VARIABLES CONTROL CHART AND CASE EXAMPLE

In section 7.6 a histogram was constructed by using the data from a case example. The data collected was the response times on each of 235 suggestions for improvements. The data was presented in the same order as the suggestions were received by the suggestion committee and the data was grouped with five measurements in each subgroup (Table 7.5).

The grouping of observations is done in order to calculate and analyse the variation between the mean response times and the variation within the groups measured by the range. The suggestion system may be out of statistical control either because of non-random patterns in the means or non-random patterns in the range. Table 7.6 shows the calculated means and ranges within each subgroup.

The number of observations within each subgroup determines the number of subgroups. The number may vary according to the total number of observations in the data set but five observations are usually recommended in each subgroup and for the first construction of the control chart which we deal with in this example, it is recommended

that there are at least 80–100 observations (16–20 subgroups). For further explanations study the literature concerning control charts.

The steps to be followed when constructing the M-R control chart are the following. Step 1: Plot the calculated means (M) and ranges (R) in two different charts (diagrams) where the abcissa is equal to subgroup number and the ordinate measure are the means and ranges respectively.

Step 2: Calculate the average range and the process average (=the average of the subgroup means):

(7.2) (7.3) Step 3: Calculate the control limits UCL (Upper Control Limit) and LCL (Lower Control Limit):

Control limits for the means (=M):

(7.4) (7.5)

Table 7.6 Response times, means and ranges for quality

suggestions

Subgroup Response time (X) Mean (M) Range (R)

1 14, 14, 11, 13, 10 12.4 4 2 19, 10, 11, 11, 14 13.0 9 3 11, 11, 17, 10, 11 12.4 7 4 13, 10, 13, 10, 13 11.8 3 5 11, 13, 10, 13, 10 11.4 3 6 11, 16, 12, 10, 13 12.4 6 7 11, 16, 10, 10, 9 11.2 7 8 9, 14, 12, 10, 13 11.6 5 9 13, 14, 10, 10, 11 11.6 4 10 10, 13, 11, 9, 11 10.8 4 11 13, 9, 11, 10, 10 10.6 4 12 10, 9, 11, 11, 10 10.2 2 13 14, 11, 11, 9, 10 11.0 5 14 10, 11, 9, 14, 11 11.0 5 15 14, 11, 17, 10, 11 12.6 7 16 11, 11, 9, 16, 10 11.4 7 17 10, 11, 10, 10, 14 11.0 4 18 14, 13, 9, 11, 14 12.2 5

19 10, 10, 10, 14, 11 11.0 4 20 11, 14, 11, 10, 11 11.4 4 21 8, 11, 11, 11, 11 10.4 3 22 9, 11, 11, 10, 10 10.2 2 23 10, 11, 9, 10, 13 10.6 4 24 11, 11, 10, 20, 14 13.2 10 25 10, 10, 11, 10, 11 10.4 1 26 11, 9, 11, 14, 11 11.2 5 27 11, 14, 17, 14, 9 13.0 5 28 9, 12, 11, 11, 14 11.4 5 29 16, 16, 13, 11, 15 14.2 5 30 16, 14, 13, 9, 16 13.6 7 31 18, 16, 14, 9, 16 14.6 9 32 15, 13, 13, 10, 10 12.2 5 33 13, 13, 11, 18, 9 12.8 9 34 11, 10, 14, 7, 14 11.2 7 35 10, 14, 9, 9, 13 11.0 5 36 11, 10, 11, 10, 9 10.2 2 37 9, 9, 10, 14, 10 10.4 5 38 13, 14, 16, 17, 14 14.8 4 39 10, 16, 19, 11, 11 13.4 9 40 9, 12, 13, 14, 11 11.8 5 41 11, 10, 14, 11, 11 11.4 4 42 11, 10, 13, 16, 10 12.0 6 43 11, 11, 11, 11, 11 11.0 0 44 9, 14, 14, 13, 13 12.6 5 45 10, 13, 16, 11, 14 12.8 6 46 13, 9, 11, 14, 14 12.2 5 47 11, 13, 14, 14, 11 12.6 3

Table 7.7 Factors for M and R charts

Number of observations in each subgroup A2 D3 D4 2 1.880 0 3.268 3 1.023 0 2.574 4 0.729 0 2.282 5 0.577 0 2.114 6 0.483 0 2.004 7 0.419 0.076 1.924 8 0.373 0.136 1.864 9 0.337 0.184 1.816 10 0.308 0.223 1.777

Control Limits for the ranges (=R):

(7.6) (7.7) The factors A2, D3 and D4 can be found in Table 7.7.

The factors in this table have been calculated in order to make the calculations of the control limits easier. The theory behind these factors is a known relationship between the standard deviation and the range when it can be assumed that the calculated means follow a normal distribution.

The following control limits can now be calculated: Control limits for the means (=M):

(7.8) (7.9) Control limits for the ranges (=R):

(7.10) (7.11) Control charts constructed with a computer package are shown in Figure 7.11.

By analyzing the control charts the following can easily be concluded:

1. In the first chart one of the means is out of control and another mean is near to the upper control limit (UCL). A specific cause has to be found and removed.

The point which is outside the control limit should be investigated. This is a signal that a specific cause has not been controlled. This specific cause should be identified and controlled so that the cause will not impact variations in the future. This data should be taken out of the data set and a revised control chart for future use should be constructed.

Fig. 7.11 Control charts (M and R) for the

response time for suggestions.

2. In the second control chart (Figure 7.12(a)) it is assumed that the specific cause has been removed so that new control limits can be calculated without the out-of-control point. But still there is one point out of control. Having found the specific cause behind the out-of-control point a revised control chart can be calculated.

3. In the third control chart (Figure 7.12(b)) there are no points out of control limits. It looks as if the mean chart is in statistical control. In the R-chart there are nine points in a row above the centre line. This is a signal that there is a specific cause behind these points which should be found and controlled. Looking at the mean chart it can be seen that most of the means are above the centre line. So the specific cause may be that the stratification principle has not been used when constructing the control chart (refer to section 7.6). It may be the complex suggestions which dominate the observations in that period. If that is the case two control charts should be constructed to control the process—one for the simple suggestions and one for the complex suggestions. The two control charts will then each have smaller variations than the combined chart. 4. The revised control charts can then be used to control the process (the suggestion

system) in the future. If the average response time in each chart is not satisfactory the suggestion system must be changed (change the system causes) and new control charts must be constructed for this new system.

Fig. 7.12 Revised control charts (M and R) for