Calibration, uncertainty and repeatability

A consistent instrument can still be biased

A balance can return nearly the same indication each time while being offset. Calibration relates indications to reference values under stated conditions, including uncertainty. A correction may then be applied; calibration is not automatically an adjustment.

Suppose a balance repeatedly indicates about 50.200 g when a reference item has an assigned mass of 50.000 g. The readings can form a tight cluster while their centre differs from the reference. Repeat scatter describes the cluster’s spread; the comparison with the reference reveals an estimated offset. These answer different questions.

An indication is the instrument’s displayed value. The measurand is the quantity being estimated: here, the mass of the specified item under the stated measurement conditions. The reference value also has uncertainty; it is not an infinitely precise glimpse of a true mass.

In this invented local model, the instrument reads high by 0.200 g. Define an additive correction C as reference value minus mean reference indication. Then C=−0.200 g. Adding this correction moves an indication downward. Reversing its sign would increase the estimated error.

Applying a correction in a calculation does not adjust the instrument. An adjustment changes its response. After adjustment the earlier calibration relationship may no longer describe it, so the relationship must be checked again. The physical object, instrument and reported corrected result should remain identifiable.

Build the measurement picture

Repeatability examines variation under specified repeated conditions. Uncertainty also includes relevant reference, resolution, correction and model contributions. Repetition can reduce some contributions to a mean but cannot erase an unrecognised common offset. Report uncertainty with its meaning rather than an unexplained plus-or-minus sign.

For repeatability, keep the relevant procedure, operator, measuring system, operating conditions and location specified, and take repeats over a short period. Leaving one object untouched and copying an unchanged display five times does not test the variability of removing and replacing it.

A common correction shifts every reading by the same amount, so it does not change differences among them. Correcting the five reference indications gives 49.980, 50.000, 50.020, 49.990 and 50.010 g. Their mean now equals the assigned reference value, but their range is still 0.040 g. Agreement at that calibration point follows from how the correction was estimated; it is not an independent demonstration of perfect performance.

The coupon readings average 50.900 g. Applying the same correction gives an estimated mass 50.700 g. This uses a local constant-offset model. Its suitability depends on the range, load arrangement and conditions; the single reference point alone cannot establish the response everywhere.

The spread of readings and the uncertainty of a mean

For the five coupon readings, subtract their mean 50.900 g. The deviations are −0.020, 0, +0.020, −0.010 and +0.010 g. Their squared deviations sum to 0.001000 g². The sample standard deviation is s=√[0.001000/(5−1)]=0.01581 g. It describes repeat scatter in individual readings under these conditions.

If the observations are independent and their distribution is sufficiently stable for this statistical estimate, the standard uncertainty contribution for their mean is u_rep=s/√n. With n=5, this is 0.007071 g. Averaging can reduce this contribution because independent positive and negative fluctuations partially cancel.

The uncertainty of the shared calibration correction does not get divided by √n when the same correction is applied to every coupon reading. Any uncertainty about that common shift remains in the mean. Drift or correlated fluctuations can also invalidate the simple independent-repeat calculation.

Type A and Type B name evaluation methods, not two physical kinds of error: statistical analysis of observations versus other information. A correction’s uncertainty can itself contain statistically evaluated contributions. A spreadsheet column labelled systematic does not, by itself, establish how a contribution should be combined.

State the measurement model before combining contributions

Use the deliberately limited additive model m=mean(I)+C +Δ_env. Here mean(I) is the mean coupon indication, C is the calibration correction, and Δ_env corrects for a specified environmental difference from calibration conditions. In this example Δ_env is estimated as zero; uncertainty about it remains.

The following invented budget supplies standard uncertainties: uncertainty values expressed on an estimated-standard-deviation scale. The supplied u_C already includes the relevant reference and calibration-evaluation contributions. They must not be added a second time as separate rows. All listed contributions are stipulated uncorrelated, and each enters this additive model with coefficient +1.

Under these assumptions, add variances rather than adding the standard uncertainties directly: u_c²=u_rep²+u_C²+u_env². Thus u_c=√(0.007071²+0.010²+0.003²)g=0.01261 g. The subscript c denotes combined standard uncertainty. This is a result for the specified model and budget, not a universal rule for every material test.

If inputs share information, covariance can matter. A force/area property also has a different measurement function and sensitivities. Reused reference data, geometry, temperature correction, sampling and model inadequacy need their own assessment. Do not add an unexplained resolution term when the same effect has already been counted elsewhere.

Say what the uncertainty statement means

An expanded uncertainty U is a chosen coverage factor k times u_c. For the illustrative choice k=2, U=0.02522 g, which can be rounded to 0.025 g. Report, for example: corrected coupon mass m=50.700 g, expanded uncertainty U=0.025 g, k=2, using the stated additive model and budget. Keep the calibration and observation conditions with that result.

This numerical statement defines the expanded-uncertainty interval approximately 50.675 g to 50.725 g after rounding. It is not a guaranteed bound on all possible mistakes. Nor does choosing k=2 by itself establish 95% coverage. Such an interpretation needs a justified distribution and adequate knowledge of the uncertainty estimates. Five repeats alone do not establish those conditions.

A smaller uncertainty requires better information about the important contributions. More independent coupon repeats can reduce u_rep; a better calibration or better control of the environmental transfer addresses different terms. Correcting a known offset and estimating uncertainty are complementary tasks.

A calibration at one load, temperature or contact arrangement is not automatically valid for another. A later reference at 100.000 g that indicates 100.400 g suggests a different offset from the 50 g comparison; the constant-offset model needs investigation across that range. For a material test, instrument calibration also does not establish specimen representativeness, correct gripping, or that the chosen test measures the intended property.

Worked examples

Try the reasoning

1. Is calibration the same as adjusting an instrument? A technician subtracts 0.200 g in a worksheet while the balance still displays the old indications. What changed?

   Show the answer and reasoning

   No. The worksheet applies a correction; the instrument response has not been adjusted.

   Calibration establishes the relationship used for interpreting indications. Applying that relationship changes the reported estimate. Adjustment changes the measuring system and can require a new calibration relationship.

2. What does a bare ±0.025 g leave unclear? Does writing k=2 automatically settle every missing detail?

   Show the answer and reasoning

   It leaves the kind of uncertainty, underlying model/conditions and any justified coverage interpretation unclear. k=2 identifies a multiplier but does not by itself establish 95% coverage.

   Specify the measured quantity and corrected value, whether the uncertainty is standard or expanded, its budget/model and the coverage factor when used. A probability statement requires its own basis.

3. A different reference has assigned mass 20.000 g and gives indications 20.080, 20.100 and 20.120 g. In a valid local constant-offset model, find C and correct a coupon mean indication 21.340 g.

   Show the answer and reasoning

   The reference mean is 20.100 g, so C=−0.100 g. The corrected coupon estimate is 21.240 g.

   Subtract the mean reference indication from the assigned reference value, then add the correction to the coupon indication. Adding +0.100 g would move a high indication farther in the wrong direction. Reference/correction uncertainty still remains.

4. A learner divides the supplied u_C=0.010 g by √20 because twenty coupon observations were taken. What would need to be different for that statistical reduction to be justified?

   Show the answer and reasoning

   Taking more coupon observations does not independently redetermine the common correction. Its contribution stays 0.010 g in the stated budget.

   Averaging independent fluctuations can reduce a mean’s repeat contribution. A shared reference/correction uncertainty remains common. A new calibration evaluation would require its own model, independent information and treatment of common components; it cannot be assumed from the coupon count.

5. The 50.000 g reference has mean indication 50.200 g. A 100.000 g reference has mean indication 100.400 g. Can one simply subtract 0.200 g throughout this larger range?

   Show the answer and reasoning

   These two central observations do not support one constant offset across the range; the observed offsets are 0.200 g and 0.400 g.

   Investigate an appropriate calibration relationship and associated uncertainty across the intended range. The pattern might motivate a scale-factor model, but two values alone do not establish its validity, time stability or behaviour elsewhere.

6. A colleague uses the calibrated balance to compare two material coupons, then declares their tensile strengths equal because their corrected masses agree within uncertainty. What additional issue remains?

   Show the answer and reasoning

   Mass agreement does not establish tensile-strength agreement, and the mass measurement’s uncertainty does not cover every other material property.

   Identify the intended property and its test, specimen geometry, preparation and conditions. Calibration supports a specified measurement relationship; it does not turn one measured quantity into evidence for every service claim.

What to carry forward

• A shared offset can remain in a highly repeatable set of readings. Estimate a justified correction with its own uncertainty.
• Averaging reduces a repeat contribution only under suitable statistical conditions; it does not erase a shared calibration uncertainty.
• Define the measurement model before combining standard uncertainties, and avoid double counting or ignoring covariance.
• Report the kind of uncertainty, coverage factor when used and conditions; transfer beyond the calibration range requires evidence.