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In this section we give a brief introduction to the methods used to characterise the accuracy of measuring instruments.

This will help you to make corrections to your instrument readings and calculate the uncertainty in your measurements. Throughout the discussion we refer to ‘the instrument’. The term includes all indicating instruments such as thermometers, voltmeters and mass balances. It also includes ‘material measures’ such as standard masses, standards of electrical quantities, and photometric reflectance.

Characterising the accuracy of an instrument

From experience we all know that measuring instruments are not perfect. One person may get nearly the same result every time a measurement is made, but another person may define the measurement differently and get a different result because the instrument is sensitive to the measurement setup. Even if a measurement is repeated by the same person with the same instrument a slightly different reading is expected. For example, when a wooden rule is used to measure a length some of the variation between measurements is due to an inability to position the rule against the object in exactly the same place every time. Some of the variation is also due to an inability to read the scale accurately, changes in the length of the rule with temperature and humidity, and similar changes in the length of the object. These variations are typical of every measurement.

Some measuring instruments are better than others. For example a steel rule may have better quality scale markings, and be less sensitive to environmental influences than a wooden rule. Repeated measurements made with the steel rule usually result in smaller variations between readings. One of the tasks of the calibration laboratory is to measure just how accurate each measuring instrument is.

To measure the variations in readings associated with a particular instrument, a number of comparisons are made with a more accurate instrument. The differences between the readings of the two instruments provide us with information for calculating the uncertainty. Figure 2 gives a pictorial view of the results of measurements we might expect for the wooden rule.

We can describe the variations in terms of the centre and the width of the distribution of readings.

  • The centre of the distribution of readings is characterised by the mean, which will usually be biased away from the reference value.
  • The width of the distribution is characterised by the standard deviation or a range of values.

The calibration certificate provides you with:

  • The means to apply corrections to the instrument’s readings, which removes the bias. For material measures the certificate will give the average value, e.g. the certificate for a standard mass will report the average mass.
  • A number which measures the width of the distributions of the measured values for the corrected instrument readings or values. This is the uncertainty.

The results supplied on the certificate are sufficient to relate the instrument readings to the SI over the entire range of interest. Note that the range of interest can be less than the full range of the instrument as specified by the manufacturer. Where non-SI units are used e.g. pounds, gallons etc., MSL will provide a conversion factor to the appropriate SI units.

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Applying Corrections to Instrument Readings

Certificates are presented in a manner that simplifies the application of the results to your measurements. Here we give some examples of how to apply corrections to instrument readings. The calibration certificate reports results at selected points over the range of interest, as it is not practical to take or report measurements at every possible reading of most instruments. The spacing and number of points selected are sufficient to allow corrections for any reading to be deduced with sufficient accuracy by linear interpolation. If linear interpolation is considered inadequate, then an appropriate equation for the correction terms will be given. As a general rule, extrapolation of corrections is not recommended.

1. A certificate reporting a table of corrections

Consider a calibration certificate reporting corrections:

Instrument reading (units)  Correction (units)
100 -0.2
110 -0.3
120 -0.3

The result of a measurement in which the instrument gave a reading of 110 units is obtained by adding the correction to the reading:

corrected result = 110 + (-0.3) = 109.7 units.

By convention a correction is added to the instrument reading

2. A certificate reporting a correction equation

The corrections may be presented as a simple polynomial equation. Consider a certificate that reports the corrections as

correction = 0.1 + 0.004´ reading - 0.001´ reading2

The result of a measurement in which the instrument gave a reading of 50.0 units is obtained by adding the correction to the result:

corrected result = 50.0 + 0.1 + 0.004´ 50 - 0.0002´ 502
                      =50.0 + 0.1 + 0.2 - 0.5
                      =49.8

As with the table of corrections the equation should be chosen to be valid over the range of interest to the user.

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3. A certificate reporting instrument readings

For instruments that are expected to be used at particular points in their range or as calibration sources (e.g. voltage sources) it is often more convenient to present the results in terms of a measured value versus the instrument reading. This simplifies the use of the certificate at the points reported.

Consider a certificate for a calibration source with the following table of results.

Measured Value (units)  Instrument Reading (units)
10 14.7
20 24.9
30 35.2

 

To obtain a value of 10 units at the output of the source, the user simply adjusts the instrument to read 14.7 units.

For values not listed on the table linear interpolation is used. There are two cases:

1. The user requires a value of 15 units:

eq1

2. The instrument gives a reading of 20 units

eqn2

4. A certificate reporting the value for a material measure

The calibration result for a material measure is conventionally reported as a measured value:

Nominal value (units)  Measured Value (units)
100 100.032

Note that a calibration is not an assurance that the measured value is equal to the nominal value or any value previously marked on the material measure.  Calibration does not necessarily involve adjustment.

Consider a material measure having a measured value as given by either of the tables above. Suppose the material measure is used with an indicating instrument and a reading of 100.001 units is obtained. The correction for the indicating instrument is then:

correction = measured value of reference standard- reading
              = 100.032 - 100.01
              = 0.02 units

where the correction has been rounded to the resolution of the instrument.

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5. Applying corrections for changes in conditions

Most instruments are sensitive to the conditions in which they are operated. These influence quantities include environmental factors, operating methods and conditions, and instrument settings. For factors that may affect the performance of the instrument, the calibration conditions will be specified on the certificate.

The results reported on the certificate apply only for the conditions specified.

Where the instrument may be operated in different conditions the calibration laboratory may report information on the sensitivity of the instrument to the influence quantity.

Consider the example of a material measure of value 100.032 units calibrated at a temperature of 20.5 ° C. The measure is used at 25 ° C and the user ascertains, by separate measurement or from the certificate or from manufacturer’s specifications, that value will increase with temperature by 0.01 units per ° C. The correction to the value reported on the certificate is then:

eqn3

The temperature coefficient may have been reported as 0.01% per ° C or as 100´ 10-6 units/unit per  ° C. In this case the correction is calculated as

eqn4

If a correction is not applied then a term accounting for the change in temperature should be included in the uncertainty.

 

Calculating the uncertainty in your measurements

1.What does the uncertainty mean?

The uncertainty statement on your certificate should say something like:

"The expanded uncertainty in the reported results is estimated to be ±0.4 mm at the 95% level of confidence. This is calculated using a coverage factor of 2.2 (see the ISO Guide to the Expression of Uncertainty in Measurement, 1993 for an explanation of terms)."

What does " expanded uncertainty … at the 95% level of confidence" mean?

 

Figure 2

Figure 2

Consider Figure 2, which shows the distribution of readings for the wooden rule. The uncertainty measures the width of this distribution. As we can see, the edges of the distribution are not very well defined. To make the definition of the uncertainty clear, the international measurement community has decided that for calibrations and most tests the uncertainty shall be that which encloses 95 % of the area under the distribution. This is the shaded area in Figure 2. The expanded uncertainty, U, is obtained by multiplying the standard deviation, u, by the coverage factor, k, which generally has a value in the range 2 to 3, but often assumed to have a value of 2.0.

eqn5

The ISO "Guide to the Expression of Uncertainty in Measurement" describes how to more accurately calculate the coverage factor.

The probability "95%" in the uncertainty statement means that there is 1 chance in 20 that the reported results are in error by more than the stated uncertainty.

The certificate should contain sufficient information, namely the measurement conditions and/or the effects of influence variables, to calculate the uncertainty in measurements made with the instrument.

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2. Calculating the uncertainty

There are two main aspects to the calculation of uncertainty: calculating the uncertainty due to each of the factors affecting the measurement, and combining these uncertainties to find the total uncertainty in the measurement. Although a detailed treatment is beyond the scope of this guide, we can provide a general outline. For more detailed information refer to the ISO "Guide to the expression of uncertainty in measurement" or to the training courses offered by MSL.

The various uncertainties that may have to be considered in your measurement fall into one of several categories:

  • Uncertainties associated with the instrument. This is the uncertainty reported on the certificate.
  • Uncertainties associated with your measurement process. These factors include departures from the method used in the calibration, operator related effects, sampling effects, and transfer media related effects such as calibration bath instability.
  • Uncertainties associated with environmental factors, including temperature, pressure and humidity.
  • Uncertainties associated with the drift of the instrument with time.

You will need to consider these factors where they are not included in the uncertainty reported on the certificate.

The expanded uncertainty in the measurement is given by

eqn6

Uncertainties combined in this way are said to be ‘added in quadrature’ or ‘root-sum-square’.

To calculate the uncertainty due to a specific factor in a measurement it is necessary to build up a description of the likely variations caused by that factor. There are many methods for gathering this information.

The simplest method for calculating an uncertainty is to make comparative measurements as described in section 3.1. Where comparative methods are impractical it is necessary to infer the distribution by other methods which may include theory, other people’s work, manufacturer’s specifications, subsidiary experiments, etc. These methods are often less objective and require assumptions to be made. To make your measurements traceable it is essential that you record these assumptions. Records provide the means for checking results and incorporating later improvements to estimates of the uncertainties.

As an example we consider a calculation of the uncertainty due to environmental factors, Uenv for a material measure with a temperature coefficient of 0.01 units per ° C. The temperature of the laboratory is not 20.5 ° C, which was the calibration condition, but fluctuates between 22 ° C and 28 ° C. If we assume that the mean temperature is 25 ° C we can apply a correction as described in section 3.2.5 above. In addition if the standard deviation in the temperature is 1.5 °C we can calculate an uncertainty based on the known temperature coefficient of the measure.

The uncertainty in the material measure due to the temperature variation is

 eqn7units.

This is combined with any other environmental uncertainties to give:

eqn8

One of the most important uncertainties you will have to estimate is that due to the drift of an instrument with time. Ideally you should have at least two calibrations carried out on the instrument separated by perhaps one year. This enables you to make estimates of the likely drift and hence predict corrections and estimate uncertainties. With only one calibration you may have to rely on the manufacturer’s specification for the drift and the uncertainty will probably be greater.

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