Uncertainty of Measurement


Introduction and Overview

Under ISO 17025 (clause 5.4.6) laboratories are required to estimate the uncertainty of measurement in all tests. At present, however, testing laboratories are not required to report uncertainty except under the following circumstances.


  • When the client requests it.

     

  • Where the uncertainty is essential in order to allow the result of the testing to be interpreted.

 

The purist might well argue that a knowledge of the uncertainty is essential for the interpretation of any measurement but in practice accreditation bodies do not take this view. Hence reporting of uncertainty is only required at the client’s request. However it must be emphasised that the laboratory still has to carry out the estimation and assessors will need to see the result of this exercise.


The general approach to uncertainty of measurement requires the following steps.


  • Consider the method and identify any areas of uncertainty not covered by the intermediate precision study and quantify them. Typical examples of this type of uncertainty will be instrument calibrations, issues of purity of standards, uncertainty in concentration of calibration solutions and other references.

     

  • In order to derive standard uncertainties from available data on calibration uncertainties and tolerances it is necessary to decide on the actual statistical distribution of the data and to carry out an appropriate conversion from the data supplied to a standard uncertainty expressed as a standard deviation.

     

Textile and leather testing involve chemical measurements, physical measurements and measurement of relevant properties such as colour fastness. The uncertainty in chemical measurements is dealt with in detail in the part of this website dedicated to chemical testing [HL here]. Some examples of uncertainty estimates for other areas of textile and leather testing are shown below.


Example Uncertainty Estimates

Weight Per Unit Area


Procedure Outline

Self-evidently this measurement will be subject to uncertainties resulting from the measurement of area and from weighing. The method, in outline, involves the following.


  • Condition the leather sample for 24 hours in an agreed standard atmosphere.

     

  • Using a ruler mark out a square of side 25 cm and cut out.

     

  • Weight the square of material on a three place balance and calculate the weight of one square meter in grams.

     

Intermediate Precision (Type A Uncertainty)

An intermediate precision exercise was carried out as part of method validation where a cloth of high homogeneity was subjected to the test 11 times using three different operators and spread over three weeks. In each case the same balance and ruler were used throughout. The result of this exercise returned a mean value of 372.346 g for the 25 cm square with a standard deviation of ±4.321 g. Expressed as a percentage this is 372.346± 1.605%.


This represents the uncertainty arising from the laboratory’s actual operations and is effectively the in-house reproducibility for the method. It will largely derive from differences between operators and over time in measurement of the sample. There is not likely to be any significant similar operator effect derived from the weighings as the operation is simple and digital.


The same balance and ruler were used throughout the contributions. Hence the uncertainty in weighing with that balance and in the calibration of the ruler need, additionally, to be taken into account.


Weighing Uncertainty(Type B )


The uncertainty in a weighing is a combination of three contributions.


  • The uncertainty in the annual balance calibration. This will be stated on the calibration certificate and will normally be expressed as a 95% confidence interval. For the purpose of this exercise assume that the value quoted on the annual calibration certificate is ±0.0003g.


  • The uncertainty which results from the tolerance allowed on the balance daily check. For a three place balance this will typically be ±0.001g, i.e. the balance is allowed to drift by no more than one digital interval. This represents a tolerance on the weighing of ±0.001g.


The uncertainty in weighing with this balance is calculated below.

Source of Uncertainty

Value (g)

Nature of Value

Distribution Type

Calculation to Provide Standard Uncertainty

Value of Standard Uncertainty (g)

Uncertainty in annual balance calibration.

±0.0003

95% confidence interval based on coverage factor of 2.

Normal

0.0003/2

0.00015

Uncertainty due to tolerance on daily check.

±0.001

Tolerance for drift

Rectangular

0.001/Ö 3

0.0005774

Uncertainty due to digital display resolution.

±0.0005

Tolerance created by limitations on display resolution.

Rectangular

0.0005/Ö 3

0.0002887

Combined Uncertainty (square root of 0.000152 + 0.0005772 + 0.00028872)

0.000662

This is the absolute uncertainty, in grams, of weighing with this balance. For the sample concerned, weight 372.346 g the percentage uncertainty is 0.000178%.


Area Uncertainty Due to Uncertainty in Ruler Calibration (Type B)

The measurement is done with a 30 cm ruler marked in millimetres. The quoted uncertainty of calibration, as a 95% confidence interval, is ±0.2 mm. The standard uncertainty will be ±0.1 mm assuming a normal distribution.


Expressed as a percentage of the 25 cm (250 mm) measurement this is ±0.04%.

In order to obtain an area two measurements are made so the 0.04% must be incorporated twice by taking the square root of the sum of the squares, i.e. square root of (0.042 + 0.042) = ± 0.0566%. The uncertainties are combined in this way and not simply added as there is no reason to suppose they would both operate with the same sign.


We now have values for all of the contributions to uncertainty as percentages. Note that expression of the uncertainties in this relative fashion as opposed to as raw values is essential as we are now going to combine them to get an overall uncertainty. We cannot do this directly as we would then be combining cm and grams which is clearly absurd !

Combined Uncertainty


Source of Uncertainty

Value %

Type A uncertainty from intermediate precision

1.605

Uncertainty in weighing

0.000178

Uncertainty due to measurement of sample area

0.0566

Combined Uncertainty (square root of 1.6052 + 0.0001782 + 0.05662)

1.606

Converted back into grams for the weighing of 372.346 g this means an absolute standard uncertainty of ±5.980 g for the 25 cm square. In terms of square meters the result is 5957.536 g per square meter with a standard uncertainty of 95.678 g per square meter. Applying a coverage factor of 2 we have a 95% confidence interval of ±191.356 g per square meter.

Note the following points.


  • The overall uncertainty is dominated by the intermediate precision. This is common in many types of testing. This is how it should be. If the uncertainty in calibration of one of the pieces of equipment dominated the uncertainty then a better level of calibration would be indicated.

     

  • The intermediate precision inevitably includes and element from sample variability since the material will never be perfectly homogeneous. Any realistic measurement of intermediate precision must use a new sample each time as the uncertainty derived from cutting the sample needs to be included.

     

  • This uncertainty estimate is only valid for measurements of the order of 6000g/sqm. Other ranges would have to be checked as intermediate precision may well be a function of the point in the measurement range. Typically intermediate precision should be estimated at 20% and 80% of the full range of measurement for the method.

     

Yarn ID

Mean of 11 Determinations (N)

Standard Deviation

(N)

A

53.2

0.91

B

102.5

1.96

C

203.4

3.74

Thus establishes the type A contribution to uncertainty. The only additional factor to be considered is the calibration uncertainty of the load cell of the universal tester since the same instrument was used throughout.

Uncertainty in Load Cell

The calibration certificate for the load cell of the tester states that the uncertainty in measurement, based on a 95% confidence interval, is 0.42N plus 0.1% of the reading.

Hence the uncertainty in the force measurements is as shown below.

Yarn ID

Mean of 11 Determinations (N)

Standard Uncertainty in Force Measurement (N)

A

53.2

0.4732

B

102.5

0.5225

C

203.4

0.6234

Combined Uncertainty

The two sources of uncertainty are now combined, to give the overall standard uncertainty for each measurement, by taking the square root of the sum of squares.

Yarn ID

Mean of 11 Determinations (N)

Standard Deviation

(N)

Standard Uncertainty in Force Measurement (N)

Combined Uncertainty

(N)

Combined Uncertainty as Percentage of Measurement

A

53.2

0.91

0.4732

1.026

1.928

B

102.5

1.96

0.5225

2.028

1.979

C

203.4

3.74

0.6234

3.792

1.864

As a percentage of the measured value the standard uncertainty is similar across the measurement range. It would be legitimate to adopt the practice, in this case, of reporting the uncertainty for measurements in the range 50 to 200N as 2% of the value. The 95% confidence interval would be 4% of the value.

Light Fastness

Procedure Outline

The procedure being used is a simple mercury/tungsten light fastness test such as BS 1006. The equipment is fully controlled as specified by the supplier and all of the requirements of the standard with respect to checking the equipment, lamp lifetime and humidity control are rigorously applied.

This means that the equipment introduces no uncertainties of significance since the controls required by the standard will ensure this.

The only uncertainty in measurement which remains is with respect to estimates of the colour fade made with a grey scale, effectively the intermediate precision.

Overall Uncertainty


 

The laboratory will need to have a programme to ensure that grey scale and similar estimations are being carried out with consistency and within controlled limits. This would produce data as shown below. The standard deviation for determinations by each technician and for measurements over the year are shown. From the range data it can be seen the laboratory can legitimately claim that the maximum range of any estimate covers no more than half a grey scale unit, i.e. ±0.25. this would be a reasonable estimate of the uncertainty in any light fastness measurement.


Clearly uncertainty for measurements such as grey scale evaluations are not susceptible to the same kind of statistical analysis as instrumental measurements. ISO 17025 is still satisfied, however, since it requires an appropriate estimate of uncertainty and, in particular, a demonstration that the laboratory has considered all potential sources of uncertainty and arrived at an estimate of the main contributions

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