How to get from a single measurement to the full picture
No matter whether you’re measuring layer thickness, electrical conductivity or the composition of a material – there are always fluctuations. Most measurements are influenced by random factors. A single value cannot describe the true quantitative property of a measured entity. For that, repeated measurements and multiple individual values are necessary. And statistical methods are required to evaluate the repeated measurements.
From a sufficiently large number of measured values, one can identify the mean value and the corresponding variance. Then it is possible to calculate the distribution of the individual values around the mean value. Using statistical distribution, it is possible to predict the thickness of the coating throughout the entire process – meaning that the process can be assessed without having to do 100 % monitoring.
With Fischer devices, statistical analysis of the measuring results is no problem. Here’s a summary of the most important statistical parameters.
The mean x is an average of the different readings. The simplest way to calculate a mean is to add all the values together and then divide that sum by the number of values. This is called the arithmetic mean. There are other ways to calculate a mean, but they’re seldom used.
The range R shows how far apart the smallest measured value is from the largest. To calculate the range, simply subtract the lowest measured value from the largest one. The range can be greatly distorted by outliers and is therefore only useful if you have just a few readings. For larger quantities of data, the standard deviation is more meaningful.
The standard deviation σ indicates how widely scattered or clumped together the readings are around the mean. A high standard deviation indicates that the measured values differ greatly from each other. But if the values are all close to the mean, the standard deviation is small. How well the mean and the standard deviation describe the reality depends, among other things, on the number of measurements: the more measurement points, the more meaningful the metrics.
In two measurement series you get the values [1, 2, 3] and [1.5, 2, 2.5]. In both cases, the mean is 2. However, the standard deviations are different: In the first case it’s 1, in the second it’s 0.5. The standard deviation makes clear that the values in the second case are more similar to each other.
Coefficient of variation
The magnitude of the standard deviation depends not only on the variance among the readings but also on the size of the values: A higher average automatically leads to a higher standard deviation. To address this problem, the relative standard deviation – that is, the coefficient of variation V – is often given as a percentage. For that, the standard deviation is divided by the arithmetic mean. As with the standard deviation, higher values indicate that the measured values are more widely scattered.
You measure a thin and a thick coating. The thin paint is uneven and has, for its average depth of 10 microns, a standard deviation of about 1 micron. That corresponds to a coefficient of variation of 10 %. The thicker coating is more even and, for its depth of 100 μm, also has a standard deviation of 1 μm. But here, the coefficient of variation is 1 %. In this case, the coefficient of variation expresses the differences in the coating quality much better than the standard deviation does.