Q: Let’s assume we have a process that is under control and we want to monitor a number of key quality characteristics expressed through small subjective scales, such as: excellent, very good, good, acceptable, poor and awful. This kind of data is typically available from customer satisfaction surveys, peer reviews, or similar sources.
In my situation, I have full historical data available and the process volume average is approximately 200 deliveries per month, giving me enough data and plenty of freedom to design the control chart I want.
What control chart would you recommend?
I don’t want to reduce my small scale data to pass/fail, since I would lose insight in the underlying data. Ideally, I’d like a chart that both provides control limits for process monitoring and gives insight on the repartition of scale items (i.e., “poor,” “good,” “excellent”).
A: You can handle this analysis a couple of ways. The most obvious choice and probably the one that would give you the most information is a Q-chart. This chart is sometimes called a quality score chart.
The Q-chart assigns a weight to each category. Using the criteria presented, values would be:
- excellent = 6
- very good =5
- good =4
- acceptable =3
- poor =2
- awful=1.
You calculate the subgroup score by taking the weight of each score and multiply it by the count and then add all of the totals for the subgroup mean.
If 100 surveys were returned with results of 20 that were excellent, 25 very good, 25 good, 15 acceptable, 12 poor, and 3 awful, the calculation is:
6(20)+5(25)+4(25)+3(15)+2(12)+3(1)= 417
This is your score for this subgroup. If you have more subgroups, you can calculate a grand mean by adding all the subgroup scores and dividing it by the number of subgroups.
If you had 10 subgroup scores of 417, 520, 395, 470, 250, 389, 530, 440, 420, and 405, the grand mean is simply:
((417+ 520+ 395+ 470+ 250+ 389+ 530+ 440+ 420+ 405)/10) = 4236/10 =423.6
The control limits would be the grand mean +/- 3 √grand mean. Again, in this example, 423.6 +/-3√423.6 = 423.6 +/-3(20.58). The lower limit is 361.86 and the upper limit is 485.34. This gives you a chance to see if things are stable or not. If there is an out of control situation, you need to investigate further to find the cause.
The other choice is similar, but the weights have to total to 1. Using the criteria presented, the values would be:
- excellent = .3
- very good = .28
- good =.25
- acceptable =.1
- poor=.05
- awful = .02.
You would calculate the numbers the same way for each subgroup:
.3(20)+.28(25)+.25(25)+.1(15)+.05(12)+.02(1)= 6+7+6.25+1.5+.6+.02=21.37
If you had 10 subgroup scores of 21.37, 19.3, 20.22, 25.7, 21.3, 17.2, 23.3, 22, 19.23, and 22.45, the grand mean is simply ((21.37+ 19.3+ 20.22+ 25.7+ 21.3+ 17.2+ 23.3+ 22+ 19.23+ 22.45)/10)= 212.07/10 =21.207.
The control limits would be the grand mean +/- 3 √grand mean. Therefore, the limits would be 21.207+/-3 √21.207= 21.207+/-3(4.605). The lower limit is 7.39 and the upper limit is 35.02.
The method is up to you. The weights I used were simply arbitrary for this example. You would have to create your own weights for this analysis to be meaningful in your situation. In the first example, I have it somewhat equally weighted. In the second example, it is biased to the high side.
I hope this helps.
Jim Bossert
SVP Process Design Manger, Process Optimization
Bank of America
ASQ Fellow, CQE, CQA, CMQ/OE, CSSBB, CSSMBB
Fort Worth, TX
For more on this topic, please visit ASQ’s website.
