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Posts tagged ‘repeatability’

The uncertainty of measuring instruments

In addition to classical analytical methods, we have several instruments that are helpful in our routine laboratory analysis.  Examples are aplenty, such as pH meter, dissolved oxygen meter, turbidity meter, Conductivity meter, UV-visible spectrometer, FT-IR spectrophotometer, etc.  Some are being used for in-situ measurements in the field.  Hence, it is important to estimate their respective measurement uncertainty.

Most measuring instruments are generally characterized by:

  • Class (depending on the precision of its measurement grading, such as Class A and Class B of burette, etc)
  • Sensitivity on instrument response
  • Discrimination threshold in identification
  • Resolution of displaying device
  • Stability as measured by drifting of its graded measurement

To evaluate the uncertainty of readings from a measuring instrument, we look for two basic uncertainty contributors, namely:

  1. The maximum permissible error provided by the supplier. 
  2. The repeatability of measuring instrument

Maximum permissible error (MPE)

By VIM definition, MPE is an extreme value of measurement error, with respect to a known reference quantity value, permitted by specifications or regulations for a given measurement, measuring instrument, or measuring system.  It is the ‘best’ accuracy confirmed by a calibration and specified by the manufacturer of the instrument during the warranty period.

MPE data can always be found in the manufacturer’s manual under the instrument specification. It is usually expressed in one of the following manners:

  1. When the MPE is constant throughout the instrument indications, it is expressed as:

MPE = +/-a

          where a is a given value for its unit. 

For example, a glass thermometer with a measuring range of 0 – 50oC with sub-divided units of 0.1oC, MPE = +/-0.2oC

  • When MPE varies with a change of instrument indications following a regression line, the maximum error tolerance can be a given relation as follows:

MPE = +/-(a + bx)

            where x is a measured value.

  • When the measuring instrument uses a constant relative standard deviation RSD, its MPE can be expressed as:

MPE = +/-RSD.x

Repeatability of measuring instrument

Repeatability is the closeness of the agreement between the results of successive measurements of the same measure carried out under the same conditions of measurement, being taken by a single person or instrument on the same item, under the same conditions, and in a short period of time. Indeed, repeatability is a measure of instrument indicator’s variation under successive measurement exercise.  It is expressed as sr, the standard deviation of a series of repeated measurements.


A breathalyzer is a device for estimating blood alcohol content (BAC) from a breath sample. A given brand breathalyzer has the following performance data:

  1. Maximum permissible error

BAC  < 0.20 g/100ml               MPE = +/- 0.025 g/100ml

BAC  0.20 – 0.40 g/100ml       MPE = +/- 0.04 g/100ml

  • Measurement repeatability expressed as standard deviation

sr = +/- 0.006 g/100ml

Evaluating measurement uncertainty of the breathalyzer

  1. The standard uncertainty of the MPE is calculated by MPE/SQRT(3) using the rectangular probability factor for a maximum bound of error estimation.  Hence, we have:

BAC  < 0.20 g/100ml               u(E) = +/- 0.014(4) g/100ml

BAC  0.20 – 0.40 g/100ml       u(E)  = +/- 0.023(1) g/100ml

  • Measurement repeatability

sr = +/- 0.006 g/100ml

The combined standard uncertainty u (Comb) = SQRT(u(E)2 + sr2) and the expanded uncertainties which are 2 x u(Comb) with 95% confidence for the two ranges are as follows:

BAC Testing range g/100ml Comb std uncertainty, u,  g/100ml Expanded uncertainty, U, g/100ml
< 0.20   0.015(6) 0.03
0.20 – 0.40   0.023(9) 0.05

Can we estimate uncertainty by replicates?

The method traditionally practiced by most test laboratories in the estimation of measurement uncertainty is by the ISO GUM (ISO/IEC Guide 98-3) approach, which is quite tedious and time consuming to study and gather uncertainty contributions from each and every step of the test method.  An alternative way of looking at uncertainty is to attempt to study the overall performance of the analytical procedure by involving replication of the whole procedure to give a direct estimate of the uncertainty for the final test result. This is the so-called ‘top-down’ approach.

We may use the data from inter-laboratory study, in-house validation or ongoing quality control. This approach is particularly appropriate where individual effects are poorly understood in terms of their quantitative theoretical models which are capable of predicting the behavior of analytical results for particular sample types.  By this approach, it is suffice to consider reproducibility from inter-laboratory data or long-term within-laboratory precision as recommended by ISO 21748, ISO 11352 and ASTM D 6299.

However, one must be aware of that by repeatedly analyzing a given sample over several times will not be a good estimate of the uncertainty unless the following conditions are fulfilled:

  1. There must be no perceptible bias or systematic error in the procedure.  That is to say that the difference between the expected results and the true or reference value must be negligible in relation to twice of the standard deviation with 95% confidence. This condition is usually (but not always) fulfilled in analytical chemistry.
  • The replication has to explore all of the possible variations in the execution of the method by engaging different analysts on different days using different equipment on a similar sample. If not, at least all of the variations of important magnitude are considered. Such condition may not be easily met by replication under repeatability conditions (i.e. repeated testing within laboratory), because such variations would be laboratory-specific to a great extent.

The conclusion is that replicated data by a single analyst on same equipment over a short period of time are not sufficient for uncertainty estimation. If the top-down approach is to be followed, we must obtain a good estimate of the long-term precision of the analytical method.  This can be done for example, by studying the precision for a typical test method used as a QC material over a reasonable period of time. We may also use a published reproducibility standard deviation for the method in use, provided we document proof that we are able to follow the procedure closely and competently.

What are the types of precision estimates?

Accuracy, Precision & Trueness


When we evaluate the validity of a test result, we are mostly concern if the performance of the test method used is precise and reproducible enough to fit for a particular purpose or to meet the customer’s requirements.  That concern also includes in some cases whether the method detection limit is low enough to meet the regulatory or specification limits required.  Read on ….Types of precision estimates