Statistics for Process Control Engineers

30Chi Distribution

The chi‐distribution is described by the PDF

(30.1)

The parameter f is often referred to as the degrees of freedom, in which case it should be restricted to integers. Figure 30.1 shows the effect of changing f with α fixed at 0 and β at 1.

Figure 30.1 Chi: Effect of f on shape

30.1 Half‐Normal

With f set to 1, Equation (30.1) becomes almost the same as Equation (5.35), describing the normal distribution.

(30.2)

The only difference is a factor of 2 – required because it only applies to half the values of x, i.e. those greater than the mean. It is therefore known as the half‐normal distribution. Process disturbances and inferential errors, which are likely to have a mean close to zero, might follow a normal distribution. Their absolute value will then follow the half‐normal distribution.

Fitting to the absolute values of the NHV disturbances gives α as −0.08 and β as 1.31. As expected, because of the high kurtosis of the data, the fit is poor with RSS at 0.3248.

Mean and variance are

(30.3)

(30.4)

These give μ as 0.96 and σ as 0.52 – somewhat away from the ...