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CHAPTER THREE: UNIT-COUNT CHARTS

Unit-Count 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 Unit-Count Charts. Proportion Charts assume that results were either "successes" or "failures." Unit-Count Charts assume that an infinite number of "bad things" or "good things" could happen per incident.

UNIT-COUNT CHARTS # of errors IRS agents make while talking to taxpayers over the phone # of deficiencies in airplanes brought in for maintenance # of errors in a day's work # of leads developed in a week by a sales force # of complaints per day by customers; (Every complaint counts even if by the same person) # of letters received per week of unsolicited praise # of interruptions in an hour # meals sold per hour # items purchased per customer Length of queues and backlogs as measured at different times of the day. PROPORTION CHARTS % of phone calls in which an IRS agent gives at least one answer to a taxpayer question that is incorrect % of airplanes brought in for maintenance that have at least one major deficiency % of a day's work that is error-free % of sales contacts generated by each sales-person % of customers who complain; ("complainers" only get counted once) % of customers who send unsolicited letters of praise % of tasks that are interruped two or more times by phone calls % of all sales that are value-meals % of customers who purchase three or more items per transaction % of time that queues and backlogs are two or more customers

Unit-Charts (U-Charts) describe activities in terms of counts per unit of a subgroup We made on average .08 errors per form. There is on average one programming "bug" for every 300 lines of code that is written. We develop 15.5 customer leads for every 1000 people who walk by our counter at the home show. There are no restrictions relative to subgroup sizes. My subgroup is a computer project. The number of lines of code vary by project. Count-Charts (C-Charts) describe activities in terms of counts per subgroup We made on average 6.8 errors per day. There are on average 107 programming "bugs" per project. We develop 22 leads per home show. All subgroups must be the same size. My subgroup is a computer project. We have the same number of lines of code every project.

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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.

In the table below, raw data has been collected in the first three columns. The numbers in "blue" have been calculated based on the raw numbers entered. If the paperwork examples doesn't work for you, then substitute for "paperwork" whatever it is that you do.

DATA FOR U-CHART EXERCISE
Results from doing an average 85 pieces of paperwork per day

NOTE: The raw data is the same as in the P-Chart example from Chapter Two, but the calculated variance and sigma are different.

STEP #1: DO YOU HAVE ENOUGH DATA TO CREATE A CHART?

Requirements A. The average subgroup must have a minimum of five errors. B. You need at least 20 subgroups A. How many is your average number of errors per subgroup? ANSWER: 340 errors divided by 20 subgroups is 17 per subgroup B. How many subgroups do you have? ANSWER: 20

NOTE: These are the same rules as for P-Charts and np-Charts

STEP #2: FIND THE CENTRAL TENDENCY

The central tendency is the mean (traditional average) errors per form. It is labeled "u." What is the central tendency for errors per form? ANSWER: 340 errors divided by 1700 forms equals .200 errors per form.

In the P-Chart example from Chapter 2, we had a failure rate of 20%. The difference between the U-Chart and the P-Chart is that the P-chart treated a paper form with five errors the same as a form with only one error (both forms were called a failure.) The U-Chart would count the five-error form as being five times as bad as the one-error form.

STEP #3: FIND THE AVERAGE SIZE OF A SUBGROUP

The size of the variance and sigma depend upon how large the subgroup is. For instance, the sigma for Day 1 with 93 opportunities will be smaller than the sigma for Day 2 with 85 opportunities. In order to simplify appearances, the average subgroup can be used instead of calculating a separate sigma for every subgroup. Divide the "Total of all opportunities" (1700) by the number of subgroups (20) to get 85 forms per subgroup.

STEP #4: FIND THE VARIANCE.

The formula for variance is: V = u/n The variance is equal to the average error rate per unit (u) divided by the size of the subgroup (n). where V = variance u = central tendency (error/form) n = size of the subgroup Since u = .200 and n = 85, the variance for an average subgroup size of 85 is .002353

For an explanation of variance, see Chapter Two.

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

Sigma is the square root of the variance. Sigma is the square root of .002353 or .0485

STEP#6: FIND THE ONE SIGMA LINES

The Upper One-Sigma Line is the central tendency plus sigma. The Lower One-Sigma Line is the central tendency less sigma The Upper One Sigma Line is .200+.049 = .249 The Lower One Sigma Line is .200-.049 = .151

STEP #7: FIND THE WARNING LINES:

The Upper Warning Line (UWL) is the central tendency plus 2 sigma. The Lower Warning Line (LWL) is the central tendency less 2 sigma. UWL = .200 + (2)(.0485) = .297 LWL = .200 - (2)(.0485) = .103

STEP #8: FIND THE CONTROL LINES:

The Upper Control Line (UCL) is the central tendency plus 3 sigma. The Lower Control Line (LCL) is the central tendency less 3 sigma. The UCL is .200 + (3)(.0485) = .346 The LCL is .200 - (3)(.0485) = .055

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 one-sigma lines. C. The daily average failure rate is drawn in black.

Note that this graph uses the same raw data as the P-Chart graph in Chapter Two. Sigma does not strictly corresond though and the control lines come out with a wider spread than we got in the P-Chart.

STEP #10: APPLY THE EIGHT RULES FOR IDENTIFYING SPECIALNESS

1. Any values outside the control lines. Freak value 2. Two out of three points in a row in the region beyond a single warning line. Freak value 3. Six points in a row steadily increasing or decreasing. Process shift 4. Nine points in a row on just one side of the central tendency. Process shift 5. Four out of five points in a row in the region beyond a single one-sigma line.Process shift 6. Fourteen points in a row which alternate directions. Shift work or overcorrection 7. Fifteen points in a row within the region bounded by plus or minus one sigma. Garbage data or overcorrection 8. Eight points in a row all outside the region bounded by plus or minus one sigma. Garbage data or overcorrection There are no indications of specialness using the eight rules even though Day 11 spikes very high and Day 8 spikes very low. Intuitively it is easy to "see" a downward trend running from Day 2 to Day 8 or an upward trend from Day 12 through Day 20, but the trends are illusory and are the result of nothing more than normal variation.

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 one-quarter 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.

C-CHARTS A different form of a U-Chart is the C-Chart (or Count-Chart.) You can use C-Charts only if the subgroups are the same size everytime. In this way, a C-Chart is to a U-Chart the same as an np-Chart is to a P-Chart. As an example here, assume that you began with the same raw data as above for the U-Chart, 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 U-Chart. 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 C-Chart 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 U-Chart 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 n-squared 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 One-Sigma 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 U-Chart 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.

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TABLE OF CONTENTS

Chapter One

Chapter Two

Chapter Three

Chapter Four

References and Photocopying Rights