How averaging changes noisy evidence
How often does an average miss the true value by more than your tolerance?
Entering midway? Revisit Why more measurements can still be wrong for the preceding worked example, or start here and use the explanation below.
Start with a concrete question
Original synthetic scenario. Every independent instrument reading measures a true value of 10 plus bias and one of five equally likely disturbances. Exact enumeration counts all possible sample sequences.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
What do you expect?
Will more readings improve precision even if the instrument remains biased?
Change one thing. Inspect what happens.
- Start with one reading, scale four, no bias and tolerance one; count which disturbances pass.
- Increase sample size and inspect the exact probability, not only a single lucky average.
- Add a bias of two units; explain why more precision may concentrate measurements around the wrong value.
- Propose both a calibration plan and a sample-size plan, naming the independence assumption needed for averaging.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
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.Follow the calculation
- 1 readings: 5 equally likely draw sequences compressed into 5 sums; 4 have |bias + mean disturbance| > 1. Probability = 4/5 = 0.800.
- 2 readings: 25 equally likely draw sequences compressed into 9 sums; 12 have |bias + mean disturbance| > 1. Probability = 12/25 = 0.480.
- 3 readings: 125 equally likely draw sequences compressed into 13 sums; 70 have |bias + mean disturbance| > 1. Probability = 70/125 = 0.560.
- 4 readings: 625 equally likely draw sequences compressed into 17 sums; 244 have |bias + mean disturbance| > 1. Probability = 244/625 = 0.390.
More independent readings shrink random variation around the biased mean. They do not remove bias, and the exact tail probability can change unevenly because the noise distribution is discrete.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
What the experiment explains
Original synthetic scenario. Every independent instrument reading measures a true value of 10 plus bias and one of five equally likely disturbances. Exact enumeration counts all possible sample sequences.
Random error varies across readings. Here disturbances are equally likely at −scale, −scale/2, zero, scale/2 and scale. We count every possible sequence rather than display one random run.
With scale four, one reading has disturbances −4, −2, 0, 2, 4. Four of five are outside a ±1 tolerance, so the chance is 80%. With two readings there are 25 equally likely pairs; 12 are outside ±1, giving 48%. The mean has less variance; because outcomes are discrete, other thresholds need not improve at every sample count.
The standard deviation describes spread. Each disturbance has variance scale²/2, so the mean of n independent readings has standard deviation scale/√(2n). Bias shifts the expected mean without shrinking; more data cannot repair it.
Read the relationship
Measured mean = 10 + bias + average disturbance. Standard deviation of mean = scale/√(2n), for n independent readings from the stated five-point distribution. Outside probability counts |bias + mean disturbance| > tolerance.
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
- Independent identically distributed discrete noise with known probabilities; correlated or drifting instruments violate this model.
- The exact probability is a model calculation, not a confidence statement about unknown real-world noise.
Return to the scientific reasoning and experimental design course and project to connect this mechanism to the broader investigation.
Try a fresh case
A new independent noise distribution has standard deviation 6 units per reading. What is the standard deviation of the mean of nine readings?
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
Enumerate the 25 pairs of two five-outcome readings and compare their mean errors.
College investigation
Validate the compressed count distribution with independent enumeration and then introduce correlated noise.
Workplace investigation
Write a measurement plan separating instrument calibration, independent repeats and a decision tolerance.
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.