Journal of The Japanese Society for Quality Control

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In most discussions on evaluation of process capability, quality characteristics are assumed to be normally distributed. In actual processes, however, we often observe the characteristics which are not normally distributed. To evaluate process capability correctly, we must clarify the shapes of distributions and use a proper evaluation method. Not only in evaluation of process capability but also in statistical method for quality control, quality characteristics are usually assumed to be normally distributed. Due consideration of the shape of distributions should be required in using statistical methods for quality control. The author has already clarified the shape of distribution of a dimension in a machining process controlled by a sizing device and the ones of positional and directional deviations. Characteristics representing form of surfaces of work-pieces are often measured in maching processes. In this paper the distribution of form deviation is derived theoretically using the theory of crossings of stochastic processes. In a case when the theory cannot be used measured data are expanded in Fourier series and the distribution is derived from the relationship between the Fourier coefficients and form deviation. It is verified by three examples of actual processes. As form deviation includes many kinds of characteristics such as out of roundness, flatness, surface roughness etc., and as the theory in this paper does not depend on processing methods, the derived distribution can be widely applied to various machining processes. To derive the distribution of form deviation in addition to those of dimensions, positional and directional deviations contributes to clarification on most parts of quality characteristics generally measured in machining processes.

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Let a random variable X, which indicates a control characteristic of a process, be distributed as N(μ, σ²) and the two parameters, μ and σ, take the values of μc and σc in a controlled state. If a point that is out of control limit is found in a control chart, we need to practically distinguish the following types of change in the process :
(1) Which parameter gets out of control ?
　 a) μ only does.
　b) σ only does.
　 c) Both of μ and σ do.
(2) Does the parameter increase or decrease compared with the controlled value (μc or σc) ?
　 a) It increases beyond the controlled value (μc or σc).
　 b) It decreases less than the controlled value ((μc or σc). Usually a pair of control charts such as x-R charts and x-s charts are originally expected to distinguish the types of change mentioned above. However, these control charts cannot necessarily do so, because each control chart is designed as a both-sided test for each parameter of μ and σ, respectively. In this paper, a simultaneous control chart for multiple decision about μ and σ is proposed, for the purpose of distinguishing these types of change in a process.

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