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4 Statistics and probability

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  1. 4.1 Statistics and probability - SL content

    1. SL 4.1Sampling and data

      • Understand population, sample, random sample, discrete and continuous data. • Assess reliability, bias, missing data, recording errors and outliers. • Use simple random, convenience, systematic, quota and stratified sampling.

    2. SL 4.2Data presentation

      • Use frequency tables, histograms and cumulative frequency graphs. • Find median, quartiles, percentiles, range and IQR. • Produce and compare box-and-whisker diagrams; mark outliers.

    3. SL 4.3Summary statistics

      • Use mean, median, mode, modal class and grouped-data mean estimates. • Use IQR, standard deviation and variance; technology may calculate SD/variance. • Understand effects of adding/subtracting or scaling all data values.

    4. SL 4.4Correlation and regression

      • Use scatter diagrams, best-fit lines and Pearson correlation coefficient r. • Distinguish correlation strength/direction and correlation vs causation. • Use y-on-x regression for prediction; interpret parameters and beware extrapolation.

    5. SL 4.5Probability basics

      • Use trial, outcome, sample space, event and relative frequency. • Calculate P(A)=n(A)/n(U), complements and expected number of occurrences. • Represent sample spaces with lists or tables.

    6. SL 4.6Combined, conditional and independent events

      • Use Venn, tree, sample-space diagrams and outcome tables. • Use combined-event, conditional-probability and independence formulae. • Handle probabilities with and without replacement.

    7. SL 4.7Discrete random variables

      • Understand discrete random variables and probability distributions. • Find expected value/mean; interpret E(X)=0 as a fair game context.

    8. SL 4.8Binomial distribution

      • Use binomial distribution as an appropriate model. • Use mean and variance of binomial distribution. • Use technology for binomial probabilities; formal proofs are not required.

    9. SL 4.9Normal distribution

      • Use normal distribution curve and its properties. • Use 68-95-99.7 rule around mean and standard deviation. • Use technology for normal and inverse-normal probabilities.

    10. SL 4.10Spearman's rank correlation

      • Use Spearman's rank correlation coefficient rs with technology. • Average ranks for tied data. • Compare Pearson vs Spearman, including monotonic relationships and outlier sensitivity.

    11. SL 4.11Hypothesis testing

      • Form null/alternative hypotheses, significance levels and p-values. • Use chi-square tests for independence and goodness of fit. • Use t-tests for comparing two population means; interpret test results. • At SL, t-test samples are unpaired and population variance is unknown.

  2. 4.2 Statistics and probability - AHL content

    1. AHL 4.12 (HL)—Data collection and validity

      • Design valid surveys/questionnaires and choose relevant variables/data. • Use unbiased, structured and precise questioning. • Categorize numerical data for chi-square tests with suitable degrees of freedom. • Distinguish reliability and validity; know test-retest, parallel forms, content and criterion-related validity.

    2. AHL 4.13 (HL)—Non-linear regression

      • Use technology for least-squares regression curves: linear, quadratic, cubic, exponential, power and sine. • Use residual sum of squares and coefficient of determination R^2 to evaluate fit. • Know R^2 alone is not enough to choose a model.

    3. AHL 4.14 (HL)—Random variable transformations

      • Use linear transformations of random variables: E(aX+b)=aE(X)+b and Var(aX+b)=a^2Var(X). • Use expected value and variance for linear combinations of independent random variables. • Use sample mean and unbiased sample variance as estimators.

    4. AHL 4.15 (HL)—Sampling distributions and CLT

      • Linear combinations of independent normal random variables are normal. • Use distribution of sample mean: Xbar ~ N(mu, sigma^2/n) for normal populations. • Use central limit theorem; n>30 is sufficient in examinations.

    5. AHL 4.16 (HL)—Confidence intervals

      • Construct confidence intervals for the mean of a normal population. • Use normal distribution when sigma is known and t-distribution when sigma is unknown. • Interpret confidence intervals in context.

    6. AHL 4.17 (HL)—Poisson distribution

      • Use Poisson distribution, mean and variance. • Sum of independent Poisson variables is Poisson. • Choose between normal, binomial and Poisson models based on context.

    7. AHL 4.18 (HL)—Advanced hypothesis testing

      • Use critical values and critical regions. • Test population means with normal or t distributions, including paired/unpaired samples. • Test proportions with binomial distribution and population means with Poisson distribution. • Test whether population correlation rho is zero; know Type I and II errors and their probabilities.

    8. AHL 4.19 (HL)—Transition matrices and Markov chains

      • Use transition matrices, powers and state matrices: s_n=T^n s_0. • Use transition diagrams for discrete dynamical systems. • Work with regular Markov chains, steady state and long-term probabilities. • Recognize steady state as eigenvector for eigenvalue 1.