METHODS FOR CONTROLLING SEVERAL RELATED QUALITY CHARACTERISTICS

There are many situations in which the simultaneous control of two or more related quality characteristics is necessary. For example, suppose that a bearing has both an inner diameter (x ₁ ) and an outer diameter (x ₂ ) that together determine the usefulness of the part. We can use a regular control chart for each of the variables , or we can use special control charts called multivariate. The pioneer for this work was Hotelling (1947). That work was followed by Hicks (1955), Jackson (1956, 1959), and Montgomery and Wadsworth (1972).

The chart, in essence, calculates the Hotelling statistic and plots the data in a normal way. The Hotelling statistic is calculated as follows :

where xbar ₁ and xbar ₂ are the sample means, and the variance of each sample, s ₁₂ the covariance of x ₁ and x ₂ , and n - 1 the degrees of freedom.

The actual control graph may be a region or an ellipse or a normal-looking control chart with the appropriate matrix transformations. If T ² > then at least one of the quality characteristics is out of control, where is the upper alpha percentage point of Hotelling's T ² distribution with 2 and n - 1 degrees of freedom.