Uncertainty of measurements – How to quantify ambiguity of your results?


How to quantify uncertainties?

Ambiguity or uncertainty refers to a degree of doubt on the results of testing. It is an accepted fact that it is never possible to report the actual value as there is always a scope for error, however small it may be, that results in variation between the true value and reported value.

Measurement is truly valid if it is accompanied by definition of degree of uncertainty. Such uncertainties arise from deviations from stipulated environmental controls, non-homogeneity of sample, method or instrumental limitations and personal operator errors. Some of these errors are measurable and can be corrected through calibration whereas others are indeterminate. In other words every measurement or test should be defined in terms of two parameters. First is the bandwidth or interval defining the accuracy of the measurement and the second is the level of confidence.

A number of factors contribute to uncertainty of measurements made in a chemical laboratory. Some of these factors are:

  • Sample homogeneity
  • Sample degradation between collection and analysis
  • Contamination during sampling, sample treatment or storage
  • Incomplete extraction of analyte from sample matrix
  • Interferences arising from other sample matrix components
  • Variations in recommended environmental controls
  • Personal bias in recording observations
  • Errors in weighing or volumetric transfer of solutions
  • Degradation of standards and reference materials over storage

Expression of uncertainty

A set of repeat observations should be made and both mean and standard deviation (SD) are reported. From these the value of uncertainty, U, is calculated as

U =\frac{SD}\sqrt{n}

where n is the number of observations.

The combined uncertainty resulting from several factors is expressed as

Combined uncertainty,

U_C = \sqrt{a^2 + b^2 + c^2 + d^2+......}

where a, b, c, etc are individual uncertainties.

Expanded uncertainty, Ue, is arrived at by multiplying the combined uncertainty with the coverage factor, k

Ue = kU_c

Average factor k of 2 is used to report expanded uncertainty at a confidence level of 95%

Finally the observation is reported to 2 significant digits as

Result = Mean value +/- Ue

In any analysis it is important to appreciate the role of factors that can contribute to uncertainty of measurements. There is a need to identify, minimize and wherever possible quantify them through frequent calibration against reference materials and inter- laboratory proficiency testing.

About Dr. Deepak Bhanot

Dr Deepak Bhanot is a seasoned professional having nearly 30 years expertise beginning from sales and product support of analytical instruments. After completing his graduation and post graduation from Delhi University and IIT Delhi he went on to Loughborough University of Technology, UK for doctorate research in analytical chemistry. His mission is to develop training programs on analytical techniques and share his experiences with broad spectrum of users ranging from professionals engaged in analytical development and research as well as young enthusiasts fresh from academics who wish to embark upon a career in analytical industry.


  1. Brief and helpful! straight points

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