It is generally not possible to measure error-free. The deviations of the measured values from their true values affect a measurement result, so this also deviates from its true value. The error analysis attempts to determine the influence of measurement errors on the measurement result quantitatively.
The term error calculation can be understood differently.
- Frequently one wants to calculate a measurement result of a measured variable or in the general case of multiple measurements using a known equation. In case of incorrect determination of the input variable (s) and the output size is determined incorrectly, because the individual deviations from the equation or transfer and lead to a deviation of the result. We call this error propagation. Under this heading formulas are given separately for the cases that the deviations are known as
- If you repeat the measurement of the variables under the same conditions, it is often found that distinguish the individual measured values ; scatter them. You then have
The scatter of measured values can be illustrated in a diagram. It divides the range of possible values in small areas with a width b and contributes to each region on how many measured values occur in this area, see example in the adjacent picture.
In the Gaussian or normal distribution (after Carl Friedrich Gauss ) we let go of the number of measurements N → ∞, while b → 0 with a diagram of the stepped curve merges into a continuous curve. This describes
- The density of the measured values in dependence on the measured value, and also
- For future measurement, whichever is the probability to be expected.
With the mathematical representation of the normal distribution can be described statistically induced many natural, economic or engineering operations. Also, random measurement errors are described in their entirety by the parameters of the normal distribution. These parameters are
- The arithmetic average of all measured values , called the expected value. This is the size of the abscissa of the maximum of the curve. At the same time he is at the point of the true value.
- The standard deviation as a measure of the width of scattering of the measured values . Is as large as the horizontal distance between a turning point of the maximum. In the region between the turning points are about 68 % of all measurements.
Uncertainty of a single measurement variable
The following applies in the absence of systematic errors and normally distributed random errors.
Estimating values of the parameters
If one has the size of several afflicted with random errors values with j = 1 ... N, so you get compared to the single value to an improved statement by taking the arithmetic mean value
The (empirical ) standard deviation is given by
These sizes are estimates for the parameters of the normal distribution. Due to the finite number of measurements is also subject to the mean nor random errors. A measure of the width of the spread of the mean value is the uncertainty
This is the smaller, the larger N is. It features together with the average a range of values in which the true value of the measurand is expected.
This expectation is fulfilled only with a certain probability. If you want to set the latter to a specific level of confidence, so you have a range ( confidence interval ) specify where the true value lies with this probability. The higher the probability is chosen, the wider the range needs to be. The factor t take into account the chosen confidence level and the number of measurements to the extent that is not yet conclusive with a small number N, the statistical treatment. If you choose the number above 68% confidence level and N> 12, then t = 1.0. For the frequently used in the art level of confidence of 95% and for N > 30, t = 2.0. A table of values of t ( Student's t-distribution) is located in DIN 1319-3.