University:
California State University, Northridge
Course:
SOM 306 | Operations Management
Academic year:

2024
Views:

344

Pages:

19
Author:

Jaslyn Bean

Chapter 6 Part 1 X-bar and R Control Charts Attribute Data Data that is discrete • Discrete data is based on “counts.” • Assumes integer values • Number of defective units  Number of customers who are “very satisfied”  Number of defects  Variables Data X-bar and R chart is used to monitor mean and variance of a process when quality characteristic is continuous. • Continuous values (variables data) can theoretically assume an infinite number of values in some interval. • Time • Weight • Ounces • Diameter • X-bar and R Chart X-bar chart monitors the process mean by using the means of small samples taken frequently • R chart monitors the process variation by using the sample ranges as the measure of variability • • Range = Maximum value – Minimum value Notation X = quality characteristic n = sample size k = number of samples X = sample mean ( X - bar) X ∑ = n Notation X = sample mean of X ( X - bar) X = mean of the Xs X is also called " X double bar, " the estimated process mean, or the overall mean Notation R = Sample Range = Maximum value - Minimum value R ∑ R = Mean of ranges = k σˆ = estimated standard deviation of X σˆ n = estimated standard error of X Example of Notation A company monitors the time (in minutes) it takes to assemble a product. • The company decides to sample 3 units of the product at three different times tomorrow: • 9 AM • 12 Noon • 3 PM • What is the sample size, n? • What is k, the number of samples? • • Suppose the following data are obtained. Hour • Assembly Time (minutes) X1 X2 X3 9:00 AM 5 12 4 12 Noon 3:00 PM 6 9 8 4 10 2 How would you compute • X-bar? • R • R-bar • X double bar Example of Notation Hour Assembly Time (minutes) X1 X2 X3 Sample Mean, X Sample Range, R 9:00 AM 5 12 4 21/3 = 7 12 – 4 = 8 12 Noon 6 8 10 24/3 = 8 10 – 6 = 4 3:00 PM 9 4 2 15/3 = 5 9–2=7 X = 20/3 = 6.7 R =19/3 =6.3 Sample Means X ∑ X = n First Sample X ∑ X = n 5 + 12 + 4 21 = = =7 3 3 Sample Means Second Sample X ∑ X = n 6 + 8 + 10 24 = =8 = 3 3 Third Sample X ∑ X = n 9 + 4 + 2 15 = = =5 3 3 Estimated Process Mean X ∑ X = k 7+8+5 = 3 20 = 3 = 6 .7 Sample Ranges R = Maximum Value - Minimum Value First Sample : R = 12 − 4 = 8 Second Sample : R = 10 - 6 = 4 Third Sample : R = 9-2 = 7 Mean of R R ∑ R= k 8+4+7 = 3 19 = 3 = 6.3 Underlying Distributions • When constructing an X-bar chart, we actually have two distributions to consider: • The distribution of the sample means X , and • The process distribution, the distribution of the quality characteristic itself, X. • The distribution of X is a distribution of averages. • The distribution of X is a distribution of ??? Underlying Distributions • These distributions have the same mean Mean of X = Mean of X Their variances (or standard deviations) are different. • Which distribution has the bigger variance? • • • Would you expect more variability among averages or among individual values? The variability among the individual values is ??? Underlying Distributions • The standard deviation among the sample means is smaller by a factor of 1 n • Therefore, Std of X = σ  1  σˆ  = Std of X = σˆ  n  n Underlying Distributions Sampling distribution of X Distribution of X X