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Chapter

Chapter

# - Univariate CUSUM Charts

DOI link for - Univariate CUSUM Charts

- Univariate CUSUM Charts book

# - Univariate CUSUM Charts

DOI link for - Univariate CUSUM Charts

- Univariate CUSUM Charts book

## ABSTRACT

Assume that we want to monitor a univariate quality characteristic X of a produc-

tion process. The Shewhart charts described in the previous chapter make decisions

about the process performance at a given time point based solely on the observed

data at that single time point. Their decision rules are constructed in a similar way

to those of the hypothesis testing procedures, by which the process is declared OC

if the observed value of the charting statistic of a Shewhart chart is beyond its lower

and upper control limits at the given time point, and the process is declared IC oth-

erwise. For simplicity of perception, let us further assume that we are interested in

monitoring the process mean in a given application, and other distributional proper-

ties of X would not change after a mean shift. In such cases, the distribution of X is

F0 before the mean shift, it changes to another distribution F1 after the shift, and the

only difference between F0 and F1 is their means. At a given time point n, if there

is no shift before n, then all observations obtained before n should provide useful

information about the IC process performance. In cases when the mean shift occurs

before n, all observations obtained between the occurrence of the shift and the time

n should all provide useful information about the OC process performance. In both

cases, the history data obtained before n contain useful information for process mon-

itoring; but, the Shewhart charts do not make use of such information at all. For this

reason, they would not be effective for detecting persistent shifts, especially in cases

when the shifts are relatively small, which is demonstrated in the following example.

Example 4.1 Assume that 10 samples of size 5 each are generated from the N(0,1) distribution, and another 10 samples of size 5 each are generated from the N(0.2,1) distribution. These samples can be regarded as observed data collected at 20 consecutive time points from a production process, the IC distribution of the process is N(0,1), and the process has a mean shift of size 0.2 at the 11th time point. Figure 4.1(a) shows the X chart constructed from the 20 samples with the control limits computed by the formulas in (3.6). In the plot, the first 10 sample means are denoted by solid dots and they are connected by solid lines, and the second 10 sample means are denoted by little circles and they are connected by dotted lines. This chart can be used for a phase I SPC. From the plot, it can be seen that the true mean shift at the 11th time point is not detected by the chart. Figure 4.1(b) shows an alternative X chart for a phase II SPC, in which the IC process distribution is assumed to be

known N(0,1), the lower and upper control limits of the chart are set to−3/√5 and

3/ √ 5, respectively, and all the symbols and notations in the plot have the same inter-

pretation as those in Figure 4.1(a). From the plot, it can be seen that the alternative X chart cannot detect the true mean shift at the 11th time point either.