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GUIDEFOREVALUATIONOFUNCERTAINTY IN CALIBRATION

September 22, 2008

Copyright 2008
IAS/CL/019
Revised January 1, 2016
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GUIDE FOR EVALUATION OF UNCERTAINTY IN CALIBRATION

Symbols and Abbreviations Used∑:

Sigma, which means Summationα:

Coefficient of thermal expansion(…µ per oC per meter, OR …µ” per oF per inch)σ:Standard deviationk:

Coverage factor

DOF:

Degrees of freedomn:Number of repeated readings% Rh:Percentage relative humidityt:

Temperaturets:

Temperature at start of exercisete:Temperature at end of exercise

UUC:

Unit under calibration

UA, UA1, UA2:

Type ‘A’ uncertainties

UB, UB1, UB2:

Type ‘B’ uncertainties

Uc:

Combined uncertainty

U95:

Expanded Uncertainty at 95% confidence level

X1, X2, X3:

Individual readings

X:

Mean (average) of readings

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METHODOLOGY FOR EVALUATION OF UNCERTAINTIES

1. Instrument details:(Unit under calibration)

Name:Identification number:
Range:
Resolution:

2. Calibration procedure:

Procedure number:
Last revised on:
Reference for procedure:

3. Date / Location of calibration:
4. Reference standards used:(a) Name/Grade/Class:Traceability / Cal Cert Number:
Uncertainty of Calibration:
Deviation from nominal value:(b) Name/Grade/Class:Traceability/Cal Cert Number:
Uncertainty of Calibration:
Deviation from nominal value:[Use (c), (d), etc., if additional reference standards are used]
5. Standard reference temperature/humidity: 68°F (20°C) / 50±5% Rh(for dimensional)
6. Actual environmental conditions:a) Temperature:

At start (ts): __˚cAt end (te): __˚cAverage: (ts + te) ∕ 2: __˚cThermometer resolution: __˚cb) Humidity% Rh

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7. Repeatability Measurements (Type ‘A’ Evaluation):

Serial No. Readings(Xi) Mean ( x )1

X1

2X2(Xi - x)2∑(Xi – x)2

Standard deviation(X1 – x)2x
(X2 – x)2

X3

isequal to4

X4∑ Xn5X56X6(X6 – x)27X7(X7 – x)28X8(X8 – x)29X9(X9 – x)2

10X10(X10 – x)23n(X3 – x)2∑ (Xi – x)2∑(Xi – x)2( n-1)

2(X4 – x)(X5 – x)2σ = Standard deviation of the above 10 repeated readings =∑(X i – x)2( n-1)
To arrive at the random uncertainty of the population, the standard deviation of the mean has
to be evaluated. This is obtained by dividing the standard deviation of the sample by √n.
Standard deviation of the mean = σ / √nThis is the type ‘A’ uncertainty orRandom Uncertainty

UA1 = σ / √nIf there is more than one contributor for type ‘A’ uncertainties, repeat the above process foreach one of them. Term them as UA2, UA3, UA4, etc.
Then, Random Uncertainty,

UA = U 2A1 + U 2A3 +….+ U 2An

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8. Type ‘B’ Evaluation:
While considering the various contributory factors towards type B uncertainties, one has toanalyze the calibration process itself, if necessary by breaking it down into a number ofsmall processes and studying the influences that are likely to affect the process. Generally,type B factors may include but are not limited to: the reference standard, the UUC itself,other supporting instruments involved in the calibration process, personnel carrying out thecalibration, and environmental factors such as temperature, pressure, humidity, air density,
local “g” value, etc. Any possible interaction between the above factors may also need tobe taken into account.

Generally, the following factors should be included in any type ‘B’ evaluation.

UB1:

Uncertainty contribution due to deviation of the reference standard from its nominalvalue. (Assume rectangular distribution and coverage factor of √3.)

UB2:

Uncertainty due to the uncertainty of calibration of reference standard (valueobtained from the calibration certificate of the reference standard from a higher level
laboratory or NMI. Coverage factor is also obtained from the certificate and is usuallyk = 2, for normal distribution).

UB3:

Uncertainty contribution due to resolution of UUC (take half resolution & k=√3).

UB4:

Uncertainty contribution due to resolution of the temperature measuring device (takehalf resolution and distribution as rectangular – use sensitivity coefficient to accountfor input/output mismatch).

UB5:

Uncertainty contribution due to uncertainty of the temperature measuring device (thisvalue & the coverage factor, k, are obtained from the calibration certificate).
UB6, UB7, UB8, UB10, etc.

(These contributory factors are considered depending on the actual calibration being carriedout and keeping in mind that there is no limit to the number of factors to be considered.)

After evaluating all the contributory factors, the combined uncertainty has to be evaluated, thedegrees of freedom must be calculated and the final expanded uncertainty determined.

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9. Uncertainty Budget:

Serial Source ofNo. Uncertainty& Value
1Type A
2Type B
3Type B
4Type B
5Type B
6Type B
7Type B
8Type B etc

Probability
Distribution
FactorNormal – 2Standard
Sensitivity Uncertainty
Uncertainty Coefficient* Contribution
1
1
1
1
1
1
1
1

UA1
UB1
UB2
UB3
UB4
UB5
UB6
UB7 etc

DegreesofFreedom(n-1)∞∞∞∞∞∞∞[*The sensitivity coefficient depends on the units for input and output quantities (for example,temperature input in °C and linear measure output in microns) into the measuring system andhas to be worked out depending on the particular calibration being carried out.]
Now, Combined Uncertainty,
Uc=

Effective Degrees of Freedom Veff =U2A + U2B1 + U2B2 + … + U2Bn
U4A
(n-1)

U4C
44+ U B1 + U B2 + …
DOF

DOF

This Veff should be a large number.

10. Refer to the “Student T” table to obtain the “k” factor for the number of repeatability
readings taken and the intended confidence level (usually 95% for calibration laboratories).
11. Expanded uncertainty at 95% confidence level is:
U95 = UC X ‘k’ Factor

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Type ‘B’ Factors to be Considered for Various Disciplines
Following are the factors that generally need to be considered for type ‘B’ evaluation ofuncertainties. However, it may be worthwhile remembering that this list is not exhaustive andalso that there is no limit to the consideration of contributory factors. From the list, choose only
the factors that are relevant for the particular calibration exercise that is being carried out.
I.

Dimensional (consider whatever factors are relevant for the exercise being done)
1)

Deviation of reference standard from its nominal value (k=√3)

2)

Uncertainty of calibration of reference standards (from certificate, k=2)

3)

Resolution of UUC (consider half resolution & k=√3)

4)

Resolution of thermometer (consider half resolution & k=√3)

5)

Uncertainty of calibration of thermometer (value from certificate and k=2)

6)

Difference between standard temperature and average temperature of calibration(k=√3)

7)

Nonuniform expansion because of temperature difference between reference
standard and UUC. (Consider this to be 20% of (6) and rectangular distribution k=√3)

8)

Uncertainty of thermal coefficient of expansion of UUC(Consider this to be 10% of the value and rectangular distribution. K=√3.)

9)

Uncertainty of thermal coefficient of expansion of reference standard(Consider this to be 10% of the value and rectangular distribution. If both UUC andreference standard are of the same material, combine (8) and (9) and take 20% k=√3.)

10) Flatness of surface plate (10-20% of value), if a surface plate is used in thecalibration (k=√3).
11) Uncertainty of flatness of surface plate, if a surface plate is used in the calibration(k=√3).
12) Flatness of anvils (e.g., a micrometer & k=√3)13) Parallelism of anvils (e.g., a micrometer & k=√3)14) Parallelism of jaws (e.g., a caliper & k=√3), etc.
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II.

III.

IV.

Hardness (Indirect calibration)(1)

Uncertainty of calibration of reference hardness block.

(2)

Resolution of hardness tester (converted to hardness value using sensitivity
coefficient).

Force
(1)

Uncertainty of calibration of the proving device(2)

Resolution of force indication(3)

Effect of temperature on proving device(4)Effect of drift of proving device if no data is available, assume a value equal to (1)i.e., uncertainty of the proving device(5)Linear interpolation between calibration points (If polynomial curve fit is used, this isnegligible.)

Mass
(1)

Uncertainty of calibration of reference weight(2)

Resolution of balance(3)

Air buoyancy(4)

Eccentricity effects(5)

Drift(6)Hysteresis effects.
**************************

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UNCERTAINTY EVALUATION FORMAT
1. Name of the laboratory:
2. Unit under calibration:

Name:
Range:
3. Standards and equipment used:
SerialNo.
1

Ref.Std / Equipment

Resolution:

Range/ Resol.

U95

Deviation

Traceability234

4. Temperature and humidity:
StartMiddle

End

AverageHumidity

5. Type ‘A’ evaluation:
SerialNo.

Readings

Mean ( X bar)

( Xi - X bar)

( Xi -X bar)

2

Std. Dev (σ)

123Σ XnX bar =σ =n4∑(Xi –X bar)2(n-1)

5678910

Ua i.e., Random Uncertainty is ( σ / √n ):Degrees of freedom: (n-1) =IAS/CL/019
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6. Type ‘B’ evaluation:
SlNo

SourceValueDistribution& ‘k’ factor

Calculations

Std.Unc1

2

3

4

5

6

7

8

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7. Uncertainty budget:

SourceofUncertainty

Estimate

Probability
Distribution

Standard
Uncertainty

Sensitivity
Coefficient

Uncertainty
Contribution

DegreesofFreedom

Effective Degrees of freedom, Veff ,=Uc4
UA4 / (n -1) + UB14/ DOF + UB24/DOF + UB34/DOF + ………..+ UBn 4/ DOF *Coverage factor for 95% confidence level =Expanded uncertainty at 95% confidence level = Uc X coverage factor =* DOF: Degrees of Freedom

Calibrated byApproved byAssessor’s comments and signature:

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