Home Page  Resource Links  Sounding Board  Books 
Chapter One  Chapter Two  Chapter Three  Chapter Four  References and Photocopying Rights 
from
How to Recognize When Special Causes Exist,
A Guide to Statistical Process Control
© Ends of the Earth Learning Group 1998
by
Linda Turner and Ron Turner
UnitCount Charts are used to count how often "bad things" or "good things" occur. They are used to identify when changes in the number of "bad things" or "good things" indicate something "special" is occurring. Don't Confuse Proportion Charts with UnitCount Charts. Proportion Charts assume that results were either "successes" or "failures." UnitCount Charts assume that an infinite number of "bad things" or "good things" could happen per incident.


******************
ASSIGNMENT: Figure out how well a system for processing paperwork is functioning.
SIZE OF THE SUBGROUP: One day's work.
MEASUREMENT: The number of errors in a day. This is translated into the number of errors per form for the day.

DATA FOR UCHART EXERCISE
Results from doing an average 85 pieces of paperwork per day
Subgroup day 
Number of Opportunities "n" = # of paper forms processed 
Count of Errors "c" = # of errors in the subgroup from all forms (some forms had more than one error) 
Unit Error Rate "u" = # errors per opportunity in the subgroup 
Day 1  93  25  .269 
Day 2  85  17  .200 
Day 3  85  18  .212 
Day 4  87  23  .264 
Day 5  86  15  .174 
Day 6  85  15  .176 
Day 7  88  16  .182 
Day 8  97  12  .124 
Day 9  80  12  .150 
Day 10  85  13  .155 
Day 11  69  23  .333 
Day 12  84  12  .143 
Day 13  82  11  .134 
Day 14  85  18  .212 
Day 15  76  12  .158 
Day 16  86  18  .209 
Day 17  91  17  .187 
Day 18  89  22  .247 
Day 19  82  22  .268 
Day 20  86  19  .221 
Total of 20 subgroups  1700  340  .200 
Average  85  17  .200 
Average Variance  0.002353  
Average Sigma  .0485 
NOTE: The raw data is the same as in the PChart example from Chapter Two, but the calculated variance and sigma are different.
STEP #1: DO YOU HAVE ENOUGH DATA TO CREATE A CHART?

STEP #2: FIND THE CENTRAL TENDENCY

In the PChart example from Chapter 2, we had a failure rate of 20%. The difference between the UChart and the PChart is that the Pchart treated a paper form with five errors the same as a form with only one error (both forms were called a failure.) The UChart would count the fiveerror form as being five times as bad as the oneerror form.
STEP #3: FIND THE AVERAGE SIZE OF A SUBGROUP

STEP #4: FIND THE VARIANCE.

STEP #5: FIND SIGMA (THE STANDARD DEVIATION):

STEP#6: FIND THE ONE SIGMA LINES

STEP #7: FIND THE WARNING LINES:

STEP #8: FIND THE CONTROL LINES:

STEP #9: CREATE AN SPC CHART
A. The vertical axis measures "Average Errors Per Form". The horizontal axis tracks data by the subgroup day.
B. The red lines are the control lines. The green lines are the warning lines. The blue lines are the onesigma lines. C. The daily average failure rate is drawn in black. 
STEP #10: APPLY THE EIGHT RULES FOR IDENTIFYING SPECIALNESS

STEP #11: ADJUST FOR SUBGROUPS WHOSE SIZES ARE SIGNIFICANTLY DIFFERENT THAN THE AVERAGE SUBGROUP
If you examine the equation for the Variance [ V = u/n ], you will notice that variance is dependent upon "n", the number of opportunities in a given subgroup. Higher "n" values will mean lower variances and standard deviations.
The rule of thumb is this: if you quadruple your "n", the variance will be reduced to onequarter of its initial value and the standard deviation will be halved. This in turn will make the gap between control lines to come closer by half.
In this data set, the size of "n" varies from 97 on Day 8 to 69 on Day 11. The Variance for these two extremes would be (.20/97 = .002062) and (.20/69 = .002899). Sigma for the two extremes would be .045 on Day 8 and .054 for Day 11.
SPC programs do these calculations automatically for you. The resulting control lines step in and out based on how large the "n" is for each day.
The control lines "step" in and out based on the number of opportunities. Day 8 is closer to the Lower Control Line once we take into account "n". 
STEP #12: REMOVE ANY SUBGROUPS THAT HAVE "SPECIAL CAUSES" AND RECALCULATE THE CENTRAL TENDENCY AND SIGMA.
Subgroups that have special causes are by definition not part of a stable process. Remove them from your data base and recalculate the central tendency and control lines. Usually this causes the control lines to come closer together. It is conceivable that after recalculating them, more subgroups will fall into a "special" category in which case you would repeat step #12.
For our sample data in this chapter, we had no indication of special causes and therefore don't have to recalculate anything. Most commercial programs for calculating SPC don't do Step #12 automatically meaning you will have to manually delete the data from your data base for the subgroups that appear special.
CCHARTS A different form of a UChart is the CChart (or CountChart.) You can use CCharts only if the subgroups are the same size everytime. In this way, a CChart is to a UChart the same as an npChart is to a PChart. As an example here, assume that you began with the same raw data as above for the UChart, but the opportunities ("n") (at 85) were the same every day.
The central tendency is the average "c" per subgroup. In this case, the central tendency was 17 rather than .200 that we used in the UChart. This means that on average, we had 17 errors per 85 forms (which we found by dividing 340 errors by 20 subgroups). Computationally, average "u" is found by dividing the average "c" (in this case 17) by the size of the subgroup (in this case 85). This means that "c" is "nu" or 85 times bigger than the average "u" of .200 or (85 times the "u" of .2 = the "c" of 17.)
The variance for the CChart is simply the same value as the average "c" per subgroup. In this case, it would be 17. Why is variance the same value as the mean? Computationally, the variance for the UChart was average u/n (or .20/85). Because variance is a summing of the square of the differences between an expected average and a particular value, the variance for average "c" should be nsquared times the size of the variance for "u". In other words the variance for average "u" which is (u/n) will become (u/n)(n ²) = nu or 17 for this sample. If you don't understand this part, don't worry and simply use the rules.
Sigma is the square root of the variance. For this example, the square root of 17 would be 4.1. The OneSigma Lines would be at 21 and 13. The Warning Lines would be at 25 and 9. And the Control Lines would be at 29 and 5.
In comparison, the UChart Control Lines were .3455 and .055. If these values are multiplied by the subgroup size of 85, you would get Control Lines of 29 and 5, the same as we calculated above.
For tips on how make your SPC charts more sensitive, see Appendix A of Chapter Two.
Top of page  Home Page  Resource Links  Sounding Board  Books 
Chapter One  Chapter Two  Chapter Three  Chapter Four  References and Photocopying Rights 