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Accuracy, Error, Precision, and Uncertainty

- Terms
- Uncertainty
- PDFs
  - Gaussian
  - Uniform
  - Triangular
- Procedure
- Combined
- Example
- References


All measurements of physical quantities are subject to uncertainties in the measurements. Variability in the results of repeated measurements arises because variables that can affect the measurement result are impossible to hold constant. Even if the "circumstances," could be precisely controlled, the result would still have an error associated with it.  This is because the scale was manufactured with a certain level of quality, it is often difficult to read the scale perfectly, fractional estimations between scale marking may be made and etc. Of course, steps can be taken to limit the amount of uncertainty but it is always there. 

In order to interpret data correctly and draw valid conclusions the uncertainty must be indicated and dealt with properly. For the result of a measurement to have clear meaning, the value cannot consist of the measured value alone.  An indication of how precise and accurate the result is must also be included.   Thus, the result of any physical measurement has two essential components: (1) A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured, and (2) the degree of uncertainty associated with this estimated value.  Uncertainty is a parameter characterizing the range of values within which the value of the measurand can be said to lie within a specified level of confidence.  For example, a measurement of the width of a table might yield a result such as 95.3 +/- 0.1 cm.  This result is basically communicating that the person making the measurement believe the value to be closest to 95.3cm but it could have been 95.2 or 95.4cm.  The uncertainty is a quantitative indication of the quality of the result.  It gives an answer to the question, "how well does the result represent the value of the quantity being measured?"

The full formal process of determining the uncertainty of a measurement is an extensive process involving identifying all of the major process and environmental variables and evaluating their effect on the measurement.  This process is detailed in the ISO Guide to the Expression of Uncertainty in Measurement (GUM) and the corresponding American National Standard ANSI/NCSL Z540-2. This material will provide a basic introduction.

The first step is to develop an understand of the terminology related to measurement quality.  The terminology can be a bit confusing, which is partly due to some of the terminology having subtle differences and partly due to the terminology being used wrongly and inconsistently.  For example, the term "accuracy" is often used when "trueness" should be used.  Using the proper terminology is key to ensuring that results are properly communicated.    

True Value

Since the true value cannot be absolutely determined, in practice an accepted reference value is used.  The accepted reference value is usually established by repeatedly measuring some NIST or ISO traceable reference standard.  This value is not the reference value that is found published in a reference book.  Such reference values are not "right" answers; they are measurements that have errors associated with them as well and may not be totally representative of the specific sample being measured

Accuracy and Error

Accuracy is the closeness of agreement between a measured value and the true value.  Error is the difference between a measurement and the true value of the measurand (the quantity being measured).  Error does not include mistakes made by the person making the measurement.  Values that result from reading the wrong value or making some other mistake should be explained and excluded from the data set.  Error is what causes values to differ when a measurement is repeated and none of the results can be preferred over the others. Although it is not possible to completely eliminate error in a measurement, it can be controlled and characterized.  Often, more effort goes into determining the error or uncertainty in a measurement than into performing the measurement itself.

The total error is usually a combination of systematic error and random error. Sometimes results are quoted with two errors. The first error quoted is usually the random error, and the second is the systematic error. If only one error is quoted it is the combined error.

Systematic error tends to shift all measurements in a systematic way so that in the course of a number of measurements the mean value is constantly displaced or varies in a predictable way.  The causes may be known or unknown but should always be corrected for when present.  For instance, no instrument can ever be calibrated perfectly so when a group of measurements systematically differ from the value of a standard reference specimen, an adjustment in the values should be made.  Systematic error can be corrected for only when an accepted reference value (such as the value assigned to a calibration or reference specimen) is known.  

Random error is a component of the total error which, in the course of a number of measurements, varies in an unpredictable way.  It is not possible to correct for random error.  Random errors can occur for a variety of reasons such as:

  • Lack of equipment sensitivity.  An instrument may not be able to respond to or indicate a change in some quantity that is too small or the observer may not be able to discern the change.
  • Noise in the measurement.  Noise is extraneous disturbances that are unpredictable or random and cannot be completely accounted for.
  • Imprecise definition.  It is difficult to exactly define the dimensions of a object.  For example, it is difficult to determine the ends of a crack with measuring its length.  Two people may likely pick two different starting and ending points.

Trueness and Bias

Trueness is the closeness of agreement between the average value obtained from a large series of test results and an accepted true.  The terminology is very similar to that used in accuracy but trueness applies to the average value of a large number of measurements.  Bias is the difference between the average value of the large series of measurements and the accepted true.  Bias is equivalent to the total systematic error in the measurement and a correction to negate the systematic error can be made by adjusting for the bias.

Precision, Repeatability and Reproducibility

Precision is the closeness of agreement between independent measurements of a quantity under the same conditions. It is a measure of how well a measurement can be made without reference to a theoretical or true value. The number of divisions on the scale of the measuring device generally affects the consistency of repeated measurements and, therefore, the precision.  Since precision is not based on a true value there is no bias or systematic error in the value, but instead it depends only on the distribution of random errors.  The precision of a measurement is usually indicated by the uncertainty or fractional relative uncertainty of a value.

Repeatability is simply the precision determined under conditions where the same methods and equipment are used by the same operator to make measurements on identical specimens.  Reproducibility is simply the precision determined under conditions where the same methods but different equipment are used by different operator to make measurements on identical specimens.     


Uncertainty is the component of a reported value that characterizes the range of values within which the true value is asserted to lie.  An uncertainty estimate should address error from all possible effects (both systematic and random) and, therefore, usually is the most appropriate means of expressing the accuracy of results.  This is consistent with ISO guidelines.  However, in many measurement situations the systematic error is not address and only random error is included in the uncertainty measurement.  When only random error is included in the uncertainty estimate, it is a reflection of the precision of the measurement. 

The process for characterizing the uncertainty of a measurement or a calculation based on multiple measurements will be discussed in the following pages.

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