P370 Final: Topic 5
types of control charts
- We focus on three: X-bar, R, p charts, as a percentage %
Special variation characteristics
-Special causes are not usually present 1) They may come and go sporadically; may be temporary or long term 2) A special cause is something special or specific that has a pronounced effect on the process 3) We can't predict when a special cause will occur or how it will affect the process 4)The process is unstable, or unpredictable, when special causes contribute to the variation 5) Also called "assignable cause" variation" -unlikely trends
Types of variation
Normal, common and abnormal, special
sampling error
sample is either too small or does not represent population
know how to find pchart with given z or desired z or percentage
see slides
know what all charts look like and how to interpret them
see slides
know control chart types and when negative or positive
see supps
X-bar and X double bar
the average, and the average of averages
R and R-bar
the range of the obervations and the average of those ranges
What to use for Z in p-chart calculation
always use Z = 3 unless told otherwise, So UCL = p-bar + 3(Sp)
In Control
"- A process is "in control" when the process variation is random and within the limits of the normal curve. Or, variation is due to chance or sampling error. - The process needs no adjustment., know why then leave alone"
Out of Control
"- A process is deemed "out of control" when the process variation is non-random or outside of the limits of the normal curve. This variation is due to some assignable or special cause. - The process needs some type of attention or adjustment., know why then fix it. also look for a pattern or a data point thats really out of control"
Special Cause Variation, the enemy to quality
"Special causes are factors that are not always present in a process but that appear because of some particular circumstance."
The basis for SPC, statistical process control
"The application of these sampling distribution principles to production processes is the basis for statistical process control! (SPC)"
Hypothesis test for control charts
- H0: The system is in control, null hypo, reject if out of control - Ha: The system is out of control, alternative hypo
What is a sample?
- How many samples are possible from a given population? (nCr) - What does the distribution look like? • A normal curve • Variation - Much around the mean; a little at the tails • Would you expect any trends? (hopefully not) - Would you believe the process was "stable"?"
Control Charts
1) A control chart is a graphical tool, that uses actual variation in observed data to determine if a process is "in control" or "out of control" . uses normal distribution This is just a statistical tool and NOTHING IS CERTAIN 2) There are several types of control charts based on the kind of data that is being collected. (used to find normal and abnormal variation, with the assumption that less variation is better, helps determine if error is from sampling error or other)"
Unlikely data patterns that might lead us to conclude the system is out of control:
1) A single sample statistic that is outside of the control limits (if its 3sigma then beyond that, check limits) 2) Two consecutive sample statistics near the control limits (majority should be near central line) 3) Five consecutive points above or below the central line 4) A trend of five consecutive points 5) Very erratic behavior (large swings or minority of points near central line) 2-5 all speak of patterns, so we need multiple samples, to chart past data
Characteristics of Control Charts
1) Center line - average (central tendency) 2) Upper control limit: + 3 standard dev 3) Lower control limit: - 3 standard dev 4) Data - collected through sampling, if between 3 deviations it common if not its special" Y= umber of units, X = time sequences between samples
x chart formula, and steps
1) Find the average of all samples, and then the average of averages 2) Find the range of each sample (MAX-MIN), then the average of all the sample ranges (RBAR) 3) Look up A2 based on n= number of observations in each sample, A2 factor is from a chart in the lecture packet -Upper Control Limit (UCL) = X-dbl-bar + A2(R-bar) -Lower Control Limit (LCL) = X-dbl-bar - A2(R-bar)
Addressing Common & Special variation actions
1) Look for differences between individual points 2) Take action based on the reported differences 3) study all the data 4) make basic changes to the process
Good actions for special variance
1) Look for differences between individual points, gain useful information 2) Take action based on the reported differences, reduce variance
Bad actions for common variance
1) Look for differences between individual points, it will waste time 2) Take action based on the reported differences, this will increase variance when adjusting when it is not needed
Interpreting Control Charts
1) The purpose of control charting is to give us an objective, statistically based tool to judge if a process is in control or out of control 2) In Control - functioning as it has historically, exhibiting only common causes or non-attributable of variation (sampling error) 3) Out of Control - process is not functioning as it has in the past, exhibiting evidence that a special or attributable cause of variation has entered the process.
why 3 standard deviations and whats the trade off and costs
1) Type one and Type 2 errors move in different directions. If you widen the width to, say, 4, then you decrease the chance of Type one errors but increase the chance of Type 2 errors. Its is difficult to compute Type two errors so we usually think in terms of Type one and set widths accordingly 2) Because that is where the combined total risk of both type 1 and 2 errors intersect and it is minimized at 3 stedev?
Common variance characteristics
1)Common causes are a part of the process 2)They contribute to output variation because they themselves vary 3) Each common cause contributes a small part of the total variation by looking at a process over time, we know how much variation to expect from common causes 4) The process is stable, or predictable, usually sampling error. its ok to have variation in a process but it needs to be controlled, should look random with half above and half below -fluctuation only occurs due to environmental factors -cant be traced back to a single cause
The relationship between Sigma and Type 1 and Type 2 errors
3 sigma minimizes the total of type 1 and type two errors! They behave oppositely with respect to the width of Z
Good actions for common variance
3) study all the data, understand the system better 4) make basic changes to the process, reduces variance
Bad actions for special variance
3) study all the data, wastes time not responding to the problem 4) make basic changes to the process, lost productivity may increase variance
What are Control Limits?
A control limit defines the bounds of common cause variation in the process, set at +/- 3sigma and dont adjust machine when not needed because that could increase variance
What are control limits used for?
A control limit is a tool we use to help us take the right actions: - If all points are between the limits, assume only common cause variation is present (unless one of the other Signals of a Special Cause is present) - If a point falls outside the limit, you treat it as a special cause - Otherwise, you do not investigate individual data points, but instead study the common cause variation in all data points, know what it looks like"
p chart formula
Center line = p-bar (average of the sample percentages) - UCL = p-bar + z * Sp - LCL = p-bar - z * Sp • Z=usually is 3 (3 std devs) unless told otherwise • also where Sp = [(pbar(1-pbar)/n)]^(.5), where n is sample size
Common Cause Variation
Common causes are the process inputs and conditions that contribute to the regular, everyday variation in a process."
Can a p-chart have a negative value? for LCL?
NO, if you do have a negative value just set it to 0, cant have negative number of defects
When to use X-bar charts
For changes in the process mean: use X-bar chart Helps determine changes in the process average this is continuous 1) Center Line (average) = the average of the sample averages (X-double-bar), 2) xbar can be negative, Rcharts can not be
when to use R charts and steps
For changes in the process variance: R chart Use for changes in the process variance 1) calculate the range for each sample 2) find average of all sample ranges 3) Look up D4 for UCL, or D3 for LCL based on n in packet Center Line (average) = the average of the sample ranges (R bar)
when to use p charts
For data that is not on a continuous scale, like XBAR or RBAR used for changes in attributes, usually limited to two classifications - (typically defective vs non defective) and will establish out upper and lower specification limits
"Using Control Charts to Decide if a Process is "In- Control" or "Out of Control"" (SLIDE)
From top left going clock wise actually 1) out of control but control chart says we are in = type 2 error, don't think we have a problem but we do 2) out of control and chart says so, we do have a problem and we know it 3) in contol but chart says were out of control, we think we have a problem but we dont = type one error 4) in contol and chart says so, we dont have a problem and we know it
Changing the width of the control limits and when and result
ONLY FOR PCHARTS THIS CAN BE DONE Not mandatory but common practice to leave at 3 1) Adjust Z, if you want a width of 2.5, then set Z = 2.5, or 2) but if you want to define Z based on the level of Type one errors you want to live with. - Given a desired Type I error level, we use the standard normal distribution table to determine the number of standard deviations away from the mean we need to be.
Topic 5
Satistical Process Control
When in control pattern
Should be random, half above and below central line, if you see a pattern could mean special variation. A trend up of down is bad because you are shifting the bell curve and basically changing mean. Also know time pattern when data is observed
What is Sp for the UCL and LCL p-bar calculation
Sp = standard deviation of p, so if given fill it in
Consumers Risk
System is out of control but we believe it is in control This is a type II error and also called a consumer risk Type II error is difficult to compute. How big to set control limit width is a crucial decision, and the cost and sensitivity to type I and type II errors is a key issue. Since it is difficult to impossible to assess type II error directly, we usually think in terms of type I error, and set our control limit widths accordingly." type two errors, this can be more flexible and widened to 4stdev but consumers will see it as out of control, we spend less on type one but more on type 2 under that scenario but hard to assess directly False negative
R-Chart, why its important
The R-chart works off of the sample range (the difference between the highest and lowest single observation). While the range of a sample is not the variance, it is an acceptable proxy, and it is much easier to compute. *XBAR cannot detect shifts from the central limit due to variance and VERY IMPORTANT because range is increasing This is continuous, and cannot be negative
How to adjust width for p-chart
To define Z based on the level of Type one errors you want to live with. Take level and look it up in normally distributed table 1) Determine the control limit desired, may be given or you have to find it from acceptable Type one error, which you take 1 minus the Type one error level 2) Divide the control limit(%) in half. This is your look-up number. 3)Using the standard normal table, find the entry that is as close to your look up number as possible. To determine the z score, read out to the row and column., and just change Z
P-bar control chart layout
UCL Central Line LCL
R-bar control chart layout
UCL Central Line LCL
X-bar control chart layout
UCL Central Line LCL
P-Bar Charts
We refer to these as the specification limits (different from the control limits). Any observation outside those limits would be classed as defective, while anything within those would be considered non-defective. We therefore turn our sample observations into a stream of 0s and 1s, with 1s representing each defective observation. We can then calculate our necessary parameters for each control chart. These calculations are quite easy, with p equal to the percent defective in a given sample binomial and have outcomes of usually, defective or non defective.
THE Relationship between increasing Z Scores and Type 1 and Type 2 errors
What is the result of this tinkering with z? If you recall our discussion of type I and type II errors you should realize that as z increases, the likelihood of committing a type I error has decreased while the likelihood of committing a type II error has increased. Type II error is difficult to determine exactly, mainly because it requires knowledge of a number of factors, some of which we don't know exactly. Type I error is easy to assess, however, because it is based primarily on the null hypothesis, which is that the system is in control and all deviations from the mean are due to random chance. We therefore often specify z based on the level of type I error we are willing to live with.
TYPE I ERROR:
When a system that is in control is judged to be out of control and adjustments are made. (a.k.a. "False Positive", or Producer's Risk—the risk a producer takes when "adjusting" a system.)
TYPE II ERROR:
When a system that is out of control is judged to be in control, and we fail to intervene in the system. (a.k.a. "False Negative, or "Consumer's Risk—the risk a consumer takes when buying a product from an out of control process)
Type I and Type II errors and width
Why 3 standard deviations for control limits? - There is the possibility that 1% of the samples will fall outside of the control limits. This is a Type I statistical error. - As you widen the control limits, p(Type I) goes down, p(Type II) goes up. - We use 3σ because it minimizes the sum of both errors...
Producers Risk
With a normally functioning system in control, and a width of plus or minus 3 standard deviations, it is natural and expected that 1% of all samples statistics will fall outside the control limit boundaries as a function of normal, random variation. This is called type I error (a false positive), or producer's risk. The wider the control limits, the less the probability of committing type I error. type one errors, we focus on this because this is what we directly control
example of variation
dripping soda at soda machine, not 100% perfect
what is n
n is the number of oberservations in each sample
What is p and p-bar equal to, and calculation (SLIDE)
p = equal to the percent defective in a given sample, and 1) determine number of defects based on criteria 2) add them up and divide by n = % p-bar = is equal to the total number of defectives divided by the total number of observations 1) add up all the total number of defects, and divide by the total inspections (samples taken * n)
