Six Sigma and Beyond: Statistical Process Control, Volume IV

Sometimes it happens that variability in test results is significantly affected by testing conditions which cannot be controlled. Under such circumstances, an Xbar chart may not be appropriate because of lack of control of testing techniques rather than lack of control in quality. When that is an issue, the difference control chart may be appropriate.

The steps for doing such a chart are as follows :

• Step 1. Take a sample from each day's output and also a sample from the reference lot.

• Step 2. In each case, calculate the mean of the sample.

• Step 3. Take the difference between the average of the standard sample and the average of the current sample and plot the difference on an -difference chart. The center line of the chart is always 0, and the control limits will fall at

  where is an estimate of the standard deviation of the reference lot and is the estimated standard deviation of the current output. These both might be estimated, for example, by computing from m samples of n each, say, 30 samples of 5 each. Then we could take

  and

• Step 4. Calculate the limits as

• Step 5. Analyze the results. The analysis for this chart is the same as for the Xbar. The R chart is the usual because the assumption is that the testing conditions from day to day do not affect test variations.

THE LOT PLOT METHOD

A special method for variable sampling plan was developed by Shainin (1950). The process is as follows:

• Step 1. A random sample of 50 items is taken from the lot, and a frequency distribution is made of these items in the manner indicated below.

• Step 2. First, a subsample of 5 is taken from the original sample of 50, and the Xbar and R are computed for this subsample.

• Step 3. The mean of the subsample of 5 is taken as a convenient arbitrary origin for constructing the frequency distribution, and the size of the class interval is determined so that twice the value of R computed in Step 2 includes from 7 to 16 intervals. The intervals are numbered 0, 1, 2, -1, -2, and so on, taking the 0 interval as that containing the arbitrary origin.

• Step 4. The initial subsample of 5 cases is distributed among the various intervals determined in Step 3, each item being indicated on the frequency chart by a 1.

• Step 5. The remaining 45 cases are divided into groups of 5 items each, and the items in each group are entered on the chart by recording the number of their group .

• Step 6. The sum of the class interval deviations and the range in class interval units is entered for each group in a supplementary table. From this table, the mean of the whole 50 items (Xbar) and the average range for the 10 groups (Rbar) are computed, both in class interval units. Then ƒ ' (in class interval units) is estimated by dividing Rbar by d ₂ = 2.326.

• Step 7. Upper and lower "control limts" are laid off by adding and subtracting 3 /d ₂ (in class interval units) from (in class interval units).

• Step 8. If the distribution of the 50 items approximates the normal form and if the "control limits" fall within the specification limits, the lot is accepted.

• Step 9. If the distribution of the 50 sample items is significantly skewed or shows significant kurtosis and if the "control limits" are near the specification limits, the data are studied further. The steps taken are discussed below.

• Step 10. If the distribution of the 50 items approximates the normal form but the control limits exceed the specification limits in either or both directions, indicating that a fraction of the lot will not conform to specifications, the data are passed on to a salvage board for review and further action.

• Step 11. In some cases, one or more points will fall beyond the control limits. In such cases an attribute plan is employed. If no points of the total of 50 lie beyond specification limits, the lot is accepted. If the number exceeds 3, the lot is submitted to 100% inspection.