Evaluation+of+Type+A+and+Type+B+Uncertainties

Back to Measurements, uncertainty and Detection Limits

Evaluation of Type A Uncertainty
Let Xi be an input quantity in the mathematical model for Y. A series of n independent measurements of Xi are made under the same experimental conditions, yielding the results Xi,1, Xi,2, …Xi,n. The appropriate value for the input estimate xi is the arithmetric mean defined as: x_I=\overline{X_i}=\frac{1}{n}\cdot\sum^{n}_{k-1}X_{i,k} math || Eqn. 1 ||
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The experimental variance of the observed values is defined as:

S^2(X_{i,k})=\frac{1}{n-1}\cdot\sum^{n}_{k-1}\left(X^2_{i,k}-\overline{X_i}\right)^2 math || Eqn. 2 ||
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The experimental standard deviation, s(Xi,k), is the square root of s2(Xi,k). The experimental standard deviation of the mean is defined by:

u(X_i)=\sqrt{\frac{1}{\cdot(n-1)}\cdot\sum^{n}_{k-1}\left(X_{i,k}-\overline{X_i}\right)^2} math || Eqn. 3 ||
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Example: 10 independent measurements of a quantity Xi are made yielding the values:


 * 15.235 || 15.333 || 15.458 || 15.641 || 15.124 ||
 * 15.280 || 15.701 || 15.427 || 15.222 || 15.564 ||

From Eqn. 1 above on determines that

math x_i=\overline{X_i}=\frac{153.985}{10}=\underline{15.3985} math The standar uncertanity of X i is:

math u(x_i) = s(\overline{X_i})=\sqrt{\frac{1}{10\cdot(10-1)}\cdot\sum^{10}_{k-1}((X_{i,k}-15.3985)^2}=0.061 math and the masured number would be denoted math \underline{\overline{X_i}=15.399\pm0.061} math or, when considering the uncertainty in he uncertainty, rather math \underline{\overline{X_i}=15.40\pm0.06} math

Evaluation of Type B Uncertainty
One example of a Type B method is the estimation of counting uncertainty using the square root of the observed counts. If the observed counts is S, when the Poisson counting model is used, the standard uncertainty of S may be evaluated as:

U(S)=\sqrt{S} math || Eqn. 4 || Sometimes a Type B evaluation of u(x) consists of estimating an upper limit a to the error based on professional judgement of best available information. If the error has a uniform distribution, u(x) may be calculated by: u(x)=\frac{a}{\sqrt{3}} math || Eqn. 5 ||
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