Learn

Learning preferences

Make learning more comfortable. Changes apply immediately.

Enlarges page content, including diagrams and their labels. You can also use your browser’s zoom.

Choose the colours that are easiest for you to distinguish. Applies to lesson diagrams and lab plots. Labels, shapes, hatching and line patterns also distinguish meanings. Source photographs and raw image brightness stay unchanged; mask-only views are available for image overlays.

ExampleComparisonSample link

Saved in this browser on this device. Preferences are separate from progress backups.

Mathematics · unit version 1

Statistics: displays, designs and differences

Construct and interpret statistical evidence, plan data collection and reason about differences between independent samples. Six focused lessons extend the school statistics sequence; probability technology and independently scored examination work remain separate.

Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.

Readiness and preparation

Check your starting point

Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.

What denominator gives the percentage of successes within group A?

A conditional display must use the group named in its question.

Compare reasoning and find a refresher

Divide the successes in A by all observed members of A, provided that group is nonempty. Dividing by all successes answers a different question.

Review tables and conditional denominators

Why does subtracting independent sample means add their variances?

Subtraction changes the center but does not cancel independent uncertainty.

Compare reasoning and find a refresher

For independent X and Y, Var(X−Y)=Var(X)+Var(Y). The coefficient −1 is squared in the variance calculation. Dependence adds a covariance term that must not be silently dropped.

Review sampling distributions

Loading saved mathematics work…

Choose the evidence and justify the comparison

The mixed assessment has separate response history and review criteria; it does not change lesson participation records.

Lesson sequence

Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.

  1. Decide what a study must answer before collecting data

    Lesson details and objectives

    Translate a decision into a population, measurable outcome, comparison and collection plan.

    • Write an investigative question with a population, observational unit, variable and time window before collection.
    • Distinguish a census, a sample and prospective or retrospective observation using actual collection actions.
    • Separate population generalization, causal conclusions and ethical data choices.

    6 practice tasks.

  2. Execute a systematic sample and a two-factor experiment

    Lesson details and objectives

    Specify random starts, treatment combinations and allocation steps that another learner can reproduce.

    • Execute a systematic sample with a uniform random start and diagnose periodic frame bias.
    • Construct all four combinations of two two-level factors and allocate replicates randomly.
    • Justify controls, blocking, blinding, ethical recruitment and the separate scopes of sampling and assignment.

    6 practice tasks.

  3. Read categorical displays with their denominators

    Lesson details and objectives

    Construct bars and pie sectors, then distinguish conditional percentages from mosaic areas when group sizes differ.

    • Read and construct one-variable categorical bars and pie charts with a stated total.
    • Compare conditional distributions in equal-width segmented bars and unequal-width mosaics.
    • Use differences in conditional proportions as evidence of association without inferring causation.

    6 practice tasks.

  4. Reconstruct keyed stems and define percentile rank

    Lesson details and objectives

    Preserve individual values in a stem-and-leaf display and state how ties enter an empirical percentile calculation.

    • Construct and reconstruct an ordered stem-and-leaf display using an explicit place-value key.
    • Calculate both strictly-below and at-or-below empirical percentile ranks.
    • Distinguish the percentile rank of a value from a percentile cutpoint under a specified quantile convention.

    6 practice tasks.

  5. Predict differences between sample proportions

    Lesson details and objectives

    Start from known population probabilities to derive the center, variability and approximate shape of a repeated-sampling difference.

    • Compute the mean and variance of a difference of independent sample proportions from known population probabilities.
    • Check within-sample, between-sample and large-count assumptions before a normal approximation.
    • Distinguish population-model variance from estimated interval variance, pooled-null variance and paired variance.

    6 practice tasks.

  6. Predict differences between sample means

    Lesson details and objectives

    Connect known population spreads to repeated mean differences and distinguish normal populations, large-sample approximations and paired designs.

    • Derive the center and standard deviation of an independent difference of means using population parameters.
    • Calculate a probability under an explicitly justified normal sampling model.
    • Identify finite-population and paired-data limitations and distinguish known parameters from estimated standard errors.

    6 practice tasks.