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Grades 9–12 · Everyday Life

What do uncertainty bars on a measurement mean?

A measurement is more than a displayed number. It includes an estimate of the quantity and an account of uncertainty: the remaining doubt associated with that estimate under a measurement model. Repeatability, calibration and other identified effects can contribute. A precise-looking display does not remove them.

J17 visual explanationFictional length estimates A 10 and B 10.3 centimeters, each with standard uncertainty 0.20 centimeters. Coverage factor 2 gives expanded uncertainty 0.4. Reported intervals are 9.6 to 10.4 and 9.9 to 10.7 centimeters; they overlap. Approximate 95 percent coverage assumes a normal model and reliably known uncertainty. Error bars alone do not supply a statistical significance test or establish covariance assumptions.An uncertainty bar needs a definitionFictional lengths · standard uncertainty u = 0.20 cmCoverage factor k = 2; U = k × uA9.60–10.40 cm10.00 cmB9.90–10.70 cm10.30 cm9101112Bars overlap; no significance test is supplied.

In the diagramAn uncertainty bar needs a definitionFictional lengths · standard uncertainty u = 0.20 cmCoverage factor k = 2; U = k × uA9.60–10.40 cm10.00 cmB9.90–10.70 cm10.30 cm9101112Bars overlap; no significance test is supplied.

Initial normal-model bars use k = 2 and u = 0.20 cm: A spans 9.60–10.40 cm and B spans 9.90–10.70 cm; overlap alone supplies no significance verdict.

Our fictional lengths are A = 10.00 cm and B = 10.30 cm. Each has standard uncertainty u = 0.20 cm. The diagram uses expanded uncertainty U = k × u, where k is a stated coverage factor. With k = 2, U = 0.40 cm, giving intervals 9.60–10.40 cm and 9.90–10.70 cm.

The approximately 95% coverage associated with k = 2 is conditional. It relies here on a normal uncertainty model and a reliably known standard uncertainty. A different distribution or poorly estimated uncertainty can change that interpretation. With k = 1 under the same stipulated model, the marginal coverage is about 68%, not 95%.

The label on a bar matters. Other figures may use standard deviations of observations, standard errors or confidence intervals. These are not interchangeable definitions. A bar cannot be interpreted correctly from its shape alone.

Overlap also is not an automatic test of whether two estimates differ statistically. A comparison requires the uncertainty of the difference, including any shared calibration effects or other dependence, and a defined testing procedure. The diagram supplies no such test. Its bars communicate conditional measurement intervals, not a verdict of significance.

About this explanation

Starting knowledge: Interpret intervals and normal-distribution coverage and distinguish uncertainty in an estimate from variation among individual observations.

Sources

  • National Institute of Standards and Technology — NIST TN 1297: 6. Expanded Uncertainty
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