Distinguish repeatability, accuracy, standard deviation, standard error, and measurement uncertainty before comparing two results.
Field diagramAccuracy and precision
Estimated reading
9 min
Estimated practice
15 min
Equipment
Calculator or spreadsheet and supplied repeated-measurement dataset.
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Objective
By the end, you can:
Separate repeatability from accuracy and identify what each requires.
Calculate and explain sample standard deviation and standard error of the mean.
Use an uncertainty-aware comparison without treating overlapping error bars as a universal test.
Several kinds of ‘close’
Repeatability concerns how close results are when the same method, operator, equipment, and conditions are repeated over a short interval. Reproducibility broadens the conditions, such as operator, day, or instrument. Accuracy describes closeness to a reference or accepted value; it cannot be inferred from tight clustering alone. A method can be repeatable but biased, or variable yet centered near a reference. State which claim your data can address.
For observations x1 through xn, the sample standard deviation s describes spread among readings. The standard error of the mean, often s/√n, describes the estimated sampling variation of the mean under assumptions about independent observations. It is not a measure of how far each reading is from truth and should not be used to make a small dataset look more precise than the measurement process supports. NIST TN 1297 treats uncertainty as a combination of components evaluated by statistical and other means.
Compare a question, not just two numbers
Before comparing methods, define the estimand: difference in means, paired differences, agreement with a reference, or repeatability. Identify independent units, paired structure, calibration or reference information, and the decision threshold. A difference of 0.5 mL may be important for one method and irrelevant for another. Error bars can represent SD, SE, confidence intervals, or expanded uncertainty; the caption must say which. Overlap or non-overlap alone is not a general decision rule.
Quick table-and-plot recap
If you have not taken the optional table lesson, use one row per observational unit, keep raw values separate from derived fields, show units in every heading, and display individual points when the dataset is small. A figure caption should state what the points and any error bars represent.
Worked example
Same readings, different claims
For 9.8, 10.0, 10.2, and 10.0 mL, mean = 10.0 mL. Deviations are −0.2, 0, +0.2, 0; sample SD = √(0.08/3) ≈ 0.163 mL; SE ≈ 0.163/√4 = 0.0816 mL.
01SD ≈ 0.163 mL: observed spread of four readings.
02SE ≈ 0.0816 mL: estimated uncertainty of the mean under the independent-observation model.
03Neither establishes accuracy without a reference value and uncertainty context.
Materials
Set out what you need.
Supplied repeated-measurement data
Formula sheet
Plot with explicit error-bar definition
Safety & stop conditions
Pause if the work no longer fits the plan.
Use representative data only; label it clearly.
Do not infer a clinical, safety, or release decision from this introductory comparison.
Ask for statistical or metrology support when the decision has material consequences.
Method
Work through the steps.
01
State the comparison
Write whether you are comparing repeatability, a reference difference, paired changes, or another defined quantity.
02
Check independence
Identify experimental units and repeated readings. Do not use technical repeats as independent evidence without a reasoned model.
03
Calculate the spread
Use the sample standard deviation for observed spread and show the denominator convention used.
04
Label the uncertainty
If reporting SE or an uncertainty interval, name the formula, coverage or confidence convention, and components included.
05
Write the limit
State whether the comparison supports repeatability, accuracy against a reference, or only a difference in this dataset.
Checkpoints
Observe, record, investigate.
Accuracy has a reference or accepted value.
SD and SE are named rather than called generic error bars.
Technical repeats are separated from independent units.
The comparison has a decision or practical threshold defined in advance.
Common mistakes
Correct the process, preserve the record.
A small SD is called accurate.
Ask for a reference; tight clustering only supports a repeatability statement.
SE is reported to describe individual reading variability.
Use SD for observed spread and explain that SE concerns the estimated mean under assumptions.
Overlapping error bars are called proof of no difference.
Use the defined comparison and uncertainty method; visual overlap is not a universal test.
More readings are treated as more independent samples.
Check whether they are repeated readings of the same unit before interpreting n.
Practice
SD versus SE
For supplied readings 9.8, 10.0, 10.2, and 10.0 mL, calculate the mean, sample SD, and SE. Explain what each says and what it does not say.
What is the experimental unit?
Which value describes reading-to-reading spread?
What assumption supports SE?
What reference would be needed to discuss accuracy?
