19.1 Type A Uncertainty Evaluation

For type A uncertainty evaluation the PDF of the measurand (\(X\)) is estimated from a series of measurements. This is typically used when measurements of the measurand varies with each measurement, sometimes due unquantifiable chaos in the physical system or some underlying randomness (as with Monte-Carlo methods - though there are GUM supplements that adress these directly). The best approximation of the measurand is determined as the mean of PDF describing the measurements, and the standard uncertainty as the standard deviation of this mean.

Consider a series of \(n\) measurements of the measurand \(X\): \(x_i\), all taken under the same conditions. Assuming these \(x_i\) are normally distributed, the best approximation of \(X\) is given by the arithmetic mean:

\[ \mu_x \approx \bar{x} = \frac{1}{n} \sum_{k-1}^{n} x_k \]

The varience (\(\sigma^2\)) of the PDF of x can be estimated using the experimental variance:

\[ \sigma \approx s^2(x) = \frac{1}{n-1} \sum_{j=1}^{n} (x_j - \bar{x})^2 \]

The standard deviation of the mean (\(\sigma_\mu\)), which is used as the standard uncertainty, can be calculated as:

\[ \sigma^2_\mu = \frac{\sigma}{n} \]

which can be estimated using the experimental variance of the mean:

\[ u^2(\bar{x}) = \sigma^2_\mu \approx s^2(\bar{x}) = \frac{s^2(x)}{n} \]

Hint

To calculate the experimental standard deviation, you can make use of the numpy.std() function with the keyword argument ddof=1.