Kurtosis Amount of peakedness or flatness Kurtosis < 0 Kurtosis > 0 Kurtosis = 0
Heteroskedasticity Sub-groups with different variances
Design Tolerances Design tolerance: Determined by users’ needs USL -- Upper Specification Limit LSL -- Lower Specification Limit Eg: specified size +/ inches No connection between tolerance and completely unrelated to natural variation.
Process Capability and 6 A “capable” process has USL and LSL 3 or more standard deviations away from the mean, or 3σ. 99.7% (or more) of product is acceptable to customers LSLUSL 33 66 LSLUSL
Process Capability LSLUSL LSL USL CapableNot Capable LSLUSL LSLUSL
Process Capability Specs: 1.5 +/ Mean: Std. Dev. = Are we in trouble?
Capability Index Capability Index (C pk ) will tell the position of the control limits relative to the design specifications. C pk >= 1.33, process is capable C pk < 1.33, process is not capable
Process Capability, C pk Tells how well parts produced fit into specs Process Specs 33 33 LSLUSL
Process Capability Tells how well parts produced fit into specs For our example: C pk = min[ 0.015/.006, 0.005/0.006] C pk = min[2.5,0.833] = < 1.33 Process not capable
Process Capability: Re-centered If process were properly centered Specs: 1.5 +/ LTL = 1.5 – 0.01 = 1.49 UTL = = 1.51 Mean: 1.5 Std. Dev. = LCL = *0.002 = UCL = = Process Specs
If re-centered, it would be Capable Process Specs
Packaged Goods What are the Tolerance Levels? What we have to do to measure capability? What are the sources of variability?
Production Process Make Candy PackagePut in big bags Make Candy Mix Mix % Candy irregularity Wrong wt.
Processes Involved Candy Manufacturing: Are M&Ms uniform size & weight? Should be easier with plain than peanut Percentage of broken items (probably from printing) Mixing: Is proper color mix in each bag? Individual packages: Are same # put in each package? Is same weight put in each package? Large bags: Are same number of packages put in each bag? Is same weight put in each bag?
Weighing Package and all candies Before placing candy on scale, press “ON/TARE” button Wait for 0.00 to appear If it doesn’t say “g”, press Cal/Mode button a few times Write weight down on form
Candy colors 1. Write Name on form 2. Write weight on form 3. Write Package # on form 4. Count # of each color and write on form 5. Count total # of candies and write on form 6. (Advanced only): Eat candies 7. Turn in forms and complete wrappers
So who cares? Dept. of Commerce National Institutes of Standards & Technology NIST Handbook 133 Fair Packaging and Labeling Act
Package Weight “Not Labeled for Individual Retail Sale” If individual is 18g MAV is 10% = 1.8g Nothing can be below 18g – 1.8g = 16.2g
Goal of Control Charts See if process is “in control” Process should show random values No trends or unlikely patterns Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily
NFL Control Chart?
Control Charts UCL LCL avg Values Sample Number
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
Control Charts NormalToo LowToo high 5 above, or belowRun of 5 Extreme variability
Control Charts UCL LCL avg 1σ1σ 2σ2σ 2σ2σ 1σ1σ
Control Charts 2 out of 3 in the outer third
Out of Control Point? Is there an “assignable cause?” Or day-to-day variability? If not usual variability, GET IT OUT Remove data point from data set, and recalculate control limits If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits
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)
Attributes vs. Variables Attributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart) Variables: measure length, weight, temperature (x-bar chart) measure variability in length (R chart)
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 Hotel Data # RoomsNo. NotProportion DaynReady p 11, /1,300 =
p Chart Control Limits
p Chart Solution
Hotel Room Readiness P-Bar
R Chart Type of variables control chart Interval or ratio scaled numerical data Shows sample ranges over time Difference between smallest & largest values in inspection sample Monitors variability in process Example: Weigh samples of coffee & compute ranges of samples; Plot
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 Chart Control Limits Sample Range at Time i # Samples Table 10.3, p.433
Control Chart Limits
R Chart Control Limits
R Chart Solution UCL
X Chart Control Limits Sample Range at Time i # Samples Sample Mean at Time i
X Chart Control Limits A 2 from Table 10-3
Table 10.3 Limits
R & X Chart Hotel Data Sample DayDelivery TimeMeanRange