Total Analytical Error Calculator
Total Analytical Error Calculator
Combine bias and imprecision into one figure for the worst a result is likely to be wrong by, then compare it with the total error your method is allowed.
Total Analytical Error
Bias, CV → total errorA glucose method with 1.5% bias and a 2.5% CV, one-sided 95% multiplier, against a 10% allowable total error
Formula
k = 1.65 for one-sided 95% · k = 2 for two-sided 95%
fit for purpose when TE ≤ TEa
- |bias|
- systematic error, taken as its absolute size. It is added, not combined in quadrature, because a bias does not average out: every result carries the whole of it, in the same direction, every time
- CV
- random error as a percentage of the mean, from your own long-term quality control at a concentration near the decision level. Repeating a measurement reduces the effect of this term and does nothing to the bias term
- k
- how much of the random error to cover. 1.65 covers 95% of results on one side of the mean and is the multiplier in the original Westgard formulation; 2 covers 95% on both sides and gives a larger figure. The choice changes the answer by about a fifth, so quote it
- TEa
- allowable total error — the room the method is given. It is a chosen specification rather than a measurement, so two laboratories can reach opposite verdicts on identical performance by choosing different sources. Name yours
- the relationship to sigma
- σ = (TEa − |bias|) ÷ CV, which is the same three numbers rearranged. TE asks 'how wrong is a result likely to be?' and sigma asks 'how many standard deviations of room are left?'. TE ≤ TEa and σ ≥ k are very nearly the same statement
- what it is not
- measurement uncertainty. That model combines the same two components in quadrature rather than by addition and treats uncorrected bias as an uncertainty component. The two give similar-sized answers and mean different things
Worked example
A glucose method with 1.5% bias and a 2.5% CV, one-sided 95% multiplier, against a 10% allowable total error
|bias| = 1.5%
1.65 × 2.5 = 4.125% from imprecision
TE = 1.5 + 4.125 = 5.6%
5.6% against a 10% allowable total error: the method meets its specification, with about 4.4% to spare
The same three numbers as a sigma metric: (10 − 1.5) ÷ 2.5 = 3.4 sigma — marginal, and needing a full multirule QC procedure. TE and sigma are the same information read two ways, and the second reading is the one that tells you how to run the controls
Choosing k = 2 instead gives 1.5 + 5.0 = 6.5%, still inside 10%. Recalibrating the bias away instead gives 0 + 4.125 = 4.1%, and lifts sigma to 4.0
Where the total error comes from
| Change | Bias | CV | TE (k = 1.65) | Sigma (TEa 10%) |
|---|---|---|---|---|
| Starting point | 1.5% | 2.5% | 5.6% | 3.4 |
| Recalibrate away the bias | 0% | 2.5% | 4.1% | 4.0 |
| Halve the imprecision instead | 1.5% | 1.25% | 3.6% | 6.8 |
| Both | 0% | 1.25% | 2.1% | 8.0 |
| Double the bias | 3.0% | 2.5% | 7.1% | 2.8 |
Three models of the same problem
| Model | How bias and imprecision combine | Answers |
|---|---|---|
| Total analytical error | |bias| + k × CV — added | How wrong is a single result likely to be, at worst? |
| Sigma metric | (TEa − |bias|) ÷ CV — subtract, then divide | How much room has the method got, and how much QC does it need? |
| Measurement uncertainty | √(uRw² + ubias²), then × k — in quadrature | What interval around the reported result plausibly contains the true value? |
| Reference change value | √2 × Z × √(CVa² + CVi²) — in quadrature, with biology | How much must two results on one patient differ before the change is real? |
One number for how wrong a result can be
A method has two kinds of error and they behave quite differently. Bias is systematic: every result carries it, in the same direction, and repeating the measurement does not help. Imprecision is random: it scatters results around the method's own mean, and averaging reduces it. Total analytical error puts the two together to answer a single practical question — how far from the truth is one result from this method likely to be?
The formula adds the absolute bias to a multiple of the CV. Using 1.65, which covers 95% of the random scatter in one direction, a method with 1.5% bias and a 2.5% CV has a total error of 1.5 + 4.125, or about 5.6%. A glucose of 5.00 mmol/L from that method could plausibly be reported when the true value is as far away as 5.28, and the answer to 'is that acceptable?' is not a matter of taste: it is a comparison with the allowable total error, which comes from a biological-variation-derived specification, a regulatory limit, or an external quality assessment scheme's criteria. Against a 10% allowable total error this method is comfortably fit for purpose.
Some authors use 2 rather than 1.65, on the grounds that the error could go either way and a two-sided 95% interval is the honest one. That raises the same example to 6.5%. The difference is about a fifth of the answer, which matters when a method is near its limit, so state the multiplier whenever you quote a figure. The addition itself is deliberately pessimistic: it assumes the bias and a large random excursion line up in the same direction at the same moment. That is the worst case, and the worst case is what a single patient result has to survive.
The most useful thing about total error is how little it is really separate from the other quality statistics. The sigma metric is the same three numbers rearranged — subtract the bias from the allowable error and divide by the CV instead of adding — and where total error asks how wrong a result might be, sigma asks how many standard deviations of room are left and converts the answer into a quality control design. A method that meets its allowable total error with room to spare is a method with high sigma, and one that scrapes past is a method that needs a full multirule procedure. It is worth calculating both from the same bias and CV figures, at the same concentration, and quoting the concentration and the source of the specification alongside them.
Frequently asked questions
How do you calculate total analytical error?
Add the absolute bias to a multiple of the CV, both as percentages. Using the one-sided 95% multiplier of 1.65, a method with 1.5% bias and a 2.5% CV has a total error of 1.5 + 1.65 × 2.5 = 5.6%. The method is fit for purpose if that figure is within the allowable total error.
Should I use 1.65 or 2 as the multiplier?
1.65 covers 95% of the random scatter in one direction and is the multiplier in the original Westgard formulation; 2 covers 95% in both directions and is the more conservative choice. The two differ by about a fifth of the answer, so the only real requirement is to say which you used.
Why is bias added rather than combined in quadrature?
Because it does not behave like random error. A bias is present in every result, in the same direction, and repeating the measurement cannot average it away. The total error model therefore treats it as a fixed offset and adds the random component on top of it.
Is total analytical error the same as measurement uncertainty?
No. They model the same two components differently. Total error adds bias to imprecision linearly and describes the worst a single result is likely to be wrong by. Measurement uncertainty combines them in quadrature, treats uncorrected bias as an uncertainty component, and describes an interval around the reported value. The figures come out similar in size and mean different things.
Where does allowable total error come from?
From a chosen specification: a goal derived from biological variation, a regulatory limit such as CLIA, or an EQA scheme's acceptance criteria. These can differ substantially for the same analyte, which is why a total error assessment is only comparable between laboratories when both state their source and the concentration they worked at.
Related calculators
References
- Westgard JO, Carey RN, Wold S. Criteria for judging precision and accuracy in method development and evaluation. Clin Chem. 1974;20(7):825–833.
- Westgard JO, Westgard SA. Six Sigma quality management system and design of risk-based statistical quality control procedures. Clin Lab Med. 2017;37(1):85–96.
- CLSI EP21-Ed2. Evaluation of Total Analytical Error for Quantitative Medical Laboratory Measurement Procedures. 2nd ed. Clinical and Laboratory Standards Institute; 2016.
- Oosterhuis WP, Theodorsson E. Total error vs. measurement uncertainty: revolution or evolution? Clin Chem Lab Med. 2016;54(2):235–239.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.