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9/20/2013 Practical Use of Statistics Dorian Shanin was a great Quality Professional I had met at the start of my career as a Quality Professional He preached to always search for the “Red X” Point Estimates vs. Confidence Intervals The “Red X” being the critical characteristic of the product Cp vs Cpk He said “Don’t ask your customer because they haven’t identified it Pp vs Ppk Cp & Cpk vs Pp & Ppk Normal vs Platikurtic vs Leptokurtic etc. I determined through the years that he was right In the immortal words of Vinnie Barbarino (a young John Travolta) 1 Relationships S l Samples n 2 Defining a Process’ Identity Every process is identified by: Population (N) (µ , σ, and σ²)Parameters “I’m so confused!” Its measures of central tendency (X1 , X X2 , etc.) etc ) (n1 , n2 , etc) Statistics Its measure of dispersion. Processes produce Populations 3 Defining a Processes Identity 4 Defining a Processes Identity The Central Tendency of a Distribution indicates at what value a distribution appears to be focused. The measure of central tendency we will be discussing this evening is the mean (average) The Dispersion of a Distribution indicates at how much a distribution varies around the mean of the distribution The measure of dispersion we will be discussing is the standard deviation. 5 6 1 9/20/2013 Sample Mean CENTRAL TENDENCY of a DISTRIBUTION The Mean (Average) of a distribution is the sum of all the values in the distribution divided by the number of values in the distribution: • Sample summary measures are called statistics • The sample mean is the sum of the values in the sample divided by the sample size, n n X= Mu is defined as the Average or Mean of the Population and as such it is a Parameter ∑X i =1 n i = X1 + X 2 + L + Xn n X Bar is a Statistic is our estimate of µ 7 Measure of Dispersion 8 Sample Standard Deviation The Standard Deviation is the most commonly used measure of Dispersion N σ= ∑(X − μ) 2 i i 1 i= N n S = Sigma is defined as the Standard Deviation of the Population and is a Parameter ∑ (X i=1 i − X )2 n -1 S is used to designate the Sample Standard Deviation. It is our estimate of Sigma and is a Statistic Shows variation about the mean Has the same units as the original data 9 10 The Rules of Thumb Concerning Normal Distribution • If the data distribution is normal, then the interval: • μ ± 1σ contains about 68% of the values in the population or the sample The Normal Distribution Also called Gaussian or Bell-shaped p 68% μ μ ± 1σ 11 12 2 9/20/2013 The Rule of Thumb cont. • μ ± 2σ contains about 95% of the values in • μ ± 3σ contains about 99.7% of the values 10 Average -3σ +3σ 9 9 9 the population or the sample 8 7 7 7 7 7 in the population or the sample Frequency 6 5 4 4 4 4 4 3 2 2 2 2 2 1 95% 1 1 1 1 99.7% 0 0 0 0 0 0 1 2 3 4 5 6 0 0 0 0 0 0 0 μ ± 2σ μ ± 3σ Remember, These Rules of Thumb apply only with a Normal Distribution Average ‐3σ 10 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 0.9 Expected Low value 13 30.1 Bin Expected High value Based on the Normal Curve 14 +3σ 9 10 Average 8 9 ‐3σ 7 +3σ 8 Frequency 6 7 Frequency 5 4 3 6 5 4 2 3 1 2 0 14 15 16 17 18 19 20 14.7 Expected Low value 21 22 23 24 25 26 1 25.3 Expected High value Bin 0 4 5 6 7 8 9 5.7 Expected Low value 10 11 12 13 14 15 16 17 16.3 Expected High value Bin 15 The Quality Professional should understand why a Process is not Normally Distributed and attempt with the rest of the Process Team to Normalize it. This falls under the umbrella of Continuous Improvement. 10 Average -3σ +3σ 9 16 The reason for the disparity is due to the calculation for the Standard Deviation: N 9 ∑(X − μ) 9 8 7 7 7 σ= 7 7 i=1 N 6 5 4 4 4 4 4 3 1 2 2 1 1 2 Frequency 2 2 1 1 0 0 0 0 0 0 1 2 3 4 5 6 0 0 0 0 0 0 0 7 8 0.9 5.7 Expected Low Expected Low value value 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 14.7 Bin16.3 Expected Expected Low value High value 25.3 Expected High value 30.1 Expected High value 10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 4 4 4 4 3 2 2 2 2 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 9 7 9 7 7 Bin 17 7 Frequency Freque ency 2 i 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Bin The Quality Professional is unjustly Penalizing the Process if the 18 Shape of the Distribution is not taken into consideration 3 9/20/2013 Average vs. Confidence Interval of the Mean The Calculation of the Average as I have so far discussed is a Point Estimate There is Error associated with this point estimate when using it to estimate µ. The magnitude of this Error depends on the Sample Size and the Standard Deviation X Bar is 18, n X= Confidence Interval of the Mean The average of 25 samples is 18 and these samples are taken from a population with a known standard deviation of 10. Calculate the 95% Confidence interval of the mean ∑X i =1 n i = σx = 10 n = 25, z = 1.96 X1 + X 2 + L + Xn n 14.08 ≤ µ ≤ 21.92 There is a 95% Confidence that the interval between14.08 to 21.92 contains µ! 19 Cp /Cpk vs Pp/Ppk Note this calculation our Risk (5%) and Error Interval (14.08 to 21.92 20 Cp /Cpk vs Pp/Ppk Pp = Process Capability = (USL-LSL) / 6σi Process Capability Indicies Cp and Cpk Cp = Process Capability = (USL-LSL) / 6σR σi = σx Note Pp doesn’t account for Measure of Central Tendency Note N t Cp C doesn’t d ’t accountt for f Measure M off Central C t l Tendency Cpk is the minimum between Cpku and Cpkl Cpkl = (µ - LSL)/3σR Cpku = (USL-µ)/3σR USL = upper specification limit LSL = lower specification limit σ = standard deviation µ = mean Ppk is the minimum between Ppku and Ppkl Ppku = (USL-µ)/3σi Ppkl = (µ - LSL)/3σi USL = upper specification limit LSL = lower specification limit σ = standard deviation µ = mean 21 Cp /Cpk vs Pp/Ppk 22 Rules of Thumb for Index Interpretation Because Cpk uses the Average Range (R Bar) to estimate sigma (Sigma hat) the Quality Professional must be sure that the Process is in control and Normally distributed otherwise Sigma Hat will be in Error I always use Ppk as a Process index because it is less likely to be in error 23 Value of Cpk or Ppk Capability Less than 1 Incapable Action Between 1 and 3 Capable Do nothing or some process improvement. Dependent on sample size. Greater than 3 Very capable Do nothing or reduce specification limits. No inspection necessary. Improve by reducing common causes of variation in process variables. Use 100% inspection. 24 4 9/20/2013 X Bar and R Charts Rational Subgroups • Key to Shewart Control Charts • The subgroup should be selected so as to minimize piece to piece variation within the subgroup • Selection of the next subgroup should be done to maximize the variation from subgroup to subgroup Where n is the size of each Subgroup and Sx = Sp = Si (Std Dev of Individuals)25 Control Chart Interpretation • Very important to ensure subgroup sequence is maintained along with timing of when the subgroup 26 was selected Process Variability • Common Causes • Have predictable effects on processes • It is present in all processes To decrease it typically requires • To decrease it typically requires an inherent change to the process • This type of change is by and large “management controllable” • Special Cause • Have unpredictable effects on processes • Is present in most processes • To decrease it requires To decrease it requires identification of the “culprit” • This type of change is best done by a Corrective Action Team Common Cause Variation is also called Chance Variation 27 Special Cause Variation is also called Assignable Cause Variation 28 X Bar and R Charts Control Chart Interpretation WESTERN ELECTRIC RULES The typical X Bar and R Chart has “3 Sigma Control Limits” In the past some quality professionals advocated “2 Sigma Control Limits” for better Control of the Process This is totally false. 3 Sigma Control Limits will tell us the Process is out of control 0.3% of the time when it is actually in control 2 Sigma Control Limits will tell us the Process is out of control 5.0% of the time when it is actually in control 29 30 5 9/20/2013 SQC vs. SPC Statistical Quality Control (SQC) connotates “Product Characteristic Control” Measuring critical dimensions of the product and determining the products acceptability This is many times required by customers but it is a reactive! Statistical Process Control (SPC) connotates “Process Characteristic Control” Identifying the “critical process nodes” (bath concentration, line speed, shut height, etc) and monitoring them This is proactive 31 6