Uncertainty of Measurement


2.Essentials of measurement uncertainties



5.ISO GUM Principles I

6.Measurement Uncertainty Estimation Protocol

7.Mathematics involved in estimating standard uncertainties

8.Expression of measurement uncertainty (for results from chemical laboratory)

9.Probability distribution

10.Case study -2

11.Pooling of Standard deviations

12.Converting information into a standard uncertainty (type B evaluations

13.Possible sources of Uncertainty in a chemical lab

14.Cause and effect analysis ( also called fish diagram)




Measurement is a process, in which a set of operations are performed to determine the value of quantity. A process is an integral set of activities that uses resources to transform inputs to outputs. In a measurement process even when all the measurement factors which can be controlled are controlled, repeated observation made during under the same condition, are rarely found identical. This is due to the variables - operator, reference standards / materials / Instrument / environment, calibration, test methods and therefore measurement results are never true value and in fact accompanied with uncertainty. Therefore the "measurement uncertainty" is a property of a measurement.
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2.Essentials of measurement uncertainties

  • 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).
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3. Scope

The purpose of this write-up is to familiarize the reader with subject of measurement uncertainty to that extent that he would be able estimate uncertainty in measurement in his laboratory.

Instead of going into theoretical aspect and in great detail this document has taken the help simple and solved examples to familiarize and develop the skill of calculating uncertainty in measurements.

The examples given cover most of the situations but the focus is on chemical testing laboratory.
Hope, those who have very little or no exposure to the estimation of uncertainty will be able to start estimating uncertainty in measurement for their kind of lab.
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Uncertainty is the parameter associated with the result of a measurement that characterizes the dispersion of the values that could be attributed to the measurand.

I……………………….I..……...……………... I


4.1Uncertainty in Measurements: Defines a range that could be reasonably be attributed to the measurement result at a given confidence. Eg 56.2 ± 9.6 mg/L

Sources of measurement uncertainties in testing There are many possible sources of uncertainty, e.g. sampling, instrument drifts and calibration, homogenization and dilution effects, human factors, environmental effects
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5.ISO GUM Principles I

  • Specify what is being measured
  • Identify what causes the result to change
  • Quantify the uncertainty components
5.1ISO GUM Principles II

Types of uncertainties

  • 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.
5.2ISO GUM Principles III
  • Convert to standard uncertainties
  • Combining the uncertainties Uc
  • Express as expanded uncertainty, Uc,
Expanded uncertainty Uc = k. Uc .
Where k is 'coverage factor'
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6.Measurement Uncertainty Estimation Protocol

Conditions of the quality assurance data are comparable with the treated problem, e.g. matrix and composition, range of values, repeatability / reproducibility contributions the presumed simplifications are appropriate and acceptable .
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7.Mathematics involved in estimating standard uncertainties

Let there be X ¡, X2, X3 , …X n measurements

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

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

iii. Combining uncertainty/deviation
U = √ ( Ui )² + ( Uii )²
u (c ) / C = √ ( u ( w )/w )² + (u (v )/v )² + ( u (p)/p )²
iv. Expanded uncertainty Uc
Uc = k. Uc ., Where k is ‘coverage factor’
7.1Example of Type A evaluation:Type A method of evaluation of standard uncertainty applies to those situations where several independent observations have been made for any of the input quantities under same condition of measurement. If there is sufficient resolution in the measurement process, there will be an observable scatter or spread of the values obtained. These uncertainties are evaluated by the statistical analysis of a series of repeat measurements.

Case study -1: Procedure for estimation of type 'A' measurement uncertainty, based on internal reproducibility standard deviation.
Given data: These are ten results measured at different times, using different analysts.

( In reality laboratory staff is required to generate such information, for a specific situation to estimate standard uncertainty. This standard uncertainty, is used for estimating expanded uncertainty for an individual measurements).

Number of Determinations Date of measurement Result mg/Kg x Result mg/Kg x- ­x­­ Result mg/Kg
( x- ­x­­־
1 4.6.07 100.0 -2.4 -5.76
2 5.6.07 103.9 +1.5 2.25
3 6.6.07 104.8 +2.4 5.76
4 7.6.07 104.0 +1.6 2.56
5 10.6.07 101.9 -0.5 0.25
6 11.6.07 103.0 +0.6 0.36
7 13.6.07 103.8 +1.4 1.96
8 14.6.07 99.5 -2.9 8.41
9 17.6.07 100.2 -2.2 4.48
10 18.6.07 102.9 +0.5 0.25

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)

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


n is the number of determinations
(n-1) is degree of freedom
S internal reproducibility standard deviation or standard uncertainty
x‾ is the mean
Relative reproducibility standard deviation = RSD
is RSD = S/ x‾ ……………………-(3)
1.897 RSD = S/ x‾ = ------ = 0.0185

The magnitude of the obtained internal reproducibility standard deviation should be evaluated in relation to the field of application of the method ( fitness for purpose) Correspondingly, the measurement uncertainty may be estimated, using results of replicate determinations carried out on reference materials, or from results of studies where the method under study was used in parallel with a well defined, established reference method. Calculations are carried out as above using the reference value in place of the mean.
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8.Expression of measurement uncertainty (for results from chemical laboratory)

The measurement uncertainty is a parameter, which describes in a quantitative manner the variation to the concentration of an analyte present in the sample. For an estimated measurement uncertainty to be of value, it must be estimated and expressed in a standardized manner.
Measurement uncertainty expressed as the standard measurement uncertainty, is defined (i) as 'measurement uncertainty of the result as one standard deviation'. And expanded measurement uncertainty, is defined as ' a quantity which defines the interval about the result, including a large portion of the variation which would result from the analyte present in the sample, and which is obtained by multiplying the standard measurement uncertainty with a coverage factor to obtain an estimate of the confidence interval for a measurement at a specific level of confidence'. This coverage factor usually equals 2; in some cases 3 may be used. Application of coverage factor of 2 corresponds to confidence level of approximately 95% , and 3 to a confidence level of more than 99%. The expanded measurement uncertainty is given by:
i. U (expanded uncertainty) = 'K.' X. 'C' X. 'RSD' -- -(4) And
ii U (uncertainty) = √ ∑ (xi -x‾)²/n-1 . X. ‘K’ -(4a)
Equation -(4a) is used to estimate standard uncertainty.
Where K is the Coverage factor,
RSD is the relative standard deviation,
and C is the concentration of analyte.
As per recommendations a Coverage factor '2' is used. ( what is a coverage factor and why a value of ' 2' has been taken will be discussed later)
The calculated expanded measurement uncertainty " U" represents half of the measurement uncertainty interval. Measurement result ± U
Example -1 ( Internal reproducibility) The measured concentration of an analyte of a solution is 99.4 mg/Kg. Determine expanded uncertainty.
U = 'K' .X 'C' X ' RSD' ……………………….Equation-(4)
U = 2 x 99.4 x 0.0185 mg/Kg = 3.7 mg/Kg
The measured result 99.4 mg/Kg accompanied by its expanded measurement uncertainty Is given by as 99.4 ± 3.7 mg/Kg Every time and for each test ( when concentration of the same analyte is measured) Equation -4 is used to determine the value of expanded Uncertainty. What is K ( Coverage factor ) and where from it has come?
  • 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.
Confidence Interval: the interval about the mean within which the true value is expected to lie with the specified level of confidence.
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9.Probability distribution

  • 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.

9.1Types of probability distribution
  • The normal distribution
  • The rectangular distribution
  • The triangular distribution
9.2 Normal distribution
The probability density function is given by =
P (x) = 1 ⁄ s √ 2π . exp[- (x- µ)²/2s²] - ∞ < x + < ….(5)
Where "µ" is the mean and "s" is the standard deviation.

Normal distribution
The normal distribution has a maximum probability at the value, which is the mean of the measurements made, and falls away on both sides towards zero probability. It never reaches zero but approaches it exponentially since there is always a finite chance that a measurement will have any value between plus and minus infinite but further away from the mean you go, lesser is the probability.
The percentage values of the covered area of the curve are the confidence level, which is an indication of Coverage factor K, as per table -1
Table-1: Confidence level and the corresponding coverage factor (K)

confidence level 67.27% 90% 95% 95.45% 99% 99.73%
Coverage factor K 1.000 1.645 1.96 2.000 2.576 3.000

9.2.1 When to use normal distribution

a. In some situations, the quoted uncertainty in an input or output quantity is stated along with level of confidence. In such cases, one has to find the value of coverage factor by using the Table-1,
b. In the absence of any specific knowledge about the type of distribution, one may assume it to be normal distribution.
c . When the uncertainty in a calibration certificate is given as a confidence interval/confidence level or in terms of a standard deviation multiplied by a coverage factor.
Like: When the calibration certificate shows calibration temp. 37°c with an uncertainty of ± 0.014°c at 95% confidence interval. Uncertainty = ? ? (xi -x?)²/n-1 . X Coverage factor In such a situation one generally assume that the data has been subjected to full statistical analysis and is generally described by a normal or Gaussian probability distribution.

9.3Rectangular distribution

The probability density function p(x) of a rectangular distribution is as follows
P (x) = 1/ 2. , a_ < x < a+ , where a = ( a + - a_ ),
Given below is the figure that is representative for rectangular distribution.

9.3.1 When to use rectangular distribution ? In those cases, where it is possible to estimate only the upper and lower limits of an input quantity (X) and there is no specific knowledge about the concentration of values of (X) within the interval, one can only assume that it is equally probable for X to lie anywhere within this interval.

9.3.2 Illustration for rectangular distribution

i. Volume

  • 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.
Conclusion: When calibration data is given in terms of tolerance interval, then in evaluating its standard uncertainty, rectangular distribution is presumed.

ii. Volt meter (digital instruments)

  • 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.
iii. Digital balance

  • 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.
9.4 The Triangular DistributionWhen the greatest concentration of the values is at the centre of the distribution, it is triangular distribution. For example when temperature is to be controlled 20 ±2°c, using thermostat which is calibrated, thermostat will hold the temperature very close to 20 °c. This shown in given fig.

  • 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.
9.4.1 Rectangular v/s triangular distribution

  • 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.
9.4.2 Conversion to Standard deviation

  • 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)
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10.Case study -2

10.1 Uncertainties Associated in pipettete and combining them
i. We have seen above that for 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.
ii. Calibration refers to 20 degree centigrade but when pipettete is not used under the same temp., it will contribute additional uncertainties due to:
  • Coefficient of thermal expansion for pipettete
  • Coefficient of thermal expansion for liquid being pipetteted
  • Temperature range in lab
For class B pipettete, the temperature coefficient for a borosilicate pipettete and water over a temperature range of 15 to 25 °c the uncertainty in the measurement is ± 0.003 cc.
iii. Random uncertainties arise as even the same operator will dispense slightly different volumes each time pipettete is used. Meniscus, number of taps on emptying may vary. In addition, each operator will have different technique. To arrive at it involves ten different operators, pipetting ten times water and weighing on four place- balance. Calculate mean and standard deviation. This would give a measure of uncertainty in the volume measurement in the laboratory due to random factors. Assuming that uncertainty due to weighing and temperature are negligible. This exercise will result in standard deviation of 0.01 cc, for pipetteting 5 c.c. In this case , we have the following contributions to uncertainty:
  • 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
10.2 Combing uncertainties

Total uncertainty = √ U² 1 + U² 2+ U² 3+ ………..
Total uncertainty = √U² 1 + U² 2+ U²3 = ? ( 0.017) ²+ (0.003) ² + (0.01) ² =± 0.0200
For class B pipettete, and for class A, cal. Uncertainty is 0.0085, which is half of the class B pipettete,
Total uncertainty= √ ( 0.0085) ²+ (0.003) ² + (0.01) ² = ± 0.0135
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11.Pooling of Standard deviations

Standard deviations obtained from two experiments are required to be pooled depending on the conditions of the measurements that are made. In addition, pooled standard deviation is estimates of data derived from two experiments.
Example-2: Replicate measurements of the two samples measured to determine lead concentration, at required temperature of 80 ± 3 °C., to know the effect of temperature on measured lead levels

77°c 83°c
84.2mg/L 85.6mg/L
85.1mg/L 86.6mg/L
84.0mg/L 85.5mg/L
82.2mg/L 84.8mg/L
84.9mg/L 86.4mg/L
Mean 84.08mg/L Mean 85.78mg/L
Std. Dev. 1.027 Std. Dev 0.652

Pooled Standard Deviation
  • 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¯¯
Pooled standard deviation = 0. 860s ppb

12.Converting information into a standard uncertainty (type B evaluations)

In those cases, where the uncertainty is quoted to be multiple of standard deviation's', the multiple becomes the specific factor ( Refer table- 1)

Example-3 : A calibration certificate states that the mass of a given body of 10 Kg is 10.000650 Kg. The uncertainty at confidence level of 95.45% is given by 300 mg. In such a case, the standard uncertainty is calculated by using equation 4 a In this case Coverage factor K is '2', ( table-1)
U (standard uncertainty) = √ ∑ (xi -x‾)²/n-1 . X. 'K'
300 = standard uncertainty X K
Standard uncertainty U(m) = 300/2 = 150 mg

Example-4: Suppose in the above example, the quoted uncertainty is at 90% level of confidence. The standard uncertainty is then: U(m) = 300/ 1.64 = 182.9 mg ( use table -1 to get 1.64 at 90% level of confidence)

Example-5: A calibration certificate states that the resistance of a standard resistor, Rs of a nominal value 10 Ωis 10.000742 Ω ± 129 µΩ at 23° C and that the quoted uncertainty of 129 µΩdefines an interval having a level of confidence of 99%. Table 1 indicates a coverage factor of 2.58 at 99% confidence level., and equation-4
U (standard uncertainty) = √ ∑ (xi -x‾)²/n-1 . X. 'K'
√ ∑ (xi -x‾)²/n-1 . = standard deviation = U (Rs)= 129/2.58 = 50 µΩ
Relative standard deviation = Standard deviation/nominal value
Relative standard deviation = Standard deviation/nominal value

Example-6 : A calibration certificate states that the length of a standard slip gauge (SG) of nominal value 50 mm is 50.000002 mm. The uncertainty of this value is 72 nm, at a confidence level of 99.7%
U(standard uncertainty) = C X Standard uncertainty
Standard uncertainty of slip gauge is = 72 nm /3 = 24 nm ( 3 is the coverage factor K, at 99.77 confidence level)

Example-7 : The manufacturer's specification for a 100 ml class A volumetric flask is ±0.08 ml. What is the standard uncertainty.
  • Answer: 0.046 ml ( rectangular distribution, 0.08/√3

Example-8: The manufacturer's specification for a 2 ml pipettete is ±0.01 ml. What is the standard uncertainty
  • Answer: 0.0058 ml (rectangular distribution, 0.01/√3)

Example-9:The calibration certificate for a four figure balance states that the measurement uncertainty is ±0.0004g with a level confidence of 95%. What is the standard uncertainty
  • Answer: 0.0002g( 95% confidence interval- divided by 1.96)

Example 10: The purity of a compound as given in the label of bottle is 99.9 ± 0.1. Determine the standard uncertainty in the purity of the compound.
  • Answer: 0.058%( rectangular distribution, 0.1/√3)

Example-11: A calibration weight is certified as 10.00000g ± 0.04 mg with a level confidence of at least 95%. What is the standard uncertainty in the weight?
  • Answer: 0.02 mg (95% confidence interval- divided by 1.96)

Example-12 : The standard deviation of repeat weighing of a 0.3g check weight is 0.00021g. What is the standard uncertainty of a single weighing?
  • Answer: 0.00021g (already a standard deviation, no conversion necessary)

Example-13: A calibration certificate for 25 ml class A pipettete quotes an uncertainty of 0.03 ml. The reported uncertainty is based on a standard uncertainty multiplied by a coverage factor k=2, providing a confidence level of 95%. What is the standard uncertainty in the volume of the liquid delivered by the pipettete.
  • Answer: 0.015 ml (divided by stated coverage factor i.e.2)
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13.Possible sources of Uncertainty in a chemical lab

  • 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
Example-14 Uncertainty in the volume of liquid in a flask

Contribution to the Uncertainty in the volume of liquid in a flask is from:

  • 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
laboratory temperature and the flask temperature is estimated as
Combining the three standard uncertainties = ∑ U²v+ U²c +U²t
Uv = ∑ 0.0173² + 0. 046² + 0.153² = 0.0016 ml

Example-15. Uncertainty in the volume of liquid delivered by a pipettete

a. Contribution to the Uncertainty in the volume of liquid in a flask is from.

The Uncertainty in the volume of liquid delivered by the pipettete is due to the very fact that each time you fill up to the mark of the pipettete the measured quantity of the liquid will differ. To evaluate this standard deviation ten fills of liquid are weighed using the 2CC class A pipettete.
The standard deviation is calculated by following the method as described in example 14, is (say) Uv = 0.0016 ml.

b. Contribution due stated tolerance for the pipettete is ± 0.01 ml.
The coefficient of volume expiation for organic solvent is 1x10 ¯³ per ° c. The uncertainty in calibration of pipettete, is calculated from manufacturer's specification which is treated as rectangular distribution Uc 0.01/ √ 3 = 0.0058 ml

c. The manufacturers due to possible difference between the laboratory temperature and the pipettete 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 difference between laboratory temperature and the pipettete calibration temperature is estimated as
Ut= 1 x 1 x 10 ¯ ³ x 3/1.96 = 0.00153 ml.
Uv = √ 0.0016² + 0. 0058² + 0.00153² = 0.0067 ml

Example -16 Uncertainty in weighing-
Contributors to the Uncertainty in weighing
a. The balance calibration certificate states a measurement uncertainty of ± 0.0004 gm with a level of confidence of not less than 95%. This converted to standard uncertainty by dividing by 1.96 , Uc= 0.0004/1.96 =0.0002g= 0.2 mg
b. Replicate weightings of a 100 mg check weigh on the 4 figure balance has a standard deviation of 0.000041g=0.041 mg
Combing uncertainties Uw= √ .0.2 ² + 0.041² = 0.201 mg

Example 17. Uncertainty in concentration of the solution
An internal standard solution is prepared by dissolving approximately 100.0 mg of material (weighed on a 4-figure balance) in an organic solvent and making it 100 ml in a volumetric flask.

  • 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
Given data: 100.5 mg of material was weighted out, in balance & its calibration certificate states a measurement uncertainty of± 0.0004g, at 95% confidence level.

i. The standard uncertainty associated with weight was calculated above in Exercise-3 Uw = 0.210 mg
ii. The purity of the material quoted by the supplier as 99.9± 0.1% standard deviation = 0.1/?3) = 0.058% ( rectangular distribution,)
iii. The standard uncertainty for the volume in 100 ml flask was calculated in exercise-1 and is = 0.16 ml.

  • Concentration of Solution = weight/ volume, C= W/V
The concentration of the solution= weight of material x purity/volume 100.5 x 99.9/ 100, this value is per 100 cc,
& to get per 1000 cc multiply by 10, the answer is = 10004.0 ml/L
Answer-1: 1004.0 mg/L

  • Standard uncertainty of the solution Concentration
(By adding individual uncertainties on account of balance, purity &, volume from calculations made & using the rule to combine)
Standard uncertainty of the solution concentration is given by
u (c ) / C = √ ( u ( w )/w )² + (u (v )/v )² + ( u (p)/p )² = 1004 x √ (0.208/100.5)² + ( .00058/0.999) ² + ( 0.16/100)² = ± 2.69 mg/L

Answer -2. Standard uncertainty of the solution is = ± 2.69 mg/L

  • The expanded uncertainty of the solution Concentration :
U (expanded uncertainty) = 'K.' X. 'SD' (taking K=2) ,
Answer-3: Expanded uncertainty = Uc = 2.x 2.69 = ± 5.38 mg/l
Concentration is therefore quoted as 1004 ± 5.4 mg /L , where k=2 and the level of confidence is about 95%.
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14.Cause and effect analysis ( also called fish diagram)

u (c ) / C = √ ( u ( w )/w )² + (u (v )/v )² + ( u (p)/p )²

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  • 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
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1. What is uncertainty?
2. How does it arise?
3. Why is it important?
4. What is done about it?
5. How is uncertainty evaluated?
6. What is best practice?

1. What is uncertainty?

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.

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2. How does it arise?

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.

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3. Why is it important?

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.

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4. What is done about it?

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.

Calibration is characterized by the facts that:

1.repeated measurements can be made.

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).

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5. How is uncertainty evaluated?

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.

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6. What is best practice?

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.

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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.



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