• The customer needs to get an idea of the applicability of the result. Measurement uncertainty has to be taken into account particularly when regarding specification limits.
• Testing laboratories need uncertainties with their calibration certificates, so that they can establish the uncertainty of their own measurements.
• An estimation of a measurement's uncertainty is required for testing and calibration laboratories complying with ISO/IEC 17025.
• A calibration laboratory ... shall have and shall apply a procedure to estimate the uncertainty of measurement for all calibrations ...",
• Testing laboratories shall have and shall apply procedures for estimating uncertainty of measurement"
• The uncertainties values have to be stated in the test report depends on requirements by the test method, requirements by the customer, or whether conformance to specification has to be assessed (ISO/IEC 17025, 5.10.3).
• In calibration, uncertainties have to be stated in the certificate (as they are required by the user of the calibrated equipment).

X
I……………………….I..……...……………... I

U U

• Specify what is being measured
• Identify what causes the result to change
• Quantify the uncertainty components

• Type A: These uncertainties are evaluated by statistical analysis of a series of observations
• Type B: These uncertainties are evaluated from any other information, such as information from past experience of the measurements, from calibration certificates, manufacturers specifications, etc.
• Type A and B uncertainties are based upon probability distribution.
• Type A uncertainties are estimated on the basis of repeat measurements, usually assuming the normal or "t" distribution for the variability in the mean of the values.
• Type B uncertainty, by and large are obtained by assuming a particular probability distribution, such as normal, a rectangular or a triangular distribution.

• Convert to standard uncertainties
• Combining the uncertainties Uc
• Express as expanded uncertainty, Uc,

i. Mean (x‾) = X¡+X2+X3+…X n = ∑Xi /n
n i

ii standard deviation is given by
/ ∑( xi − X‾)²
S= √ ——————
n-1

Mean (x‾) = X¡+ X2 + X3 + X n= ∑Xi /n …………………. (1)
n i
Where x­­־ is the mean of ten readings, & its value is = 102.4

1. And the standard deviation/standard uncertainty is given by

/ ∑( xi − X‾)²
S= √ —————— = ………………………………..–(2)
n-1

/ ∑( xi − X‾)² / 32.4
S= √ —————— = √ —————— = 1.897
n-1 10-1

• By definition it is a numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded uncertainty.
• it signifies to what extent the value of probability is associated with a confidence interval or a statistical coverage interval.

• Uncertainty can only be defined in terms of probability/ level of confidence.
• Any measurement will have some uncertainty and the quoted interval will be the range within which the true value lies at a certain level of confidence.
• The relationship between the standard deviation and the confidence interval actually depends on the type of distribution to which measurement result conforms.

• The normal distribution
• The rectangular distribution
• The triangular distribution

• 5 cc class bulb pipettete has tolerance of ± 0.03 c.c. Tolerance means that a particular pipettete has a volume between 4.970 cc to 5.030 cc at 20°c. It is due to the control in manufacturing process, and most of the pipettete made are close to 5.00 cc, and only few are outliers.
• But we can not say, in what way this volume, 5 c.c is distributed in all the pipettetes manufactured and available in lab, but one can assume that true volume in the pipettete of the lab has an equal probability of being any value in the range 4.970 cc to 5.030 c.c.
• The probability distribution for the pipettete volumes then must be assumed to be such that there is constant probability throughout the tolerance range, 4.97 cc to 5.030 cc, and zero probability out side this range.
• The distribution shows that any capacity between 4.97 cc and 5.03 cc has equal probability and any capacity out side this range zero probability. The actual probability of any capacity with in the range is 1/2a, where a is the half width of the distribution. This is a simple requirement arising from the fact that the area under the distribution, which represents the total probability of all capacities, must be unity, i.e. 2a x 1/2a.
• And for the rectangular distribution the standard deviation is given by a/?3
• When dispersion of measurement is represented by a rectangular distribution, then the standard uncertainty is the half width of the distribution divided by the square root of three, i.e 1.732 .
• Though the tolerance of a 5 cc class B pipettete is ± 0.03 cc, the standard uncertainty in the 5.00 cc capacity pipettete is 0.03/?3 = 0.03/1.732= 0.017 cc.

• Suppose in a lab there is a digital volt meter which reads to the nearest milli-volts and it is reading 5.000 volt. As the voltage applied increases we would expect the meter to change to reading 5.001 volts when the actual voltage passes 5.0005 volts. A similar situation will apply on a decreasing voltage. This means that when the meter reads exactly 5.000 volts the true voltage may be anywhere between 4.9995 volts to 5.0005 volts.
• Since the meter is digital and provides no information across the interval between changes on the digital display we must assume that, when the meter reads 5.000 volts, the true voltage is equally likely to be anywhere in the interval of 4.9995 to 5.0005 volts.
• This means that the uncertainty in the reading, resulting from the resolution of the meter display is represented by a rectangular probability distribution of half width 0.5 millivolts so the uncertainty in the meter reading resulting from readability of the meter is 0.5/?3= ±0.29 millivolts.

• The other commonest device in a laboratory with a digital scale is a balance. The principle is exactly the same as for voltmeter. For example a three place balance will read to the nearest milligram so the half width of the rectangular distribution is 0.5 milligram and the uncertainty due to the readability is 0.5/?3 = ±0.29 milligram.
• The above examples give a general result for a digital device with a scale which registers in unit steps, the uncertainty resulting from the readability of the display is given by 0.29 in the units of the digital interval. And uncertainty due to readability of a four place balance is 0.29 tenths of a milligram, e.i. 0.029 milligrams.

• The distribution implied by this picture is that the most likely value for the temperature is 20°c with decreasing probability across the range 18°c to 22°c and zero probability outside this range.
• The distribution shows the centre weighing of the temperature, with zero probability of the temperature being outside the ±2°c tolerance. The probability of a temperature of exactly 20°c is 1/a, where a is the half width of the distribution and the total probability is the area of the triangle, half of the base times the height, a x 1/a = 1.
• The probability of any particular temperature in the range decreases linearly as we move away from 20°c until it reaches zero at the ±2°c limits.
• The standard deviation for the triangular distribution, i.e. the standard uncertainty, is a/?6, in this case 2/2.45= 0.82 °c.

• Often the choice to be made, in evaluating an uncertainty, is as to whether the rectangular or triangular distribution is most appropriate. In case of the effect of temperature variation on the uncertainty of the volume dispensed by 5 cc pipettete discussed above a rectangular distribution was assumed, i.e. that all temperatures between 15°c and 25°c were equally likely.
• The volume variation will also have a rectangular distribution. The half width of this distribution, derived from the calibration tables, is 0.005 cc for a 5 cc pipettete. Hence the standard uncertainty is 0.005/?3 = ±0.003 cc.
• If the laboratory were air conditioned with a target of 20°c and reasonable time were allowed to enable any solutions which were to be pipetteted to equilibrate to the lab temperature. A view might be taken that triangular distribution is more appropriate since the probability of the temperature near 20 °c would be higher than the probability of temperature at the 15 °c and 25 °c extremes. This would then make the standard uncertainty in the volume due to temperature effects 0.005/?6 = 0.002 cc.
• In practice the difference between the standard uncertainty compounded from rectangular and triangular distributions with the same width is small; the difference between dividing by ?3 =1.73 and ?6= 2.45. This means that, unless the uncertainty being dealt with is a major contributor to the combined uncertainty it may not be worth spending time to arrive at defensible conclusion to use triangular distribution.
• The rectangular distribution can always be justified as it represents the worst case scenario.

• Uncertainty information from rectangular distribution , U = a/ ?3
• Uncertainty information from triangular distribution , U = a/ ?6 For uncertainty associated from single result , use relative standard deviation RSD. U = K X RSD X C ( where C is a single result)

• Coefficient of thermal expansion for pipettete
• Coefficient of thermal expansion for liquid being pipetteted
• Temperature range in lab

• The calibration Uncertainty in the pipettete, ± 0.017 cc
• The Uncertainty due to temperature variation ± 0.003 cc
• The Uncertainty due to random effect ± 0.01 cc

• Spool = √/ S1²( n 1 -1)+ S2² (n2-1)/ / n1+ n2 -2
• Spool = √ / 1.027²( 5 -1)+ 0.652² (5-1)/ / 5+ 5 -2
• Spool st. de. = √/4.22+ 1.7 = 0.860 ppb
  ¯¯ 8¯¯

• Answer: 0.046 ml ( rectangular distribution, 0.08/√3

• Answer: 0.0058 ml (rectangular distribution, 0.01/√3)

• Answer: 0.0002g( 95% confidence interval- divided by 1.96)

• Answer: 0.058%( rectangular distribution, 0.1/√3)

• Answer: 0.02 mg (95% confidence interval- divided by 1.96)

• Answer: 0.00021g (already a standard deviation, no conversion necessary)

• Answer: 0.015 ml (divided by stated coverage factor i.e.2)

• Uncertainty in the volume of liquid in a flask
• Uncertainty in the volume of liquid delivered by a pipettete
• Uncertainty in weighing
• Uncertainty in concentration of the solution

• The Uncertainty in the volume of liquid in a flask is due to the very fact that each time you fill up to the mark of the flask the measured quantity of the liquid will differ. To evaluate this standard deviation ten fills of organic solvent are weighed using the 100 CC class A volumetric flask. The standard deviation is calculated by following the method as described in example 1, and for this flask it is (say) Up = 0.01732 ml.
• The manufacturers specification for flask ± 0.08 ml Assuming it to be rectangular distribution Uc = 0.08/?3 = 0.0 46 ml
• The possible temperature difference between the laboratory temperature and the flask calibration temperature is estimated as ±3 ° with 95% confidence. The Coefficient of volumetric expansion of organic solvent is 1x 10 ¯ ³/ ° c. Uncertainty due to possible temperature difference between

• Q-1 Calculate the concentration of solution in mg/L
• Q-2 What is the standard uncertainty of the solution Concentration.
• Q-3 What is the expanded uncertainty? Where k=2

• Concentration of Solution = weight/ volume, C= W/V

• Standard uncertainty of the solution Concentration

• The expanded uncertainty of the solution Concentration :

• ISO-( GUM) "Guide to the expression of uncertainty in measurement".
• NABL Guidelines for estimation and expression of un-certainty in measurement
• Two-day training- Dr Bernard King
• Evaluating uncertainties for laboratories Dr Alan G Rowley
• EURACHEM / CITEC GUIDE Quantifying uncertainty in Analytical Measurements

I used to be uncertain - now I'm not so sure. In ordinary use the word 'uncertainty' does not inspire confidence. However, when used in a technical sense as in 'measurement uncertainty' or 'uncertainty of a test result' it carries a specific meaning. It is a parameter, associated with the result of a measurement (eg a calibration or test) that defines the range of the values that could reasonably be attributed to the measured quantity. When uncertainty is evaluated and reported in a specified way it indicates the level of confidence that the value actually lies within the range defined by the uncertainty interval.

Any measurement is subject to imperfections; some of these are due to random effects, such as short-term fluctuations in temperature, humidity and air-pressure or variability in the performance of the measurer. Repeated measurements will show variation because of these random effects. Other imperfections are due to the practical limits to which correction can be made for systematic effects, such as offset of a measuring instrument, drift in its characteristics between calibrations, personal bias in reading an analogue scale or the uncertainty of the value of a reference standard.

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? It allows users of the result to assess its reliability, for example for the purposes of comparison of results from different sources or with reference values. Confidence in the comparability of results can help to reduce barriers to trade. Often, a result is compared with a limiting value defined in a specification or regulation. In this case, knowledge of the uncertainty shows whether the result is well within the the acceptable limits or only just makes it.

Occasionally a result is so close to the limit that the risk associated with the possibility that the property that was measured may not fall within the limit, once the uncertainty has been allowed for, must be considered. Suppose that a customer has the same test done in more than one laboratory, perhaps on the same sample, more likely on what they may regard as an identical sample of the same product. Would we expect the laboratories to get identical results? Only within limits, we may answer, but when theresults are close to the specification limit it may be that one laboratory indicates failure whereas another indicates a pass. From time to time accreditation bodies have to investigate complaints concerning such differences. This can involve much time and effort for all parties, which in many cases could have been avoided if the uncertainty of the result had been known by the customer.

The standard ISO/IEC 17025:2005 [General requirements for the competence of testing and calibration laboratories] specifies requirements for reporting and evaluating uncertainty of measurement. The problems presented by these requirements vary in nature and severity depending on the technical field and whether the measurement is a calibration or test.

2.uncertainty of reference instruments is provided at each stage down the calibration chain, starting with the national standard and

3.customers are aware of the need for a statement of uncertainty in order to ensure that the instrument meets their requirements.

Consequently, calibration laboratories are used to evaluating and reporting uncertainty. In accredited laboratories the uncertainty evaluation is subject to assessment by the accreditation body and is quoted on calibration certificates issued by the laboratory.

The situation in testing is not as well-developed and particular difficulties are encountered. For example, in destructive tests the opportunity to repeat the test is limited to another sample, often at significant extra cost and with the additional uncertainty due to sample variation. Even when repeat tests are technically feasible such an approach may be uneconomic. In some cases a test may not be defined well enough by the standard, leading to potentially inconsistent application and thus another source of uncertainty. In many tests there will be uncertainty components that need to be evaluated on the basis of previous data and experience, in addition to those evaluated from calibration certificates and manufacturers, specifications. International and accreditation aspects.

Accreditation bodies are responsible for ensuring that accredited laboratories meet the requirements of ISO/IEC 17025. The standard requires appropriate methods of analysis to be used for estimating uncertainty of measurement. These methods are considered to be those based on the Guide to the expression of uncertainty of measurement, published by ISO and endorsed by the major international professional bodies. It is a weighty document and the international accreditation community has taken up its principles and, along with other bodies such as EURACHEM/CITAC, has produced simplified or more specific guidance based on them. Accreditation bodies are harmonizing their implementation of the requirements for expressing uncertainty of measurement through organizations such as the European co-operation for Accreditation (EA) and the International Laboratory Accreditation Co-operation (ILAC).

Uncertainty is a consequence of the unknown sign of random effects and limits to corrections for systematic effects and is therefore expressed as a quantity, i.e. an interval about the result. It is evaluated by combining a number of uncertainty components. The components are quantified either by evaluation of the results of several repeated measurements or by estimation based on data from records, previous measurements, knowledge of the equipment and experience of the measurement.

In most cases, repeated measurement results are distributed about the average in the familiar bell-shaped curve or normal distribution, in which there is a greater probability that the value lies closer to the mean than to the extremes. The evaluation from repeated measurements is done by applying a relatively simple mathematical formula. This is derived from statistical theory and the parameter that is determined is the standard deviation.

Uncertainty components quantified by means other than repeated measurements are also expressed as standard deviations, although they may not always be characterized by the normal distribution. For example, it may be possible only to estimate that the value of a quantity lies within bounds (upper and lower limits) such that there is an equal probability of it lying anywhere within those bounds. This is known as a rectangular distribution. There are simple mathematical expressions to evaluate the standard deviation for this and a number of other distributions encountered in measurement. An interesting one that is sometimes encountered, eg in EMC measurements, is the U-shaped distribution.

The method of combining the uncertainty components is aimed at producing a realistic rather than pessimistic combined uncertainty. This usually means working out the square root of the sum of the squares of the separate components (the root sum square method). The combined standard uncertainty may be reported as it stands (the one standard deviation level), or, usually, an expanded uncertainty is reported. This is the combined standard uncertainty multiplied by what is known as a coverage factor. The greater this factor the larger the uncertainty interval and, correspondingly, the higher the level of confidence that the value lies within that interval. For a level of confidence of approximately 95% a coverage factor of 2 is used. When reporting uncertainty it is important to indicate the coverage factor or state the level of confidence, or both.

Sector-specific guidance is still needed in several fields in order to enable laboratories to evaluate uncertainty consistently. Laboratories are being encouraged to evaluate uncertainty, even when reporting is not required; they will then be able to assess the quality of their own results and will be aware whether the result is close to any specified limit. The process of evaluation highlights those aspects of a test or calibration that produce the greatest uncertainty components, thus indicating where improvements could be beneficial. Conversely, it can be seen whether larger uncertainty contributions could be accepted from some sources without significantly increasing the overall interval. This could give the opportunity to use cheaper, less sensitive equipment or provide justification for extending calibration intervals.

Uncertainty evaluation is best done by personnel who are thoroughly familiar with the test or calibration and understand the limitations of the measuring equipment and the influences of external factors, eg environment. Records should be kept showing the assumptions that were made, eg concerning the distribution functions referred to above, and the sources of information for the estimation of component uncertainty values, eg calibration certificates, previous data, experience of the behavior of relevant materials. Statements of compliance - effect of uncertainty

This is a difficult area and what is to be reported must be considered in the context of the client's needs. In particular, consideration must be given to the possible consequences and risks associated with a result that is close to the specification limit. The uncertainty may be such as to raise real doubt about the reliability of pass/fail statements. When uncertainty is not taken into account, then the larger the uncertainty, the greater are the chances of passing failures and failing passes. A lower uncertainty is usually attained by using better equipment, better control of environment, and ensuring consistent performance of the test.

For some products it may be appropriate for the user to make a judgment of compliance, based on whether the result is within the specified limits with no allowance made for uncertainty. This is often referred to as shared risk, since the end user takes some of the risk of the product not meeting specification. The implications of that risk may vary considerably. Shared risk may be acceptable in non-safety critical performance, for example the EMC characteristics of a domestic radio or TV. However, when testing a heart pacemaker or components for aerospace purposes, the user may require that the risk of the product not complying has to be negligible and would need uncertainty to be taken into account. An important aspect of shared risk is that the parties concerned agree on the uncertainty that is acceptable; otherwise disputes could arise later.

Uncertainty is an unavoidable part of any measurement and it starts to matter when results are close to a specified limit. A proper evaluation of uncertainty is good professional practice and can provide laboratories and customers with valuable information about the quality and reliability of the result. Although common practice in calibration, there is some way to go with expression of uncertainty in testing, but there is growing activity in the area and, in time, uncertainty statements will be the norm.