Does increasing the sample size eliminate both sources of error?
Experiment
Compare 1, 4 and 16 measurements at bias 2, then set bias to 0.
Separate variability from bias
Worked teaching model · synthetic values · Independent readings share standard deviation 4 cm and a fixed calibration bias. The standard error formula assumes independence and equal variance; it does not estimate the bias.
Reset keeps your prediction, captures and practice history.
Calculated values on the same fixed scale
Row
Quantity
Value (cm)
1
Single-reading SD
4
2
Mean standard error
2
3
Systematic bias
2
4
Root mean squared error
2.8284
Standard error: 2.000 cm; bias remains 2 cm.
SE = 4 / √n. RMSE = √(SE² + bias²). These are model-derived long-run quantities, not an interval around a particular observed mean. Repetition reduces independent random variation, not a shared offset.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Averaging reduces one kind of error
Assume independent readings have the same standard deviation of 4 cm. The standard error of their mean is 4 / square root of n, where n is the number of independent readings.
For n = 1, standard error is 4 cm. For n = 4 it is 2 cm; for n = 16 it is 1 cm. Reducing this random component by a factor of two needs four times as many independent readings under these assumptions.
A shared offset of 2 cm remains 2 cm regardless of n. Repeating a biased measurement more precisely does not calibrate it.
Combine errors without confusing their meanings
The model’s root mean squared error is the square root of standard error squared plus bias squared. At n = 4 and bias 2 cm, it is √(2² + 2²), approximately 2.828 cm.
These quantities describe a long-run measurement model. They are not measured errors from a specific sample, and they do not form a confidence interval around an observed mean.
Copying one reading sixteen times violates independence. Shared drift and environmental conditions can also create dependence. Compare sample-size changes with bias removal and explain which intervention addresses which source of error. Keep the assumptions in your notebook.
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Explain which assumption would fail if the same reading were copied repeatedly.
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
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