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Learn through experiments

Why more measurements can still be wrong

Does repeating a measurement remove every kind of error?

Entering midway? Revisit Does the treatment work, or did the groups differ? for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Original synthetic instrument: measure a known 10 cm reference with a fixed offset and a repeating disturbance pattern.

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?

Predict the raw mean after four readings with a +2 cm offset. Will twelve readings fix it?

3 · Experiment

Change one thing. Inspect what happens.

  1. Compare 1, 2, 4 and 12 readings while keeping the offset fixed.
  2. Set disturbance to zero. Explain why perfectly repeatable readings can be inaccurate.
  3. Use the reference calibration and compare raw and corrected means.
  4. Propose a second reference length to check whether one offset correction is sufficient for a different instrument.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Raw mean12 cm
Calibrated mean10 cm
Raw absolute error2 cm

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.
012
  • True length10
    True length: 10Horizontal bar from zero to 10, on the shared zero to 12 numeric axis.
  • Raw mean12
    Raw mean: 12Horizontal bar from zero to 12, on the shared zero to 12 numeric axis.
  • Calibrated mean10
    Calibrated mean: 10Horizontal bar from zero to 10, on the shared zero to 12 numeric axis.

Follow the calculation

  1. Reading 1: 10 + bias 2 + disturbance -2.00 = 10.00 cm.
  2. Reading 2: 10 + bias 2 + disturbance 2.00 = 14.00 cm.
  3. Reading 3: 10 + bias 2 + disturbance -1.00 = 11.00 cm.
  4. Reading 4: 10 + bias 2 + disturbance 1.00 = 13.00 cm.
  5. Mean = sum 48.00 / 4 = 12.00 cm.
  6. Reference calibration removes known bias: 12.00 - (2) = 10.00 cm.

Repeating a biased instrument does not remove its offset. This fixed disturbance pattern is a controlled example, not a random sample or confidence interval.

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 instrument: measure a known 10 cm reference with a fixed offset and a repeating disturbance pattern.

Precision describes repeatability; accuracy describes closeness to the reference. An instrument can be precise and consistently wrong.

The default four readings are 10, 14, 11, 13 cm. Their mean is 48 / 4 = 12 cm. The disturbances cancel but the +2 cm offset remains.

Subtracting a known offset gives 10 cm. Real calibration estimates an offset with uncertainty; it can also reveal scale errors that this model does not include.

Read the relationship

Reading = reference + bias + disturbance. Mean = sum of readings / reading count. Corrected mean = mean − known bias.

Use the trace as evidence

The trace recomputes from your current controls. Change one control at a time, record what changed, and compare the result with your prediction. The bars use the same quantities as the numerical trace; a larger bar is not automatically a better outcome.

Boundaries of the model

  • The disturbance pattern −1, +1, −0.5, +0.5 repeats; it is not independent random noise.
  • The reference and offset are known exactly in this teaching model; no confidence interval is implied.

Continue with Does the treatment work, or did the groups differ? to investigate a different mechanism.

5 · Apply

Try a fresh case

A new 20 cm reference gives readings 22, 24, 23, 23 cm. A calibration establishes a +3 cm offset. What is the corrected mean in cm?

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

Compare repeated ruler measurements and describe resolution separately from accuracy.

College investigation

Add random noise and estimate uncertainty without treating repeated bias as random error.

Workplace investigation

Write an instrument acceptance check with reference standards, tolerance, and recalibration triggers.

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.