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Learn through experiments

How precise is a sample percentage?

How much more data is needed to halve random uncertainty?

Entering midway? Revisit Design a fair comparison for the preceding worked example, or start here and use the explanation below.

1 · Try

Start with a concrete question

Inspect a normal-approximation interval for an invented independent binary sample. The percentage is an estimated sample proportion; this is not a medical or policy recommendation.

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 whether doubling sample size halves the interval margin.

3 · Experiment

Change one thing. Inspect what happens.

  1. Compare sample sizes 100 and 400 at 50% success.
  2. At fixed size compare 10%, 50% and 90%; inspect the variance term.
  3. Explain why collecting 400 biased responses does not repair the survey-bias experiment.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Estimated proportion50 %
Approximate margin9.8 percentage points
Interval lower40.2 %
Interval upper59.8 %

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.
059.8
  • Lower interval endpoint (%)40.2
    Lower interval endpoint (%): 40.2Horizontal bar from zero to 40.2, on the shared zero to 59.8 numeric axis.
  • Estimate (%)50
    Estimate (%): 50Horizontal bar from zero to 50, on the shared zero to 59.8 numeric axis.
  • Upper interval endpoint (%)59.8
    Upper interval endpoint (%): 59.8Horizontal bar from zero to 59.8, on the shared zero to 59.8 numeric axis.

Follow the calculation

  1. Independent Bernoulli model: estimated p = 0.5, sample size n = 100.
  2. Standard error = √(0.5 × 0.5 / 100) = 0.05.
  3. Normal-approximation margin = 1.96 × 0.05 = 0.098.
  4. Clip p ± margin to [0,1]: [0.402, 0.598].

This approximate repeated-sampling interval is not a probability that this fixed population parameter lies in this particular interval. It does not cover sampling bias.

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

How precise is a sample percentage?

Inspect a normal-approximation interval for an invented independent binary sample. The percentage is an estimated sample proportion; this is not a medical or policy recommendation.

Predict before changing inputs

Predict whether doubling sample size halves the interval margin.

Understand the mechanism

Independent observations provide separate information. Copying the same rows does not increase effective sample size.

At p = 0.5 and n = 100, estimated standard error is √(0.25/100) = 0.05. Multiplying by 1.96 gives a 9.8-percentage-point margin.

At n = 400 the standard error halves to 0.025. A fourfold sample increase is needed for that reduction under the assumed model.

Calculate it yourself

Approximate standard error = √(p(1−p)/n); margin = 1.96 × standard error. p is a fraction, n independent observations.

Guided investigation

  1. Compare sample sizes 100 and 400 at 50% success.

  2. At fixed size compare 10%, 50% and 90%; inspect the variance term.

  3. Explain why collecting 400 biased responses does not repair the survey-bias experiment.

Assumptions and limitations

  • Normal intervals can perform poorly near boundaries and in small samples; the minimum n=20 allows cases where expected success counts are small. Use this to identify the approximation limitation.

  • The model excludes survey bias, clustering, optional stopping and uncertainty from data cleaning.

Further investigations

High school: Compare uncertainty from different numbers of independent coin trials.

College: Implement Wilson intervals and compare their coverage by simulation.

Working professionals: Write a dashboard note distinguishing sample uncertainty from measurement and selection bias.

5 · Apply

Try a fresh case

For p=0.5 and n=400, what is the standard error in percentage points, before multiplying by 1.96?

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 uncertainty from different numbers of independent coin trials.

College investigation

Implement Wilson intervals and compare their coverage by simulation.

Workplace investigation

Write a dashboard note distinguishing sample uncertainty from measurement and selection bias.

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.