MCAT content library / Research & data / Accuracy, precision, and sources of measurement error
Use a reference and repeated readings to decide whether a measurement problem concerns its center, its spread, or both.
About 15 minutes, including the worked examples and checks.
From a dataset to a supported claim · Lesson 3 of 6 · Continue in this path
Useful foundations
On this page
- Concept: close together and close to the reference are different claims
- Pattern: compare instruments on the same standard
- Rule: match the correction to the source of error
- Application: an offset correction fails at another level
- Use controls to locate the problem
- Check your reasoning
- Continue learning
- Sources
Concept: close together and close to the reference are different claims
A laboratory obtains nearly identical readings each time it measures a standard. That result supports repeatability under those conditions. It does not establish that the readings are close to the standard’s reference value. A balance can repeatedly report the same excess mass because its zero setting is wrong.
Accuracy concerns agreement with the quantity being measured, judged against an appropriate reference. Precision concerns the dispersion of repeated results under specified conditions. NIST treats accuracy as qualitative; a numerical difference from a reference is an error estimate, not a universal “accuracy score.” A reference itself can have uncertainty, even when a teaching calculation treats it as exact.1
To identify the problem, ask two questions: Where is the center relative to the reference? and How much do repeated readings vary? AAMC includes both systematic and random error in its research and data reasoning scope. Their practical value is that they suggest different next actions.2
Pattern: compare instruments on the same standard
The following original measurements are illustrative. Both instruments measure a stable 100-unit reference under the same controlled conditions. Treat the reference uncertainty as negligible for these calculations. The final row is an arithmetic correction of B’s three original readings, not a new set of observations.
| Record | Readings (units) | Mean (units) | Sample SD (units) |
|---|---|---|---|
| Instrument A | 98, 100, 102 | 100 | 2 |
| Instrument B | 104, 105, 106 | 105 | 1 |
| B after subtracting 5 | 99, 100, 101 | 100 | 1 |
A’s three readings have deviations −2, 0, and +2 from their mean; the sample SD is √[(4 + 0 + 4)/2] = 2 units. B’s deviations are −1, 0, and +1, giving an SD of 1 unit. B is more precise in these observations, but its readings are all farther above the reference. Its mean is 5 units high.
The observed pattern is consistent with a positive offset in B. It does not prove that B is exactly 5 units high on every sample or every day. Three measurements provide limited evidence, and a single reference level cannot reveal whether error changes across the instrument’s measurement range. A’s sample mean equals 100, but that alone does not prove its long-run bias is zero.
Rule: match the correction to the source of error
A useful simplified model is reading = target value + systematic component + random component. A systematic component shifts results in a consistent way under the specified conditions. Random variation changes across repetitions. These are features of the measurement process, not labels for whether a person made a careless mistake.3
If a stable additive offset of +5 is established for the relevant range, subtracting 5 corrects that offset. It moves B’s mean from 105 to 100 without changing the differences between its readings. Therefore its SD remains 1. Calibration can improve agreement with the reference while leaving precision unchanged. The corrected result still has residual uncertainty.
Averaging more independent, stable measurements can reduce the random uncertainty in their mean. Under the usual finite-variance model, the estimated standard error is s/√n. It does not divide a fixed +5 offset by √n: the mean of many readings with that same offset remains shifted. More repetitions and a better calibration address different problems.4
The independence condition matters. Re-reading one stored digital value five times produces five copies, not five independent measurements. A drift that affects all readings in the same direction also cannot be treated as unrelated random noise simply because the measurements occur at different times.
Application: an offset correction fails at another level
A follow-up check gives average B readings of 55, 105, and 160 for references of 50, 100, and 150 units. Their differences are +5, +5, and +10. Subtracting 5 would give 50, 100, and 155. The correction works at the first two levels but leaves a 5-unit discrepancy at the third.
The evidence therefore argues against applying one constant offset across this entire range. It does not identify the cause of the extra high-range error. Possible explanations include a changing response relationship or a condition affecting the high-level measurements. Appropriate next work would test additional reference levels and relevant operating conditions, not merely collect more repetitions at 100 units.
Distinguish an additive offset from a proportional error. If a device reports 1.10 times each reference, dividing by 1.10 addresses the stated proportional relationship; subtracting the error found at one concentration generally will not. If the relationship is nonlinear, neither a single subtraction nor a single multiplier is automatically adequate. These are model choices that need evidence across the range of use.
Use controls to locate the problem
An unknown sample cannot establish instrument bias by itself because its target value is unknown. A suitable reference or independently characterized check sample makes that comparison possible. A blank can reveal background signal, but a correct blank does not establish that the instrument responds correctly to nonzero quantities. Zero-point and scale errors are separate possibilities.
Calibration conditions also need to resemble the conditions of use. A device may perform well on a reference in one environment yet shift with temperature or sample composition. An established calibration is evidence for a specified relationship, not a guarantee that all future conditions preserve it.1
When comparing experimental groups, ask whether both were measured by the same process. If every control is read on A and every treatment on B, an instrument difference can masquerade as a treatment difference. Randomizing measurement order, measuring appropriate standards, and crossing conditions across instruments can help identify that alternative explanation.
The useful conclusion states both the pattern and its limit: B’s readings are more tightly clustered in this trial, but their center is displaced from the reference. That supports checking a systematic component. It does not justify claiming that B is universally worse, that A has no error, or that repetition alone will repair the discrepancy.
ACTIVE PRACTICE · COMPARE AND TRANSFER
Separate Offset, Repeatability, and Range Dependence
Original 100-unit reference repeats: A = 98,100,102 (mean 100, sample SD 2); B = 104,105,106 (mean 105, SD 1). Subtracting five from B gives 99,100,101. On additional references 50,100,150, B’s means are 55,105,160.
Examine the original evidence
Results are bounded to the original supplied observations. Instructional checks do not establish assessment calibration or clinical effectiveness.
Check your reasoning
Choose an answer before opening its feedback. Each option explains the reasoning behind that choice.
Check 1
A device gives readings of 41, 42, and 43 units for a stable 40-unit reference. Further testing establishes a constant +2-unit offset over the range being used. Which result follows from subtracting 2 from every reading?
A. The mean and sample SD both decrease by 2 units.
Review this choice. A constant subtraction shifts the center, but each deviation from the shifted mean is unchanged. It cannot reduce this sample SD from 1 to −1.
B. The mean becomes 40 units and the sample SD remains 1 unit.
Correct. The corrected readings are 39, 40, 41. Their mean is 40 and their deviations remain −1, 0,+1, so the sample SD remains 1. The stated offset is corrected; random spread remains.
C. The mean remains 42 units and the sample SD becomes zero.
Review this choice. The correction changes the mean. It does not collapse distinct readings into one value.
D. The readings become independent of their original measurement conditions.
Review this choice. An arithmetic correction cannot remove dependence on untested environmental conditions or establish independence of observations.
The principle to carry forward
An additive calibration correction shifts a center without changing the spread around it.
Check 2
A sensor gives mean readings of 22, 44, and 66 for reference values of 20, 40, and 60. Repeated readings have very little scatter. Assume this exact proportional relationship holds throughout the stated range. Which correction best matches the evidence?
A. Average additional readings without changing the scale.
Review this choice. Averaging can address random uncertainty; it leaves the stated proportional discrepancy in the mean.
B. Divide each reading by 1.10.
Correct. Each mean is 1.10 times its reference. Dividing the observed value by 1.10 recovers 20, 40, and 60 under the supplied model.
C. Subtract 2 from each reading.
Review this choice. Subtracting 2 repairs 22 but leaves 42 and 64 for the other references. The error grows with the quantity.
D. Subtract 10% of the reference value from every unknown reading.
Review this choice. A genuinely unknown sample has no known reference value to use in that operation. A usable calibration maps the observed reading to the estimated quantity.
The principle to carry forward
Check whether the discrepancy is constant, proportional, or more complex before choosing a correction.
Continue learning
From a dataset to a supported claim
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See every step in From a dataset to a supported claim
From a dataset to a supported claim
Interpret variability and uncertainty before combining evidence from multiple figures.
- Reading graphs: axes, scales, and the comparison that matters
- Mean, median, and spread: describing a dataset
- Accuracy, precision, and sources of measurement error · You are here
- Confidence intervals, error bars, and sample size
- Statistical significance and effect size: what the result means
- Reading a multi-figure experiment from claim to conclusion
Related lessons
- Mean, median, and spread: describing a dataset
- Psychological Measurement: What Does the Score Mean?
- Confidence intervals, error bars, and sample size
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Sources
1. NIST/SEMATECH: Bias and Accuracy