Control Charts

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)

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 ^

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….

p-Chart – Example Sample Number Sample Number Number of Errors Number of Errors 1 6 11 6 2 5 12 1 3 0 13 8 4 1 14 7 5 4 15 5 6 2 16 4 7 5 17 11 8 3 18 3 9 3 19 0 10 2 20 4

p-Chart – Example Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective 1 6 .06 11 6 .06 2 5 .05 12 1 .01 3 0 .00 13 8 .08 4 1 .01 14 7 .07 5 4 .04 15 5 .05 6 2 .02 16 4 .04 7 5 .05 17 11 .11 8 3 .03 18 3 .03 9 3 .03 19 0 .00 10 2 .02 20 4 .04 Total = 80

p-Chart – Example Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective 1 6 .06 11 6 .06 2 5 .05 12 1 .01 3 0 .00 13 8 .08 4 1 .01 14 7 .07 5 4 .04 15 5 .05 6 2 .02 16 4 .04 7 5 .05 17 11 .11 8 3 .03 18 3 .03 9 3 .03 19 0 .00 10 2 .02 20 4 .04 Total = 80 p = = .04 80 (100)(20) (.04)(1 - .04) 100 s p = = .02 ^

p-Chart - Example UCLp = p + z s p = .04 + 3(.02) = .10 ^ LCLp = p – z s p = .04 - 3(.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 UCLp = 0.10 LCLp = 0.00 p = 0.04

Possible assignable causes present p-Chart - Example UCLp = p + z s p = .04 + 3(.02) = .10 ^ Possible assignable causes present LCLp = p – z s p = .04 - 3(.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 UCLp = 0.10 LCLp = 0.00 p = 0.04

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

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.

c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c = 6 + 3 6 = 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 = 6 - 3 6 = 0

Which Control Chart to Use Variables Data Using an x-Chart and R-Chart Observations are variables Collect 20 - 25 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.

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.

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.