Summary of Six Sigma Analysis - Cheat Sheet

I have put together all the basic statistical analysis used in Measure and Analyze phase of DMAIC methodology of a Six Sigma project in a way that is succinct, convenient and can be used as a cheat sheet for quick reference

Measure Phase – Continuous Data

→ Confidence Interval 

 (1-α)% CI = Sample mean ± Constant * Std error          

 CI = Xbar ± Zα/2 * (SD/√n)     

 Zα/2 = 1.96 at 95% CL from Z table    

 

→ Sample Size 

 n = (Zα/2 * SD / Δ)^2           

 Δ=precision     

 If n/N > 5%, then sample size to be modified by calculating n finite 

 n finite = n / (1+(n/N))         

 

→ Basic Statistics 

Measures of central tendency: Mean, Median, Mode

Measures of dispersion: Range, Variance, Std Deviation

Std dev = √(Sum of Deviations from mean)²/(n-1) 

In excel use stdev(data) function

Variance = SD²

Histogram

Number of classes = √n

Class interval = (Max-Min)/no of classes

→ Probability and Distributions 

 Normal distribution 

 Data is normal when normality test in Minitab shows p value>=0.05          

 Standard normal dist: Mean = 0 and SD = 1              

 Z=(x-mean) / SD              

 Excel formula for calc probability = normdist(x,mean,SD,cumulative)

→ Gage R&R (Reproducibility, Repeatability) 

 б² total = б² part-part + б² R&R                   

 Use ANOVA or Xbar-R method to conduct the test     

       

 → Conduct Gage R&R study (crossed) testing in Minitab     

 if %R&R < 20%, measurement is acceptable, else not acceptable     

 In Xbar chart for all appraisers should have almost the same pattern with most of the points falling outside control limits 

 All points in R chart should be in control limits for all appraisers

→ Process Capability 

 Cp = Tolerance / 6SD                      

 Tolerance = USL - LSL    

      

 Cpk = Min[ (USL - mean)/3SD , (mean - LSL)/3SD ]    

 Cpk<=Cp    

 Zlt = 3*Cpk ; Zst = Zlt + 1.5    

 If Cpk>1.33, process is capable    

      

 Check if the data is normal, if not, conduct 'Box Cox' transformation, if still not normal, consider data as Discrete and calculate DPMO (capability)

Measure Phase – Discrete Data

→ Types of data: Binary, Count, Ordinal  

  

→ Confidence Interval  

 CI = p ± Zα/2 * √p(1-p)/n      

 Zα/2 = 1.96 at 95% CL from Z table    

  

→ Sample Size  

 n = (Zα/2)² * p(1-p) / Δ²    

 Δ=precision     

→ Probability and Distributions

Binomial distribution (Binary data)

Excel formula: binomdist(numbers, trials, probability, cumulative)

eg: if p=0.02 or 2%

audited errors prob of n or less number of defectives

400              0   0.031%

400              2         1.312%

400              4         9.733%

400              6         31.090%

400              8         59.255%

400            10         81.790%

Poisson distribution (Count data)

Excel formula: poisson(x, mean, cumulative)

Avg repeat calls repeat calls d prob of getting d defects or less

20                                        0 0.00%

20                                        5 0.01%

20                                      10 1.08%

20                                      15 15.65%

20                                      20 55.91%

20                                      24 84.32%

→ Process Capability Zlt = -normsinv(DPO) Zst = Zlt + 1.5

Analyze Phase - Continous Data

→ Hypothesis tests 

1. 1t Test (Mean equal to a specified value µ=µ₀)  

 Test statistic t = (Mean -Specified Value) / (SD/√n)                

 If t is close to zero (to be determined by p value) mean = specified value      

 use tdist() in excel; or 1-Sample t test in Minitab (Stats>Basic stats) by selecting a confidence level and alternate hypothesis  

 if p>=0.05, accept Ho                      

  

 2. Paired t test (Two means equal or not µ₁=µ₂)  

 Paired t test is done for dependent samples like before and after improvement of the same process  

 Use ttest() in excel; or Paired t test in Minitab (Stats>Basic stats)        

  

 3. 2t test (Two means equal or not µ₁=µ₂)  

 2t test is done for two independent samples       

 Use ttest() in excel; or 2-Sample test in Minitab (Stats>Basic stats)  

  

 4. F test (Two variances are equal or not б₁=б₂)  

 used when x is discrete and Y is continuous      

 here H₀: S1²=S2²; and Ha: S1² <>S2²      

 F = S1²/S2² = Variance1 / Variance2      

 if p>=0.05, then F value is closer to 1, hence accept Ho    

 Use Ftest() in excel or 2variance test in Minitab (Stats>Basic Stats)  

  

 5. One Sample Wilcoxen test  

 Used for non normal data to test if median is equal to a given value  

 Use Minitab (Stats> Non parametric)        

  

 6. Mann - Whitney test  

 Used for non normal data to test if two medians are equal     

 Use Minitab (Stats> Non parametric)        

  

 7. Mood's Median test  

 Used for non normal data to test if three medians are equal     

 Here H₀: all medians are equal; Ha: all medians are not equal    

  

→ ANOVA  

 H₀: All means are equal; Ha: All means are not equal; if p>=0.05, then all means are equal  

  

 1. One Way ANOVA  

 Here X is discrete (categorical grouped) and Y is continuous    

 eg: Effect of Experience (12,6,0) on processing time      

 Use Minitab (Stat>ANOVA>One way) to calculate p value    

 if p>=0.05 means the factor doesn’t have significant impact on response  

  

 2. Two Way ANOVA  

 Here X is discrete (categorical grouped) and Y is continuous              

 Here the number of factors are 2 and replication > = 2      

 eg. Effect of vintage (Low, med , high) and Speed of data entry (low, high) and their interaction on AHT (2 observation each)  

 Use Minitab (Stat>ANOVA>One way) to calculate p value      

 Draw Main effects plot and Interaction plot (Stat>ANOVA) for further inferences          

  

→ Correlation  

 X and Y need to be continuous                

 Draw scatter plot and check the kind of correlation      

 Use correl() in excel to enter the 2 data arrays to find coff of correlation 'r'      

 Strong +ve correlation r>=0.8; Strong -ve correlation r<=0.8; no correlation when r is close to zero  

 r=0 means no linear relationship but there still could be a relation in a non linear form    

  

→ Regression  

 Provides mathematical model for Xs(continuous) effect on Y(continuous)      

  

 1. Simple linear regression  

  

 Draw a scatter plot and calculate r                   

 Use Minitab (Stat>Regression>Fitted line plot) and select linear, Quadratic or cubic model (whichever gives a higher Rsquare value) 

 Rsq should be>=65%; if p<0.05 then factor has significant effect             

  

 2. Multiple Linear regression  

 Fitted linear relation between Y and multiple Xs                 

 Draw scatter plot for all Xs and calculate r     

 Use Minitab (Stat>Basic stats>Correlation); if there is strong relation between any of the factors, one of the factors need to be ignored 

 Use Minitab (Stat>Regression)      

 Rsq (adj) should be>=65%; if p<0.05 then factor has significant effect             

  

 3. Polynomial Regression  

 Fitted non linear relation between Y and X                 

 Use Minitab (Stat>Regression>Fitted line plot) and select linear, Quadratic or cubic model (whichever gives a higher Rsquare value) 

 Rsq should be>=65%; if p<0.05 then factor has significant effect   

Analyze Phase - Discrete Data

→ Hypothesis tests

1. 1P test (Proportion equal to a specified value p=p₀)  

 Here X and Y both are binary        

 Test statistic z = (p - specified value) / √(p(1-p)/n)    

 Z will be closer to zero if p>=0.05      

 Use one proportion test in Minitab (Stats>Basic stats)  

  

 2. 2P test (Two proportions equal p₁=p₂)  

 Here X and Y both are binary        

 Test statistic z calculated by standardization (p1-p2)  

 Z will be closer to zero if p>=0.05      

 Use two proportion test in Minitab (Stats>Basic stats) 

→ Regression  
 Provides mathematical model for Xs(continuous) effect on Y(continuous)

 Logistic regression
 Here X is continuous and Y is discrete       
 Draw scatter plot for all Xs and calculate r   
 eg. Regression model for Vintage and Accuracy score   
 Use Minitab (Stat>Regression>Binary logistic regression)   
 In Minitab window: Success = total pass; Trial = total audited; Model = vintage
 Minitab will generate values: a(const coeff) and b(vintage coeff) 
 P value should be <=0.05   
 Regression equation: Accuracy score = {e^(a+b.vintage)} / {1+e(a+b.vintage)}