When does a measurement disagree with a model?
A difference between a prediction and an observation is a place to ask better questions—not an automatic victory for either the model or the measurement.
Every observation below is invented for teaching. No pendulum was built, timed, or measured, and no personal experience is implied.
Begin with two different objects
A prediction is produced by a model and stated inputs. An observation is a recorded result from a measurement procedure. Their difference, using the sign convention here, is the residual = observation − prediction.
For an ideal simple pendulum at small angles, T = 2π√(L/g). With the illustrative inputs L = 1.00 m and g = 9.81 m/s², the predicted period is 2.006 s. OpenStax University Physics §15.4 derives this approximation and states its small-angle boundary.
A tiny invented record
Suppose a fictional timer displays five periods to 0.01 s. These values were chosen by hand; the display resolution is part of the scenario, not a measured uncertainty.
| Trial | Model prediction | Invented observation | Residual |
|---|---|---|---|
| 1 | 2.006 s | 2.03 s | +0.024 s |
| 2 | 2.006 s | 2.01 s | +0.004 s |
| 3 | 2.006 s | 2.04 s | +0.034 s |
| 4 | 2.006 s | 2.02 s | +0.014 s |
| 5 | 2.006 s | 2.03 s | +0.024 s |
The invented mean is 2.026 s. Its mean residual is +0.020 s: the observations sit above the model prediction on average. Their range is 0.03 s and their sample standard deviation is 0.011 s. Those last two numbers describe spread in this fabricated set; they are not a confidence interval and do not quantify every source of measurement uncertainty.
Download the invented record and calculation fields →
Precision, accuracy, uncertainty, and residuals answer different questions
- Precision
- How closely repeated results agree under stated conditions. The tight grouping here illustrates repeatability, but fabricated numbers cannot demonstrate an instrument’s precision.
- Accuracy
- How well a result agrees with a relevant reference or true value. A model prediction is not automatically truth, so this table does not establish accuracy.
- Measurement uncertainty
- A reasoned statement about the dispersion of values that could reasonably be attributed to the measurand. It needs a measurement model and identified contributions; the timer’s displayed decimal places are not enough.
- Residual
- The observed value minus the model prediction under the sign convention declared above. A residual records disagreement; it does not identify its cause.
NIST cautions that accuracy, precision, repeatability, variability, and uncertainty are qualitative concepts whose terminology should be used carefully. Its statistical glossary defines a residual as the observed value minus the predicted or fitted value. See the NIST TN 1297 terminology appendix and NIST/SEMATECH statistical glossary.
Four explanations can coexist
- Random variation: repeated timing can vary around a stable average.
- Systematic measurement effect: the procedure or instrument can shift results in one direction.
- Input mismatch: the actual length or local value of g may differ from the values supplied to the equation.
- Model inadequacy: finite amplitude, string mass, drag, pivot friction, or bob geometry may matter at the desired resolution.
The five residuals alone cannot choose among those explanations. More repetitions can characterize some repeatability, but they do not remove a shared bias or repair the wrong model.
A useful disagreement workflow
- Write the model, input values, units, approximations, and prediction before looking at the observations.
- Describe the measurement procedure and resolution without upgrading resolution into uncertainty.
- Declare the residual sign convention and retain each observation rather than only the mean.
- Plot or inspect residuals for offset, trend, curvature, changing spread, and outliers.
- Revise the procedure, inputs, or model one reasoned change at a time, then collect new evidence.
A residual becomes scientifically useful when it remains traceable to both sides of the comparison. “Close” and “far” require a purpose and a defensible uncertainty model; this synthetic example supplies neither, so it makes no quantified agreement or confidence claim.
Reproduce the arithmetic
The prediction uses 2 * Math.PI * Math.sqrt(1 / 9.81). Each residual subtracts that unrounded prediction from its invented observation; summary values are rounded only for display. Repository tests assert the prediction, mean, residuals, range, sample standard deviation, and explicit evidence boundaries. They verify arithmetic and copy—not a physical result.