SPC Introduction to SPCReview of normal (Gaussian) distributions. (VERY IMPORTANT for SPC)For a certain population of people, the average male height is 68 inches, with a standard deviation of 3 inches.What is the probability that the next male from this population has a height greater than 70 inches? (what information here is missing?)

SPC Normal Review (continued)What is the probability that the average height of the next nine males is greater than 70 inches?Central Limit Theorem (see next slide)If the mean of the population shifted to 72 inches, what is the probability that the next male is less than 70 inches (assume normal distribution)?If a sample of nine came from the new population, what is the probability the average of the sample will be less than 70 inches?

SPC Central Limit Theorem: Example of the Roll of the DiceTheoretical Frequency Distribution based on 216 rolls... One Diem=3.5, s=1.87 Average of Two Dicem=3.5, s=1.23 Average of Three Dicem=3.5, s=0.99

SPC Statistical Process ControlSample output of process and make inferences about its stateDemonstrate that the distribution of process output is known and unchangingPlot and monitor over timeUse statistical tests to detect shifts and anomalies and react to them quicklyUse statistical evidence to guide and confirm process improvements

SPC Evolution from Inspection to SPCSPC.SPC.SPC.

SPC Statistical Process Control TopicsIntroduction to VariabilityControl ChartsGeneral infox-bar Charts, R Charts p Charts, np Charts Type 1 and Type 2 ErrorsProcess Capability Analysis

SPC VariationAnalyzeand ActImproveCapabilitiesThe less variation, the better off we are.Common cause variation: Inherent in the system

Assignable cause variation: Event-related, special(assignable ~ special)

SPC Analyze and Act: React to Assignable CausesNote unusual variation diagnosed by using a common test to evaluate individual data pointsIdentify cause by noting what change in the process occurred at that point in timeEliminate cause or build in the causeMonitor performance to verify the effect of the fixGenerally, assignable causes cause points outside of control limits!

SPC Improve Process Capabilities: Drive Out Common CausesVariation is inherent in the systemDont react to individual points (this is tampering)Analyze possible factors affecting variation (use Cause and Effect Diagram, Pareto Analysis)Work to reduce variation: Make an improvement, that is, introduce a special causeMonitor performance to verify the effect of the intended improvement

SPC X-bar and R chartsSample output of process - parameter of interest is continuously variable

Plot one chart to track sample means and another one to track sample ranges (variation)

Use statistical evidence to detect changes and improve the process: to better position the mean and to reduce variation

SPC Underlying Assumptions

process mean m and standard deviation s when the process is in controlprocess may go out of control in two possible waysmean shifts to m1, with standard deviation unchangedstandard deviation shifts to s1, with mean unchanged sample means are normally distributed (when in or out of control, because either:process output measurements on individual units are normally distributed when in or out of controlOR Central Limit Theorem applies:n > 30 ORIf distribution unimodal or symmetric, then much smaller ns are acceptable to assume normality (n on the order of 4).

SPC Basic Probabilities Concerning the Distribution of Sample MeansStd. dev. of the sample means:

SPC Estimation of Mean and Std. Dev. of the Underlying Processuse historical data taken from the process when it was known to be in controlusually data is in the form of samples (preferably with fixed sample size) taken at regular intervalsprocess mean m estimated as the average of the sample means (the grand mean)process standard deviation s estimated by:standard deviation of all individual samples OR mean of sample range R/d2, where sample range R = max. in sample minus min. in sample and d2 = value from look-up table (appendix A-7)

SPC Example: Estimation of Mean and Std. Dev. of the Underlying ProcessEstimate of the process mean = m = 2.3Estimate of the process std. dev.: (1) Combined std. dev. of all 30 points: s = 1.1OR (2) s = R/d2 (n=5) = 2.7/2.326 = 1.2

SPC Determination of Control LimitsFor the x-bar chart:- Center Line = grand mean- Control Limits: Co's usually use - Can analyze process capability based on the specification limitsFor the R chart:- Center Line = average range = - Control Limits:

Alternative: Use an Economic Approach: - Consider the cost impact of out-of-control detection delay (Type 2 error), false alarm (Type 1 error) and sampling costs - Difficult to estimate costs

SPC Ex. Two Machines -Process Capability Analysis and x-bar and R-charts

SPC X-bar vs. R chartsR charts monitor variability: Is the variability of the process stable over time? Do the items come from one distribution?X-bar charts monitor centering (once the R chart is in control): Is the mean stable over time?

>> Bring the R-chart under control, then look at the x-bar chart

SPC How to Construct a Control Chart1. Take samples and measure them.2. For each subgroup, calculate the sample average and range. 3. Set trial center line and control limits.4. Plot the R chart. Remove out-of-control points and revise control limits.5. Plot x-bar chart. Remove out-of-control points and revise control limits.6. Implement - sample and plot points at standard intervals. Monitor the chart.

SPC X-bar and R chart example:Look at handout: R Chart.R-bar = sum( R )/num. samples = 87/25 = 3.48.UCL = D4R-bar = 2.114*3.48 = 7.357.LCL = D3R-bar = 0Review samples, eliminate sample 3.Do over! New R-bar = 3.29, UCL = 6.95R-bar chart now in control, proceed to X-bar!

SPC X-bar chartGrand mean, X-bar = 500.6/24 = 20.86.Control limits = 20.86 +/- A2R-bar = 20.86 +- (.5777)*3.29 UCL = 20.86 + 1.9 = 22.76LCL = 20.86 1.9 = 18.96Bring X-bar chart under controleliminate points 15, 22, 23.

SPC Conclusion of problemRedo R chart without samples 15, 22, and 23 (and 3 is out as well).R-bar = 3.24Control limits (repeat previous procedure = [0, 6.845].Grand mean (center line for x-bar) = 20.77Control limits = 20.77 +/- (.5777)(3.24) = [18.90, 22.64]Our control limits for both charts are now set.

SPC Type 1 and Type 2 Error

SPC Common Tests to Determine if the Process is Out of ControlOne point outside of either control limit2 out of 3 points beyond UCL - 2 sigma 7 successive points on same side of the central lineof 11 successive points, at least 10 on the same side of the central lineof 20 successive points, at least 16 on the same side of the central line

SPC Type 1 Errors for these Tests Test Probability Type 1 Error2/37/710/1116/201/12(0.00135)0.00270.00052(0.5)70.00780.005860.0059

SPC Type 2 ErrorSuppose m1 > m

Type 2 Error =

[This is the probability of a sample average being below the upper control limit. We have not examined possibility of being below LCL, why?]

Power = 1- Type 2 Error. Power increases as n increases, as (m1-m) increases, and as s decreases.

Extension to m1 < m is straightforward

SPC Sensitivity of Type I and Type II ErrorsTo (UCL-LCL)/sTo nTo sTo m1 - m

SPC Example of Type 1 and 2 ErrorsSuppose: m = 100m1= 102s = 4n = 9[assume 3 sigma control limits]

Find Type 2 Error:

SPC Example of Type 1 and 2 Errors (cont.)Type 2 Error = Type 1 Error =

Prob{Shift detected in third sample after shift occurred}=Average number of samples taken before the shift is detected=

Prob{no false alarm for first 32 samples, but then false alarm occurs in 33rd sample}Average number of samples before a false alarm (ARL)