Annex J *

Glossary of principal symbols

a half‑width of a rectangular distribution of possible values of input quantity Xi:
eqn_j1

a+ upper bound, or upper limit, of input quantity Xi

a lower bound, or lower limit, of input quantity Xi

b+ upper bound, or upper limit, of the deviation of input quantity Xi from its estimate xi:
eqn_j2

b lower bound, or lower limit, of the deviation of input quantity Xi from its estimate xi:
eqn_j3

ci partial derivative or sensitivity coefficient:
eqn_j4

f functional relationship between measurand Y and input quantities Xi on which Y depends, and between output estimate y and input estimates xi on which y depends

fxi partial derivative with respect to input quantity Xi of functional relationship f between measurand Y and input quantities Xi on which Y depends, evaluated with estimates xi for the Xi:
eqn_j5

k coverage factor used to calculate expanded uncertainty U = kuc(y) of output estimate y from its combined standard uncertainty uc(y), where U defines an interval Y = y ± U having a high level of confidence

kp coverage factor used to calculate expanded uncertainty Up = kpuc(y) of output estimate y from its combined standard uncertainty uc(y), where Up defines an interval Y = y ± Up having a high, specified level of confidence p

n number of repeated observations

N number of input quantities Xi on which measurand Y depends

p probability; level of confidence:
eqn_j6

q randomly varying quantity described by a probability distribution

q‾‾‾ arithmetic mean or average of n independent repeated observations qk of randomly‑varying quantity q

estimate of the expectation or mean μq of the probability distribution of q

qk kth independent repeated observation of randomly‑varying quantity q

r(xixj) estimated correlation coefficient associated with input estimates xi and xj that estimate input quantities Xi and Xj:
eqn_j7

r(X‾‾‾iX‾‾‾j) estimated correlation coefficient of input means X‾‾‾i and X‾‾‾j, determined from n independent pairs of repeated simultaneous observations Xik and Xjk of Xi and Xj:
eqn_j8

r(yiyj) estimated correlation coefficient associated with output estimates yi and yj when two or more measurands or output quantities are determined in the same measurement

s2p combined or pooled estimate of variance

sp pooled experimental standard deviation, equal to the positive square root of s2p

s2(q‾‾‾) experimental variance of the mean q‾‾‾

estimate of the variance σ2n of q‾‾‾:
eqn_j9

estimated variance obtained from a Type A evaluation

s(q‾‾‾) experimental standard deviation of the mean q‾‾‾, equal to the positive square root of s2(q‾‾‾)

biased estimator of σ(q‾‾‾) (see C.2.21, note)

standard uncertainty obtained from a Type A evaluation

s2(qk) experimental variance determined from n independent repeated observations qk of q

estimate of the variance σ2 of the probability distribution of q

s(qk) experimental standard deviation, equal to the positive square root of s2(qk)

biased estimator of the standard deviation σ of the probability distribution of q

s2(X‾‾‾i) experimental variance of input mean X‾‾‾i, determined from n independent repeated observations Xik of Xi

estimated variance obtained from a Type A evaluation

s(X‾‾‾i) experimental standard deviation of input mean X‾‾‾i, equal to the positive square root of s2(X‾‾‾i)

standard uncertainty obtained from a Type A evaluation

s(q‾‾‾r‾‾‾) estimate of the covariance of means q‾‾‾ and r‾‾‾ that estimate the expectations μq and μr of two randomly‑varying quantities q and r, determined from n independent pairs of repeated simultaneous observations qk and rk of q and r

estimated covariance obtained from a Type A evaluation

s(X‾‾‾iX‾‾‾j) estimate of the covariance of input means X‾‾‾i and X‾‾‾j, determined from n independent pairs of repeated simultaneous observations Xik and Xjk of Xi and Xj

estimated covariance obtained from a Type A evaluation

tp(v) t‑factor from the t‑distribution for v degrees of freedom corresponding to a given probability p

tp(veff) t‑factor from the t‑distribution for veff degrees of freedom corresponding to a given probability p, used to calculate expanded uncertainty Up

u2(xi) estimated variance associated with input estimate xi that estimates input quantity Xi

NOTE   When xi is determined from the arithmetic mean or average of n independent repeated observations, u2(xi) = s2(X‾‾‾i) is an estimated variance obtained from a Type A evaluation.

u(xi) standard uncertainty of input estimate xi that estimates input quantity Xi, equal to the positive square root of u2(xi)

NOTE   When xi is determined from the arithmetic mean or average of n independent repeated observations, u(xi) = s(X‾‾‾i) is a standard uncertainty obtained from a Type A evaluation.

u(xixj) estimated covariance associated with two input estimates xi and xj that estimate input quantities Xi and Xj

NOTE   When xi and xj are determined from n independent pairs of repeated simultaneous observations, u(xixj) = s(X‾‾‾iX‾‾‾j) is an estimated covariance obtained from a Type A evaluation.

u2c(y) combined variance associated with output estimate y

uc(y) combined standard uncertainty of output estimate y, equal to the positive square root of u2c(y)

ucA(y) combined standard uncertainty of output estimate y determined from standard uncertainties and estimated covariances obtained from Type A evaluations alone

ucB(y) combined standard uncertainty of output estimate y determined from standard uncertainties and estimated covariances obtained from Type B evaluations alone

uc(yi) combined standard uncertainty of output estimate yi when two or more measurands or output quantities are determined in the same measurement

u2i(y) component of combined variance u2c(y) associated with output estimate y generated by estimated variance u2(xi) associated with input estimate xi:
eqn_j10

ui(y) component of combined standard uncertainty uc(y) of output estimate y generated by the standard uncertainty of input estimate xi:
eqn_j11

u(yiyj) estimated covariance associated with output estimates yi and yj determined in the same measurement

u(xi) xi relative standard uncertainty of input estimate xi

uc(y) y relative combined standard uncertainty of output estimate y

[u(xi)xi]2 estimated relative variance associated with input estimate xi

[uc(y)y]2 relative combined variance associated with output estimate y

en_j12
estimated relative covariance associated with input estimates xi and xj

U expanded uncertainty of output estimate y that defines an interval Y = y ± U having a high level of confidence, equal to coverage factor k times the combined standard uncertainty uc(y) of y:
eqn_j13

Up expanded uncertainty of output estimate y that defines an interval Y = y ± Up having a high, specified level of confidence p, equal to coverage factor kp times the combined standard uncertainty uc(y) of y:
eqn_j14

xi estimate of input quantity Xi

NOTE   When xi is determined from the arithmetic mean or average of n independent repeated observations, xi = X‾‾‾i.

Xi ith input quantity on which measurand Y depends

NOTE    Xi may be the physical quantity or the random variable (see 4.1.1, Note 1).

X‾‾‾i estimate of the value of input quantity Xi, equal to the arithmetic mean or average of n independent repeated observations Xik of Xi

Xik kth independent repeated observation of Xi

y estimate of measurand Y

result of a measurement

output estimate

yi estimate of measurand Yi when two or more measurands are determined in the same measurement

Y a measurand

eqn_j15
estimated relative uncertainty of standard uncertainty u(xi) of input estimate xi

μq expectation or mean of the probability distribution of randomly‑varying quantity q

v degrees of freedom (general)

vi degrees of freedom, or effective degrees of freedom, of standard uncertainty u(xi) of input estimate xi

veff effective degrees of freedom of uc(y), used to obtain tp(veff) for calculating expanded uncertainty Up

veffA effective degrees of freedom of a combined standard uncertainty determined from standard uncertainties obtained from Type A evaluations alone

veffB effective degrees of freedom of a combined standard uncertainty determined from standard uncertainties obtained from Type B evaluations alone

σ2 variance of a probabiIity distribution of (for example) a randomly‑varying quantity q, estimated by s2(qk)

σ standard deviation of a probability distribution, equal to the positive square root of σ2

s(qk) is a biased estimator of σ

σ2(q‾‾‾) variance of q‾‾‾, equal to σ2n, estimated by s2(q‾‾‾) = s2(qk)n

σ(q‾‾‾) standard deviation of q‾‾‾, equal to the positive square root of σ2(q‾‾‾)

s(q‾‾‾) is a biased estimator of σ(q‾‾‾)

σ2[s(q‾‾‾)] variance of experimental standard deviation s(q‾‾‾) of q‾‾‾

σ[s(q‾‾‾)] standard deviation of experimental standard deviation s(q‾‾‾) of q‾‾‾, equal to the positive square root of σ2[s(q‾‾‾)]


*   Footnote to the 2008 version:
When the GUM was first published, there was an editorial rule in effect which prohibited the use of an Annex I. That is why the annexes go from Annex H directly to Annex J.