1. Control Charts

2. Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no, acceptable/unacceptable
Measurement is typically counting defectives
Charts may measure
Percent defective (p-chart)
Number of defects (c-chart)

3. Control Limits for p-Charts
Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLp = p + z sp ^
p (1 - p) n
sp = ^
LCLp = p – z sp ^
Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population
where p = mean fraction defective in the sample
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size ^

4. p-Chart - Example
Clerks at Mosier Data systems key in thousands of insurance records each day for a variety of client firms. The CEO wants to set control limits to include 99.73% of the random variation in the data entry process when it is in control.
Sample of the work of 20 clerks are gathered and shown in the following table (Next slide). Mosier carefully examined 100 records entered by each clerk and counts the number of errors. She also computes the fraction defective in each sample.
Calculate the control limits….

5. p-Chart – Example
Sample Number Sample Number
Number of Errors Number of Errors

6. p-Chart – Example
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
Total = 80

7. p-Chart – Example
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
Total = 80
p = = .04 80
(100)(20)
(.04)( )
100
s p = = .02 ^

8. p-Chart - Example UCLp = p + z s p = .04 + 3(.02) = .10^
LCLp = p – z s p = (.02) = 0 ^
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = 0.10
LCLp = 0.00
p = 0.04

9. Possible assignable causes present
p-Chart - Example
UCLp = p + z s p = (.02) = .10 ^
Possible assignable causes present
LCLp = p – z s p = (.02) = 0 ^
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = 0.10
LCLp = 0.00
p = 0.04

10. Control Limits for c-Charts
Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLc = c + z c
LCLc = c – z c
where c = mean number of defects per unit
Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean

11. c-Chart for Cab Company
Red Top Cab company receives several complaints per day about the behavior of its drivers. Over a 9-day period (where days are the units of measure), the owner received the following number of calls from irate passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8 for a total of 54 complaints. The owner wants to compute 99.73% control limits.

12. c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day
UCLc = c + 3 c =
= 13.35 | 1 2 3 4 5 6 7 8 9
Day
Number defective
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
UCLc = 13.35
LCLc = 0
c = 6
LCLc = c - 3 c =
= 0

13. Which Control Chart to Use
Variables Data Using an x-Chart and R-Chart
Observations are variables
Collect samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart
Track samples of n observations each.

14. Which Control Chart to Use
Attribute Data Using the p-Chart
Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states.
We deal with fraction, proportion, or percent defectives.
There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each.

15. Which Control Chart to Use
Attribute Data Using a c-Chart
Observations are attributes whose defects per unit of output can be counted.
We deal with the number counted, which is a small part of the possible occurrences.
Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth.