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Chapter 6 Part 3. 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.

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chapter 6 part 3

Chapter 6 Part 3

X-bar and R Control Charts

attribute data
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
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 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
example of notation
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?
slide9
Suppose the following data are obtained.

How would you compute

X-bar?

R

R-bar

X double bar

sample means
Sample Means

First Sample

slide12

Sample Means

Second Sample

Third Sample

slide16

Underlying Distributions

  • When constructing an X-bar chart, we actually have two distributions to consider:
    • The distribution of the sample means , and
    • The process distribution, the distribution of the quality characteristic itself, X.
  • The distribution of is a distribution of averages.
  • The distribution of X is a distribution of ???
underlying distributions
Underlying Distributions

These distributions have the same mean

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 distributions18
Underlying Distributions

The standard deviation among the sample means is smaller by a factor of

Therefore,

slide19

Underlying Distributions

Samplingdistribution of

Distribution of X

slide20

X

Distribution of X

M

M

m

M

M

m

LCL

UCL

M

M

m

The distribution of X is assumed to be normal. This assumption needs to be tested in practice.

distribution of x bar
Distribution of X-bar

M

M

m

M

M

m

LCL

UCL

M

M

m

If the distribution of X is normal, the distribution of X-bar will be normal for any sample size.

control limits for x bar chart
Control Limits for X-bar Chart

Since we are plotting sample means on the X-bar chart, the control limits are based on the distribution of the sample means.

The control limits are therefore

control limits for x bar chart23
Control Limits for X-bar Chart

Distribution of

LCL

UCL

control limits for x bar chart25
Control Limits for X-bar Chart
  • A2 is a factor that depends on the n, the sample size,
  • and will be given in a table.
slide26

Example of X-bar Chart

  • A company that makes soft drinks wants to monitor the sugar content of its drinks.
  • The sugar content (X) is normally distributed, but the means and variances are unknown.
  • The target sugar level for one of its drinks is 15 grams.
  • The lower spec limit is 10 grams.
  • The upper spec limit is 20 grams.
example of x bar chart
Example of X-bar Chart
  • The company wants to know how much sugar on average is being put into this soft drink and how much variability there is in the sugar content in each bottle.
  • The company also wants to know if the mean sugar content is on target.
  • Lastly, the company wants to know the percentage of drinks that are too sweet and the percentage that are not sweet enough. (Next section)
example of x bar chart28
Example of X-bar Chart
  • To obtain this information, the company decides to sample 3 bottles of the soft drink at 3 different time each day:
    • 10 A.M,
    • 1:00 P.M. and
    • 4:00 P.M.
  • The company will use this data to construct an X-bar and R chart. (In practice, you need 25-30 samples to construct the control limits.)
  • For the past two days, the following data were collected:
slide29

Example of X-bar Chart

What is n?

What is the k?

What is the next step?

slide36

Interpretation of X-bar Chart

  • The X-bar chart is in control because ????
  • This means that the only source of variation among the sample mean is due to random causes.
  • The process mean is therefore stable and predictable and, consequently, we can estimate it.
slide37

Interpretation of X-bar Chart

  • Our best estimate of the mean is the center line on the control chart, which is the overall mean (X-double bar) of 15.33 grams.
  • If the process remains in control, the company can predict that all bottles of this soft drink produced in the future will have a sugar content of, on average, 15.33 grams.
interpretation of x bar chart
Interpretation of X-bar Chart
  • This prediction, however, indicates that there is a problem with the location of the mean.
  • The process mean is off target by 0.33 grams (15.33 -15.00).
  • The process mean, although stable and predictable, is at the wrong level and should be corrected to the target.
interpretation of x bar chart39
Interpretation of X-bar Chart
  • Since the process mean is in control, there are no special causes of variation that may be responsible for the mean being off target.
  • Since the operators are responsible for correcting problems due to special causes and managementis responsible for correcting problems due to random causes of variation, management action is required to fix this problem.
interpretation of x bar chart40
Interpretation of X-bar Chart
  • The reason is that, because the process is in control, the filling machines require more than a simple adjustments (typically due to special causes) which can be made by the operators.
  • The machines may require
    • new parts,
    • a complete overhaul, or
    • they may simply not be capable of operating on target, in which case a new machine is required.
interpretation of x bar chart41
Interpretation of X-bar Chart
  • Expecting the operators to adjust the mean to the target when the process is in control is analogous to requiring that you produce zero heads (head = defective unit) if you are hired to toss a fair coin 100 times each day.
  • Why?
r chart
R Chart
  • Monitors the process variability (the variability of X)
  • Tells us when the process variability has changed or is about to change.
  • R chart must be in control before we can use the X-bar chart.
r chart43
R Chart
  • Rules for detecting changes in variance:
    • If at least one sample range falls above the upper control limit, or there is an upward trend within the control limits, process variability has increased.
    • If at least one sample range falls on or below the lower control limit, or there is a downward trend within the control limits, process variability has decreased.
interpretation of r chart
Interpretation of R Chart
  • Since all of the sample ranges fall within the control limits, the R chart is in control.
  • The standard deviation is stable and predictable and can be estimated—done in next section.
  • This does not necessarily mean that the amount of variation in the process is acceptable.
interpretation of r chart49
Interpretation of R Chart
  • Continuous improvement means the company should continuously reduce the variance.
  • Since the process variation is in control, management action is required to reduce the variation.
slide50

X-bar Chart

Expected Pattern in a Stable Process

UCL

LCL

Time

Expected pattern is a normal distribution

slide51

x-Chart

How Non-Random Patterns Show Up

Sampling

Distribution

(process variability is increasing)

UCL

Does notreveal increase

LCL

UCL

R-chart

Reveals increase

LCL

how non random patterns show up

x-Chart

How Non-Random Patterns Show Up

(process mean is

shifting upward)

Sampling

Distribution

UCL

Detects shift

LCL

UCL

Does notdetect shift

R-chart

LCL

is a stable process a good process
Is a Stable Process a Good Process?
  • “In control” indicates that the process mean is stable and hence predictable.
  • A stable process, however, is not necessary a “good” (defect free) process.
  • The process mean, although stable, may be far off target, resulting in the production of defective product.
  • In this case, we have, as Deming puts it, “A stable process for the production of defective product.”
control limits vs spec limits
Control Limits vs. Spec. Limits
  • Control limits apply to sample means, not individual values.
    • Mean diameter of sample of 5 parts, X-bar
  • Spec limits apply to individual values
    • Diameter of an individual part, X
control limits vs spec limits55

Mean=

Target

Control Limits vs. Spec. Limits

Samplingdistribution, X-bar

Processdistribution, X

USL

LSL

Lowercontrollimit

Uppercontrollimit

underlying distributions56
Underlying Distributions

Samplingdistribution of

Distribution ofX

LSL

USL

LCL

UCL

Control limits are put on distribution of X-bar

Spec limits apply to the distribution of X

benefits of control charts
Benefits of Control Charts
  • Control charts prevent unnecessary adjustments.
    • If process is in control, do not adjust it.
    • Adjustments will increase the variance.
    • Management action is required to improve process.
    • Adjustments should be made only when special causes occur.
benefits of control charts59
Benefits of Control Charts
  • Control charts assign responsibility for corrective action.
  • Control charts are the only statistical valid way to estimate the mean and variance of a process or product.
  • Control charts make it possible to predict future performance of a process and thereby take early corrective action.