core31 min studyLesson 105 in Physics

Separate scatter from persistent measurement effects

Distinguish random variation, systematic effects, and Type A or Type B uncertainty evaluation without confusing their classifications.

Before you begin

On this page

The question for today

What you’ll understand

Explain which uncertainty contributions averaging can reduce.
Separate random or systematic effects from Type A or Type B evaluation methods.
Build a simple uncertainty budget for a corrected mean.

Repeated readings often scatter, but their average can still be displaced by a shared calibration error. These are different ways a measurement can fail to reveal its intended quantity. Random and systematic describe behavior of measurement effects under specified conditions. Type A and Type B describe how uncertainty components are evaluated. Keeping those two distinctions separate prevents the mistaken belief that repeated trials automatically remove every important uncertainty.

Model the readings, not just their histogram

A simple model writes each indication as the intended value plus a common offset plus a varying term. Independent, zero-mean varying terms become less influential in an average. A common offset remains in every reading and survives that averaging unchanged. The classification depends on conditions: an offset may be effectively fixed during one short run but vary among instruments or days. Specify the ensemble of repetitions before calling an effect random, shared, or independent.

A trend is not harmless scatter. If an instrument warms steadily, consecutive readings may be correlated and their mean may refer to a different effective condition than intended. Plot measurements in time order, interleave references, or randomize an appropriate experimental order to diagnose such structure. Do not apply the familiar inverse-square-root rule merely because many numbers exist. That rule needs a sufficiently stable mean and the stated independence or a justified treatment of temporal covariance.

r_i=x+b+\varepsilon_i,\qquad u_A(\overline r)=\frac{s}{\sqrt n}

The sample standard deviation s estimates individual-reading scatter. The displayed standard uncertainty of the mean assumes independent readings with a common variance and a stable underlying condition. It does not estimate the shared offset b.↔ Scroll wide equations horizontally.

Classify the evaluation method separately

Type A evaluation uses statistical analysis of observations. Type B evaluation uses other available information, such as a calibration certificate, manufacturer specification, or a justified physical bound. Both can yield standard uncertainties that enter the same propagation calculation. Neither label means more scientific, and neither is synonymous with random or systematic. For example, a shared offset uncertainty can be evaluated statistically in a separate calibration campaign and still affect every reading in the later run together.

When a bound is converted to a standard uncertainty, the probability model must be stated. If a correction is plausibly uniform between minus a and plus a, its standard uncertainty is a divided by the square root of three. A different shape or additional information gives a different result. This distribution expresses the information available about the input quantity; it does not claim that a fixed physical offset redraws a fresh value independently on each repeated observation.

u_c^2=u_A^2+u_b^2\quad\text{for independent contributions},\qquad u_b=\frac{a}{\sqrt3}\quad\text{for a uniform bound}

All quantities here are standard uncertainties in the same output unit. The bounded-offset expression requires a justified rectangular distribution over [−a,a]. A covariance term is necessary when the contributions are correlated.↔ Scroll wide equations horizontally.

Correct effects and retain their uncertainty

A known significant systematic effect should be corrected when an adequate correction is available. The correction does not become exact simply because it has been applied; retain uncertainty in its estimated value and in the model supporting it. Avoid reporting a correction as though it were a signed uncertainty. A correction changes the estimated value, whereas a standard uncertainty is a nonnegative measure of dispersion associated with the result under the available information.

An uncertainty budget identifies input estimates, standard uncertainties, sensitivity coefficients, covariance, and physical origin. Use it to choose improvements. If calibration dominates, taking a hundred times as many rapid readings can leave the final uncertainty nearly unchanged. If repeatability dominates and independence is credible, more repetitions may help. Budget thinking connects statistics to apparatus: it tells you whether to improve the reference, stabilize the environment, lengthen the experiment, or reconsider the model rather than simply collecting more digits.

Work it through

A corrected mean with a shared uncertainty

Sixteen independent readings have mean $10.40\,\mathrm{mm}$ and sample standard deviation $0.20\,\mathrm{mm}$ . A calibration supplies an additive correction $-0.30\,\mathrm{mm}$ with standard uncertainty $0.12\,\mathrm{mm}$ , independent of the reading scatter. Find the corrected mean and combined standard uncertainty.

Apply the correction to the estimated mean: $10.40-0.30=10.10\,\mathrm{mm}$ . This changes the central value rather than the number of repeated observations.
The standard uncertainty from independent repeatability is $u_A=0.20/\sqrt{16}=0.05\,\mathrm{mm}$ . The shared calibration uncertainty remains $0.12\,\mathrm{mm}$ ; it is not divided by four.
Combine independent standard contributions: $u_c=\sqrt{0.05^2+0.12^2}=0.13\,\mathrm{mm}$ . This is a standard uncertainty, not a guaranteed maximum error and not an automatically assigned 95 percent interval.

The result

Report $(10.10\pm0.13)\,\mathrm{mm}$ with combined standard uncertainty. The calibration component dominates, so additional readings alone have limited benefit.

The living notebook

Explore the model

Interactive lab

Interactive lab · probability

Check the independence model behind averaging

The probability and trial-count controls show an exact binomial distribution, with mean np and variance np(1−p). Set p = 0.5 and predict that increasing n from 10 to 20 reduces the standard deviation of the success fraction, sqrt[p(1−p)/n], then infer it from the readouts. This is a model of independent binary trials; it does not resample measurements or include a shared calibration bias.

Exact probabilityHighlighted outcomeMean np

Mean · np: 5
Variance · np(1−p): 2.5
Total probability: 1

Success probability p0.5Trials n10

t = 0 s

These are exact binomial probabilities for independent trials with one fixed success probability. Playback tours the distribution; it does not resample outcomes. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAveraging reduces independent trial fluctuations under a specified model; a common offset is a separate contribution.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Type A identifies statistical evaluation, not how the resulting input affects a later set of readings. A shared calibration component can remain fully correlated across that set and survive averaging, regardless of how it was originally evaluated.

Your turn

Try it yourself

An experiment has repeatability standard uncertainty $0.08\,\mathrm V$ for its current mean and an independent calibration standard uncertainty $0.15\,\mathrm V$ . Suppose four times as many independent readings reduce only the repeatability component. Compare the combined uncertainties before and after.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The repeatability uncertainty halves; the calibration contribution stays fixed.

Compare with the solution

Initially $u_c=\sqrt{0.08^2+0.15^2}=0.17\,\mathrm V$ . Four times as many independent readings give a repeatability contribution of $0.04\,\mathrm V$ , so the new combined value is $\sqrt{0.04^2+0.15^2}\approx0.155\,\mathrm V$ . The reduction is modest because calibration dominates. This calculation assumes a stable process and no new correlations or drift during the longer acquisition. A more useful improvement might address the calibration contribution, rather than collecting still more repetitions.

Keep the idea

Take it with you

A shared systematic contribution can survive averaging of arbitrarily many readings.
Type A and Type B classify evaluation methods rather than random and systematic behavior.
Apply justified corrections, then retain uncertainty in those corrections.

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