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    Glossary of Definitions and Symbols

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    Sonja Kostić
    This document is a manual on determining uncertainties in mechanical tests on metallic materials. It contains definitions of key terms related to measurement uncertainty such as measurement, error, uncertainty, and definitions of specific uncertainty terms like standard deviation, combined standard uncertainty, and expanded uncertainty. It also lists relevant equations and provides a table of coverage factors for different effective degrees of freedom based on Student's t-distribution.

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    Glossary of Definitions and Symbols

    Uploaded by

    Sonja Kostić
    29 views5 pages
    This document is a manual on determining uncertainties in mechanical tests on metallic materials. It contains definitions of key terms related to measurement uncertainty such as measurement, error, uncertainty, and definitions of specific uncertainty terms like standard deviation, combined standard uncertainty, and expanded uncertainty. It also lists relevant equations and provides a table of coverage factors for different effective degrees of freedom based on Student's t-distribution.

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    This document is a manual on determining uncertainties in mechanical tests on metallic materials. It contains definitions of key terms related to measurement uncertainty such as measurement, error, uncertainty, and definitions of specific uncertainty terms like standard deviation, combined standard uncertainty, and expanded uncertainty. It also lists relevant equations and provides a table of coverage factors for different effective degrees of freedom based on Student's t-distribution.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    29 views5 pages

    Glossary of Definitions and Symbols

    Uploaded by

    Sonja Kostić
    This document is a manual on determining uncertainties in mechanical tests on metallic materials. It contains definitions of key terms related to measurement uncertainty such as measurement, error, uncertainty, and definitions of specific uncertainty terms like standard deviation, combined standard uncertainty, and expanded uncertainty. It also lists relevant equations and provides a table of coverage factors for different effective degrees of freedom based on Student's t-distribution.

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    of 5

    SM&T Section 2

    Standards Measurement & Testing Project No. SMT4-CT97-2165


    UNCERT MANUAL: 2000

    Manual of Codes of Practice for the Determination of Uncertainties in


    Mechanical Tests on Metallic Materials

    SECTION 2

    Glossary of definitions and symbols

    F A Kandil

    National Physical Laboratory


    Queens Road
    Teddington, Middlesex TW11 0LW
    UNITED KINGDOM

    Issue 1

    September 2000
    SM&T Section 2
    Standards Measurement & Testing Project No. SMT4-CT97-2165
    UNCERT MANUAL: 2000

    2.1 DEFINITIONS

    Coverage factor

    A number that, when multiplied by the combined standard uncertainty, produces the
    expanded uncertainty. It is dependent on the confidence level (e.g. 95% probability).

    Error of measurement

    The result of a measurement minus the true value of the measurand (not precisely
    quantifiable because the true value is unknown and lies somewhere within the range
    of uncertainty).

    Level of confidence

    The probability that the value of the measurand lies within the quoted range of
    uncertainty.

    Measurand

    The specific quantity being reported as the measurement result. A measurand can be
    a direct test reading or an estimate of a material property from other readings.

    Measurement

    A set of operations having the object of determining a value of the measurand.

    Result of a measurement

    Value attributed to the measurand, obtained by measurement.

    Uncorrected result
    Result of a measurement before correction for systematic error.

    Corrected result
    Result of a measurement after correction for systematic error.

    Standard deviation

    The positive square root of the variance.

    Uncertainty of measurement

    A parameter, associated with the result of a measurement, that defines the range
    within which the true value of a measurand is estimated to fall (within a given
    confidence).

    Page 1 of 4
    SM&T Section 2
    Standards Measurement & Testing Project No. SMT4-CT97-2165
    UNCERT MANUAL: 2000

    Standard uncertainty
    The estimated standard deviation.

    Combined standard uncertainty


    The result of the combination of standard uncertainty components.

    Expanded uncertainty
    The value obtained by multiplying the combined standard uncertainty by a coverage
    factor.

    Variance

    A measure of the dispersion of a set of n measurement results. It is the sum of the


    square of the deviation of the measurement result from the average, divided by n-1.

    2.2 SYMBOLS

    ci Sensitivity coefficient.

    dv Divisor used to calculate the standard uncertainty

    =1 (for normal probability distribution)


    =2 (for normal probability distribution, k = 2)
    = 3 (for rectangular probability distribution)
    = 6 (for triangular probability distribution)
    = 2 (for U-shaped probability distribution)

    f Functional relationship between the estimated value of the measurand, y,


    and the input parameters xi.

    k Coverage factor used to calculate expanded uncertainty U for a normal


    distribution.

    kp Coverage factor used to calculate an expanded uncertainty for a specified


    level of confidence p where a normal probability distribution cannot be
    assumed (see table in Section 2.4).

    n Number of repeat measurements.

    m Number of input parameters on which the measurand depends.

    p Probability or level of confidence expressed in percentage terms or in the


    range 0 to 1.

    q Random variable.

    Page 2 of 4
    SM&T Section 2
    Standards Measurement & Testing Project No. SMT4-CT97-2165
    UNCERT MANUAL: 2000

    q Arithmetic mean or average of n repeated measurements of randomly


    varying quantity q. [Eq. (2)]

    s(qj) Experimental standard deviation of a random variable q determined from n


    repeat measurements, when n is a relatively small number. [Eq. (3)]

    s( q ) Experimental standard deviation of arithmetic mean q . [Eq. (4)]

    u(xi) Standard uncertainty of input parameter xi. [Eq. (5)]

    uc(y) Combined standard uncertainty of the measurand, y. [Eq. (6)]

    U Expanded uncertainty of the measurand, y. [Eq. (8)]

    V Value of the measurand.

    xi Estimate of input quantity Xi.

    y Estimate of the measurand V (V= y ± U). [Eq. (1)]

    νi Degrees of freedom of standard uncertainty u(xi) of input parameter, xi.

    νeff Effective degrees of freedom of uc(y) used to obtain kP (t- distribution).


    [Eq. (7)]

    2.3 EQUATIONS FOR UNCERTAINTY CALCULATIONS

    y = f ( x1 ,x 2 ,.............., xm ) (1)

    1 n
    n∑
    q= qj (2)
    j =1

    1 n
    s (q j )= ∑
    (n − 1) j =1
    (q j − q )2 (3)

    s( q j )
    s (q ) = (4)
    n

    u( x i ) = s ( q ) [Type A uncertainty] (5a)

    Page 3 of 4
    SM&T Section 2
    Standards Measurement & Testing Project No. SMT4-CT97-2165
    UNCERT MANUAL: 2000

    tolerance
    u( xi ) = [Type B uncertainty] (5b)
    dv

    m
    uc ( y ) = ∑i =1
    [ci u ( xi )]2 (6)

    uc4 ( y )
    ν eff = (7)
    m
    ui4 ( y )
    ∑
    i =1 νi

    U = k uc ( y ) (8)

    2.4 Student’s t-Distribution Table

    νeff 1 2 3 4 5 6 7 8 10 12 14 14
    k95 13.97 4.53 3.31 2.87 2.65 2.52 2.43 2.37 2.28 2.23 2.20 2.17

    νeff 18 20 25 30 35 40 45 50 60 80 100 ∞
    k95 2.15 2.13 2.11 2.09 2.07 2.06 2.06 2.05 2.04 2.03 2.02 2.00

    NOTE: The above values are for a level of confidence of 95%. Values for other levels of confidence can
    be found in the Guide.

    Page 4 of 4

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