Back to Measurement, Uncertainty and Detection Limits

### Standard Uncertainty Classes

Standard uncertainties u(xi) are here classified as either Type A or Type B: Type A is a statistical evaluation based on repeated observations.

One typical example of a Type A evaluation involves making a series of independent measurements of a quantity Xi, and calculating the arithmetric mean and the experimental standard deviation of the mean.

The arithmetric mean is used as the input estimate, xi, and the experimental standard deviation of the mean is used as the standard uncertainty u(xi).

Any evaluation of standard uncertainty that is not a Type A evaluation is a Type B evaluation.

### Expanded Uncertainty

A measurement may be reported with the combined standard uncertainty uc(y) or it may be multiplied by a coverage factor, k, to produce an expanded uncertainty denoted U, such that the interval y ± U has a specific high probability p of containing the true value of the measurand.

The specific probability p is called the level of confidence or the coverage probability and is generally only an approximation of the true probability of coverage.

The coverage factor often chosen for approximately normal distributions of y is k = 2. If uc(y) represents one standard deviation, U then corresponds to a confidence level of 95 %.

### Significant Figures

The number of significant figures of a measured result y depends on the uncertainty of the result. Since there is uncertainty also in the uncertainty estimates, a common convention is to round off the uncertainty to one or two significant figures.

Examples:

### Procedure for Estimating Uncertainty

One typical example of a Type A evaluation involves making a series of independent measurements of a quantity Xi, and calculating the arithmetric mean and the experimental standard deviation of the mean.

The arithmetric mean is used as the input estimate, xi, and the experimental standard deviation of the mean is used as the standard uncertainty u(xi).

Any evaluation of standard uncertainty that is not a Type A evaluation is a Type B evaluation.

The specific probability p is called the level of confidence or the coverage probability and is generally only an approximation of the true probability of coverage.

The coverage factor often chosen for approximately normal distributions of y is k = 2. If uc(y) represents one standard deviation, U then corresponds to a confidence level of 95 %.

Examples:

Measured Value y |
Expanded Uncertainty U=kuc(y) |
Reported Result |

15.235 |
0.121 |
15.24±0.12 |

15.235 |
1.213 |
15.2±1.2 |

15.235 |
12.134 |
15±12 |

15.235 |
121.343 |
20±12 |

15.235 |
1213.432 |
0±1200 |

- Identify the measurand Y and all input quantities Xi and express the mathematical relationship Y = f(X1, X2,..XN)
- Determine estimates xi of Xi
- Evaluate standard uncertainty uc(xi) for each xi
- Calculate estimate y of Y from y = f(x1, x2, ….xN)
- Determine combined standard uncertainty uc(y)
- Decide on a coverage factor k and multiply uc(y) with k to determine the expanded uncertainty U.
- Report the result as y ± U and state the coverage factor used and the confidence level.