(Z > z)$.
5. Interpret in context — never just say “reject $H_0$”; say “there is sufficient evidence at the 5% significance level to suggest that the mean has increased.”

How to Use These Notes

Follow the sidebar order. Each page provides definitions, worked examples with full calculations, and exam-style problems. Start with data representation and probability before moving to distributions and hypothesis testing.