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Archive for December, 2019

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