SCIENCE · INVESTIGATE AND EXPLAIN

Separate sample spread from accuracy and common bias

Derive mean, sample spread and mean uncertainty while keeping a shared bias outside the repeatability estimate.

Learning outcome: Compare bias 0 and .20 g with five readings; then compare 3, 4, 5-reading prefixes at fixed bias. Retain the raw table, mean, sample SD and explanation of what is not represented.

Starting recap

Use algebra and dimensional consistency; distinguish a measured reading, a model estimate and the evidence needed to accept the model.

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Five synthetic readings around 10.1 g have sample SD≈.01581 g and standard-error estimate≈.007071 g, but their mean is biased+.10 g from the stipulated 10 g truth.

Open the experiment

Predict

Can increasing a common offset change sample standard deviation?

Your prediction and reason

Experiment

Compare bias 0 and .20 g with five readings; then compare 3, 4, 5-reading prefixes at fixed bias. Retain the raw table, mean, sample SD and explanation of what is not represented.

Separate repeatability from common bias

Worked teaching model · synthetic values · Synthetic readings are 10 g plus selected prefix of residuals−.02, +.01, 0, +.02, −.01 g plus one common bias. Sample SD and s/√n describe this selected dataset under an independent random-error interpretation; they do not include the common bias or establish a confidence interval.

Common indicated-mass bias: 0.1 g Replicate count: 5 readings

Reset keeps your prediction, captures and practice history.

Axes rescale with controls. Compare numerical coordinates, not apparent screen slopes. Replicate points are discrete observations; no between-reading interpolation is implied.

Synthetic reference or repeatability packet
Reading number	Indicated mass (g)
1	10.08
2	10.11
3	10.1
4	10.12
5	10.09

Conditional calculations; not physical measurements
Quantity	Value	Unit
Mean indicated mass	10.1	g
Sample standard deviation	0.01581139	g
Standard error estimate s/sqrt(n)	0.007071068	g
Known synthetic mean error	0.1	g
Assigned common bias	0.1	g

Repeated readings may be tightly clustered while all are biased.

Subtracting a sample mean removes the same bias from deviations, so SD cannot diagnose it. An independent reference or changed measurement method is needed. Adding selected readings need not monotonically reduce this finite-sample SD.

Predict before changing controls, retain contrasting captures, and explain a changed case. Displayed digits are computational precision, not an uncertainty claim. Flags check stipulated numerical compatibility, not identity, reasoning quality or practical proficiency.

Repeated readings may be tightly clustered while all are biased.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Read the actual data, not only a summary

The five original residuals relative to a 10 g reference are −0.02, +0.01, 0, +0.02 and −0.01 g. A control adds one common bias to every reading; another selects the first three, four or five observations. These are supplied synthetic values, not a random sample newly measured by the learner. Their ordering matters when a prefix is selected.

Derive the summaries

The sample mean is the sum divided by n. The sample standard deviation s is the square root of the sum of squared deviations from the sample mean divided by n − 1. Under an independent random-error interpretation, s/√n estimates the standard error of the mean. Neither the label “standard error” nor division by √n accounts for an unmodeled common offset.

Work the five-reading case

At bias 0.10 g, readings are 10.08, 10.11, 10.10, 10.12 and 10.09 g. Their mean is 10.10 g. Squared deviations sum to 0.0010 g²; dividing by four and taking the square root gives s ≈ 0.01581 g. Dividing by √5 gives about 0.007071 g. The mean is still 0.10 g above the stipulated reference. A narrow spread is not evidence of accuracy.

Prove common-shift invariance

Replace every reading xi by xi + c. The mean becomes mean + c, so each deviation is (xi + c) − (mean + c) = xi − mean. Every squared deviation is unchanged. Test the algebra with two bias settings and identical replicate count. The plot and table should move together while the spread statistic remains fixed.

Do not overinterpret replication

Adding one observation can increase the estimated sample SD when that observation is farther from the earlier mean. Even the standard-error estimate for a selected finite prefix need not decrease monotonically. More importantly, repeated measurements with the same calibration cannot establish that common bias is absent. An independent reference or genuinely different method is a stronger diagnostic.

Transfer to an uncertainty argument

Suppose a report quotes mean ± s/√n as “the entire error.” Ask what distribution, independence, calibration and stability assumptions justify that statement. A coverage probability requires additional statistical reasoning; this lesson does not automatically attach 68% or 95% to its displayed intervals. Write a revised report that separates repeatability evidence, reference disagreement and an untested systematic contribution.

Source and evidence boundary

These explanations, calculations and diagrams are original. Conceptual references checked 30 October 2026: OpenStax measurement uncertainty, NIST uncertainty framework. The linked textbook is a reference, not copied lesson text or imagery. All displayed datasets are synthetic teaching packets. Saved numerical agreement or prose is not a performed practical, independent reasoning review or laboratory certification. No handling, heating or chemical preparation procedure is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why standard error is not a complete uncertainty budget or a universal confidence interval.

Your explanation and limitations

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.

Previous: Derive exact density bounds and conditional identification Next: Carry named entities through mass and mole conversions

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