Statistical Process Control Dr. Ron Lembke

Statistics

Measures of Variability  Range: difference between largest and smallest values in a sample Very simple measure of dispersion R = max - min  Variance: Average squared distance from the mean Population (the entire universe of values) variance: divide by N Sample (a sample of the universe) var.: divide by N-1  Standard deviation: square root of variance

Skewness  Lack of symmetry  Pearson’s coefficient of skewness: Skewness = 0 Negative Skew < 0 Positive Skew > 0

Kurtosis  Amount of peakedness or flatness Kurtosis < 0 Kurtosis > 0 Kurtosis = 0

Table 14.2  Is everything OK?  How can we decide?  Does it look normal?

Histogram of 150 data points  Histogram looks pretty normal, but definitely not perfect.  Looks like 2 peaks, actually, but pretty normal.

Is it Normal?  Sorted all 150 data points, plotted  Mean = , stdv =

Compute 1,2,3 sigma limits

Is it Normal?  F(m-3s) = 0F(m+1s) = 129/150 = 0.86  F(m-2s) = 7/150 = F(m+2s) = 149/150 =  F(m-1s) = 24/150 = 0.16F(m+3s) = 150/150 = 1.00  F(m) = 71/150 = 0.473

Is it Normal?  DataTheoretical  F(m-3s) =  F(m-2s) =  F(m-1s) =  F(m) =  F(m+1s) =  F(m+2s) =  F(m+3s) =  Values we get are pretty close to a normal distribution.

Real Test of Normality  Kolmologorov-Smirnov  Anderson-Darling Sadly, we don’t have time for either today You need SAS or something like it. Excel can’t do everything.

Process Capability  UTL = 1.0  LTL = 0.5  Cpk > 1.0  Process capable, but barely  Is everything OK?

Plot Data over time  No significant trend to data?

Plot Data over time  Data is in sets of 5, all taken at same time.  Plotting individual points makes us see trends that aren’t really there.  Solution – plot averages of each sample

Sample Means

Control Chart  Control Limits are mean +/- 3 std. dev.

We have Out-of-Control Points  Looks like the mean has shifted.  Something is definitely wrong.

Control Limits catch early  In fact, we should compute control limits using first 17 data points

Revise Control Limits  New control limits using first 16 data points.

Control Chart Usage  Only data from one process on each chart  Putting multiple processes on one chart only causes confusion  10 identical machines: all on same chart or not?

In Control  A process is “in control” if it is not affected by any unusual forces  Compute Control Limits, Plot points

X-bar Chart for Averages

Definitions of Out of Control 1. No points outside control limits 2. Same number above & below center line 3. Points seem to fall randomly above and below center line 4. Most are near the center line, only a few are close to control limits 1. 8 Consecutive pts on one side of centerline 2. 2 of 3 points in outer third 3. 4 of 5 in outer two-thirds region

You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control? Hotel Example

Hotel Data DayDelivery Time

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange Sample Mean =

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange Sample Range = LargestSmallest

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange

R R Chart Control Limits R k i i k      

R Chart Control Limits Solution From B-1 (n = 5) R R k UCLDR LCLDR i i k R R        (2.114)(3.894)8232 (0)(3.894) 

R Chart Control Chart Solution UCL

 X Chart Control Limits Sample Range at Time i # Samples Sample Mean at Time i

 X Chart Control Limits From Table B-1

R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange

 X Chart Control Limits X X k R R k i i k i i k            

 X Chart Control Limits From B-1 (n = 5) X X k R R k UCLXAR i i k i i k X            *  

 X Chart Control Limits Solution From Table B-1 (n = 5) X X k R R k UCLXAR LCLXAR i i k i i k X X             (058) 5813(058) (3.894) =   (3.894) = 8.060

 X Chart Control Chart Solution*  X, Minutes Day UCL LCL

Subgroup Size  All data plotted on a control chart represents the information about a small number of data points, called a subgroup.  Variability occurs within each group  Only plot average, range, etc. of subgroup  Usually do not plot individual data points  Larger group: more variability  Smaller group: less variability  Control limits adjusted to compensate  Larger groups mean more data collection costs

General Considerations, X-bar, R  Operational definitions of measuring techniques & equipment important, as is calibration of equipment  X-bar and R used with subgroups of 4-9 most frequently 2-3 if sampling is very expensive 6-14 ideal  Sample sizes >= 10 use s chart instead of R chart.

Single Data Points?  What if you only have one data point on a process? Inspect every single item There is no range. R=0?  Charts for Individuals (x-Chart) CL: x-bar +/- 3R-bar/d2 R = difference between consecutive units Draw control limits on the chart We can also put User specification limits on, for reference purposes Doesn’t catch trends as quickly Normality assumption must hold

Attribute Control Charts  Tell us whether points in tolerance or not p chart: percentage with given characteristic (usually whether defective or not) np chart: number of units with characteristic c chart: count # of occurrences in a fixed area of opportunity (defects per car) u chart: # of events in a changeable area of opportunity (sq. yards of paper drawn from a machine)

p Chart Control Limits # Defective Items in Sample i Sample i Size

p Chart Control Limits # Defective Items in Sample i Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits # Samples

p Chart Control Limits # Defective Items in Sample i # Samples Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits

p Chart Example You’re manager of a 500- room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)? © 1995 Corel Corp.

p Chart Hotel Data No.No. Not DayRoomsReady Proportion /200 =

p Chart Control Limits

p Chart Control Limits Solution

p Chart Control Limits Solution

p Chart Control Chart Solution UCL LCL

P-Chart Example  Enter the data, compute the average, calculate standard deviation, plot lines

Dealing with out of control  Two points were out of control. Were there any “assignable causes?”  Can we blame these two on anything special? Different guy drove the truck just those 2 days. Remove 1 and 14 from calculations. p-bar down to 5.5% from 6.1%, st dev, UCL, LCL, new graph

Revised p-Chart

Different Sample sizes  Standard error varies inversely with sample size  Only difference is re-compute  for each data point, using its sample size, n. Why do this? The bigger the sample is, the more variability we expect to see in the sample. So, larger samples should have wider control limits. If we use the same limits for all points, there could be small-sample-size points that are really out of control, but don’t look that way, or huge sample-size point that are not out of control, but look like they are. Judging high school players by Olympic/NBA/NFL standards.

Different Sample Sizes

How not to do it  If we calculate n-bar, average sample size, and use that to calculate a standard deviation value which we use for every period, we get: One point that really is out of control, does not appear to be OOC 4 points appear to be OOC, and really are not.  Only potentially do this if all values fall within 25% of the average But with computers, why not do it right?

5 false readings

np Charts – Number Nonconforming  Counts number of defectives per sample  Sample size must be constant

C-Chart Control Limits  # defects per item needs a new chart  How many possible paint defects could you have on a car?  C = average number defects / unit  Each unit has to have same number of “chances” or “opportunities” for failure  UCL c z C   c LCLz C  c c

Paint Blemishes

Number of data points  Ideally have at least 2 defective points per sample for p, c charts  Need to have enough from each shift, etc., to get a clear picture of that environment  At least 25 separate subgroups for p or np charts

Small Average Counts  For small averages, data likely not symmetrical.  Use Table 7.8 to avoid calculating UCL, LCL for averages < 20 defects per sample  Aside: Everyone has to have same definitions of “defect” and “defective” Operational Definitions: we all have to agree on what terms mean, exactly.

U charts: areas of opportunity vary  Like C chart, counts number of defects per sample  No. opportunities per sample may differ  Calculate defects / opportunity, plot this.  Number of opportunities is different for every data point  n i = # square feet in sample i