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.

Learn through experiments

Same people, different order

Does comparing each participant with themselves remove every bias?

Entering midway? Revisit How averaging changes noisy evidence for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Original synthetic scenario. Six fictional participants try methods A and B. Stable baseline scores differ; practice in the second period changes performance.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review basic mathematics

2 · Predict

What do you expect?

If B is always second, will the mean paired difference equal its actual +3-point benefit?

3 · Experiment

Change one thing. Inspect what happens.

  1. At the defaults calculate the mean B−A difference and compare it with the programmed benefit.
  2. Alternate order and explain which contribution disappears from the mean.
  3. Increase disturbance while keeping balanced order; compare individual differences with the group mean.
  4. Write a protocol specifying order assignment, washout or rest, and how you would handle learning carried between periods.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Mean B minus A7 points
Programmed B benefit3 points
Order contribution to mean4 points

Compare the current results

Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.
089
  • Participant 1 B score (points)35
    Participant 1 B score (points): 35Horizontal bar from zero to 35, on the shared zero to 89 numeric axis.
  • Participant 2 B score (points)59
    Participant 2 B score (points): 59Horizontal bar from zero to 59, on the shared zero to 89 numeric axis.
  • Participant 3 B score (points)46
    Participant 3 B score (points): 46Horizontal bar from zero to 46, on the shared zero to 89 numeric axis.
  • Participant 4 B score (points)78
    Participant 4 B score (points): 78Horizontal bar from zero to 78, on the shared zero to 89 numeric axis.
  • Participant 5 B score (points)65
    Participant 5 B score (points): 65Horizontal bar from zero to 65, on the shared zero to 89 numeric axis.
  • Participant 6 B score (points)89
    Participant 6 B score (points): 89Horizontal bar from zero to 89, on the shared zero to 89 numeric axis.

Follow the calculation

  1. Participant 1: A reference 30; effect 3 + order 4 + disturbance -2 gives B−A = 5.000.
  2. Participant 2: A reference 50; effect 3 + order 4 + disturbance 2 gives B−A = 9.000.
  3. Participant 3: A reference 40; effect 3 + order 4 + disturbance -1 gives B−A = 6.000.
  4. Participant 4: A reference 70; effect 3 + order 4 + disturbance 1 gives B−A = 8.000.
  5. Participant 5: A reference 60; effect 3 + order 4 + disturbance -2 gives B−A = 5.000.
  6. Participant 6: A reference 80; effect 3 + order 4 + disturbance 2 gives B−A = 9.000.
  7. Mean paired difference = 42.000 / 6 = 7.000.

Pairing controls stable participant differences, but a common second-period improvement is still confounded with treatment unless order is balanced.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

What the experiment explains

Original synthetic scenario. Six fictional participants try methods A and B. Stable baseline scores differ; practice in the second period changes performance.

Pairing removes stable between-person differences by subtracting each person’s A reference. It cannot automatically distinguish treatment from time or order.

With B always second, the mean is benefit 3 + practice 4 + mean disturbance 0 = 7 points. With balanced order, practice contributes +4 for half and −4 for half, so the mean returns to 3.

The disturbance pattern is deliberately balanced. In real data it will not cancel exactly, and carryover from the first method may remain even with balanced order.

Read the relationship

Paired difference = B benefit + signed order effect + disturbance. Mean paired difference = sum of six differences / 6.

Inspect the evidence

The numerical trace is recalculated from your current settings. Compare a controlled pair of trials before changing several inputs together. Inspect intermediate quantities as well as final metrics; state which assumptions make the comparison meaningful.

Boundaries of the model

  • Fixed balanced disturbances, not a statistical sampling model.
  • Order effect is symmetric and constant; carryover and missing participants are not modeled.

Return to the scientific reasoning and experimental design course and project to connect this mechanism to the broader investigation.

5 · Apply

Try a fresh case

Four paired differences are 8, 2, 6 and 0 points. What is their mean?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Design a counterbalanced comparison of two sorting instructions using paper cards.

College investigation

Estimate uncertainty for paired differences and test sensitivity to carryover.

Workplace investigation

Choose between parallel and crossover usability studies with explicit reasons and operational constraints.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.