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 sp̂

LCLp = p – z sp̂

where p = mean fraction defective in the samplez = number of standard deviations sp = standard deviation of the sampling distribution

n = sample size

^

p (1 - p)nsp= ^

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 NumberNumber of Errors Number of Errors

1 6 11 62 5 12 13 0 13 84 1 14 75 4 15 56 2 16 47 5 17 118 3 18 39 3 19 0

10 2 20 4

p-Chart – Example

Sample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective

1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .084 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .039 3 .03 19 0 .00

10 2 .02 20 4 .04Total = 80

p-Chart – Example

Sample Number Fraction Sample Number FractionNumber of Errors Defective Number of Errors Defective

1 6 .06 11 6 .062 5 .05 12 1 .013 0 .00 13 8 .084 1 .01 14 7 .075 4 .04 15 5 .056 2 .02 16 4 .047 5 .05 17 11 .118 3 .03 18 3 .039 3 .03 19 0 .00

10 2 .02 20 4 .04Total = 80

.(04()1. - 04)100

s p = = .02^p = = .0480(100)(20)

.11–

.10–

.09–

.08–

.07–

.06–

.05–

.04–

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.02–

.01–

.00–

Sample number

Fra

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2468101214161820

p-Chart - Example

UCLp = p + z s p = .04 + 3(.02) = .10^

LCLp = p – z s p = .04 - 3(.02) = 0^

UCLp = 0.10

LCLp = 0.00

p = 0.04

.11–

.10–

.09–

.08–

.07–

.06–

.05–

.04–

.03–

.02–

.01–

.00–

Sample number

Fra

cti

on

defe

cti

ve

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2468101214161820

p-Chart - Example

UCLp = p + z s p = .04 + 3(.02) = .10^

LCLp = p – z s p = .04 - 3(.02) = 0^

UCLp = 0.10

LCLp = 0.00

p = 0.04

Possible assignable

causes present

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

where c = mean number of defects per unit

UCLc = c + z c LCLc = c – z c

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

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Day

Nu

mb

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defe

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12– 10–

8– 6– 4– 2– 0–

UCLc = c + 3 c= 6 + 3 6= 13.35

LCLc = c - 3 c= 6 - 3 6= 0

UCLc = 13.35

LCLc = 0

c = 6

Which Control Chart to Use

Variables Data Using an x-Chart and R-Chart

1. Observations are variables

2. 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

3. Track samples of n observations each.

Attribute Data Using the p-Chart

1. Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states.

2. We deal with fraction, proportion, or percent defectives.

3. 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

1. Observations are attributes whose defects per unit of output can be counted.

2. We deal with the number counted, which is a small part of the possible occurrences.

3. 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.

Which Control Chart to Use