Greenbelt 7
Calculating Pp
(USL - LSL) / 6s - or - VOC / VOP
Calculating Ppk: Closer to USL
(USL - Xbar) / 3s Section 46-3
Calculating Ppk: Closer to USL
(Xbar - LSL) / 3s Section 46-3
Capability Analysis
1. Assumes a stable process. 2. More conservative than Process Potential Analysis 3. Should be used once process is in statistical control 4. Should usually be used instead of Pp and Ppk. 5. Pp and Ppk can be used during setup, but Cp and Cpk preferred, since Control Chart will tell us the process is in control.
Hypergeometric Distribution: Conditions for Use
1. Sampling without replacement 2. n > 0.1N, where n = # of trails, N = entire population 3. There can only be two states (success/failure), (defective/not defective). [same as for binomial] 4. Must be fixed number of trials. [same as for binomial] Section 48-7
Binomial Distribution: Conditions for Use
1. There can only be two states (success/failure), (defective/not defective). 2. Must be fixed number of trials. 3. Each trial has to be independent. (Output of Trial x cannot affect the output of trial y). 4. The probability of success on every trial is the same (p). Section 48-3
Pp : Process Potential
A comparison between the process spread (Voice of the Process) and the specification spread (Voice of the Customer). If the process spread variation is less than the spec spread, the process spread can POTENTIALLY fit into the spec spread. This is an estimate.
D2 value
A constant used in calculations of Capability Analysis. The correct D2 value must be used depending on the subgroup size. For example for a subgroup size of 2, D2 = 1.128, while for a subgroup size of 5, D2 = 2.326
Hypergeometric Distribution: Definition
A discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population WITHOUT replacement, just as the binomial distribution describes the number of successes for draws WITH replacement.
Ppk: Centering of the process
A performance index which reflects the current process mean's proximity to either the USL or LSL. How close the process is to exceeding specification limits. A Ppk of 1.00 indicates that the bell curve end is pressed against the USL or LSL, and there is 1/2 a bell between the mean and the USL or LSL.
Process Shift
A process will naturally shift about 1.5 sigma up or down over time. This must be accounted for to keep our 6 sigma within customer specs. Without this shift, 6 sigma would represent 2 DPBO (Defects per Billion Opportunities)
Z-Score
A random variable that is normally distributed. Calculations found in the Normal tables. Used to determine the probability of parts in a certain range. Number of standard deviations from the mean
Point Estimate: Definition
A statistic (from the sample) used to estimate a parameter (of the population). It is a single value estimate, such as xbar to infer μ Section 55-5
Statistic vs. Parameter Defintions
A statistic is a value optained from a sample A parameter is a value obtained from the population (found or estimated) Section 55-4
Areas Under the Normal Curve
All the areas under the normal curve at up to 100% ± 1 Sigma = 68.26% ± 2 Sigma = 95.46% ± 3 Sigma = 99.73% The area under the entire curve = 100% or 1.000
Alpha α, One Tailed Test
Alpha (signficance level) is the area under curve for one tail in the normal distribution. Section 50-7
Estimating Standard Deviation
An estimate of standard deviation can be found using Rbar/D2. Section 46-7
Chi-Square Distribution: Degress of Freedom
As the df's increase, the shape becomes more bell-like, looks more symmetric, and widens. Section 53-3
Binomial Distribution in Excel
BINOMDIST Cumulative with this Excel function toggles the % of finding the condition true or finding the condition false. Section 48-5
3 Major Distributions Related to Probability
Binomial, Hypergeometric, F Distribution Section 48-1
3 Lesser Distributions Related to Probability
Bivariate, Exponential, Lognormal Section 49-1
Cp
Capability of the Process. Uses a different (estimate) of STDEV, Rbar/D2 Section 46-7
Cpk
Centering of the Capability of the Process. Uses a different (estimate) STDEV, Rbar/D2 Section 46-7
Multi-Vari Chart
Chart that simultaneously shows multiple measurements agains time or another variable. More than one measurement is shown within a single time frame. Section 54-2
Scatter Diagrams
Chart which plots two variables against one another, to demonstrate the correlation between the x and y axis. Section 54-3
Binomial Distribution: Example 2
Circuit Boards: History shows that 15% of all circuit boards are defective. What is the probability that at least 16 out of 20 are good? (4 or less are defective). p = 15% Section 48-5
F Distribution: Definition
Comparison of 2 variances, created by ratios of sample variances (s^2) taken from normal distributions. Larger sample variance always goes in numerator. Because variance is stdev squared, ratio is always positive. Section 48-9
Confidence and Significance Levels
Confidence Level and Significance Level add up to 100%, they are complementary. Section 55-7
Correlation is Not Necessarily Causation
Correlation is Not Necessarily Causation
D2 for Subgroup size of 2
D2 = 1.128 Section 46-7
D2 for Subgroup size of 3
D2 = 1.693 Section 46-7
D2 for Subgroup size of 4
D2 = 2.053 Section 46-7
D2 for Subgroup size of 5
D2 = 2.326 Section 46-7
Descriptive Statistics: Definition
Descriptive (or Enumerative) Statistics are a way to organize and summarize our information, such as Mean, Median, Mode, Graphs, Charts, Plots. Typically we are looking at the entire population. Section 55-2
Hypergeometric Example
Every employee has 90% service call success rate. What is the probability that if boss monitors 3 of 20 random calls, at least 2 of the calls will be a success? What is the average # of successes we would find for samples of 3 calls? What is the stdev for 3 calls? Yes, hypergeometric can be used.
Binomial Table Styles
Exact Probability Cumulative Probability Complimentary Probability
F Distribution: Formula
F = S1^2 / S2^2 (S1 is always larger than S2). S1 means S subscript 1, S2 means S subscript 2. F also = variance 1 / variance 2. Section 48-11
Weibull Distribution: Method
Gather life data on 6 parts or more. Select the lifetime distribution. Choose a relevant failure mode which is important to the customer. Generate a probability plot of failures. Make estimates about the population. B(X) life ("B10", the failure point of 10% of the population). Section 52-7
Binomial Distribution: Example 1
Guessing on a Test: Each test has 25 questions. Does it meet the criteria for being a binomial distribution? Yes, meets all criteria for binomial distribution. Section 48-4
Cpk: Higher Cpks
Higher Cpk's indicates there is extra room between the bell curve extent and the mean.
Ppk: Higher Ppks
Higher Ppk's indicates there is extra room between the bell curve extent and the mean.
Confidence Interval
How confident that a statistic is giving us an accurate picture of a parameter. We take a sample to develop a confidence interval of where approximately the parameter lies. Section 55-6, 9
Exponential Distribution: Example
If a call center receives 100 calls a year, what are the chances that a call will be received during the 1 week a year when the receptionist is out? In this case, we want the mean time between calls. Section 49-6
Alpha α, Two Tailed Test
If the distributions are two tailed, look up α / 2 in the table. Section 50-7
Lognormal Distribution
If you have a dataset which is not normal, you can take the (usually natural) log of each datum, and use the transformation formula to yield a set which is APPROXIMATELY normal. Section 49-7
Pp generally acceptable ratio
In 6 Sigma, Pp should usually be 1.5 or greater. That is, the specification is 1.5 greater than the process capability. 45-5
6 Sigma
In a normal curve 3 to the left, and 3 to the right of the centerline represents what your process is capable of producing 99.73% of the time, IF it is normally distributed. The goal is to fit 6 sigma of the process AT LEAST between the customer specs. Better would be more for "wiggle room". If acheived it means 3.4 DPMO.
Normal Curve: 1 Sigma / 1 Std Deviation
In a normal curve, 1 sigma from the center left and right (2 sigma total) represents 68% of what we expect to see.
Normal Curve 2 Sigma / 2 Std Deviations
In a normal curve, 2 sigma from the center left and right (4 sigma total) represents 95% of what we expect to see.
Normal Curve 3 Sigma / 3 Std Deviations
In a normal curve, 3 sigma from the center left and right (6 sigma total) represents 99.73% of what we expect to see.
t-statistic
Is the number from the Student t table. Section 50-7
Bivariate Distribution: Definition
Looks at 2 variables at once. Joint distribution of 2 different variables. Variables can be independent or covariate (affecting one another). Graph is 3D. X and Y are used for variables, Z axis is used for frequency (discrete data) or probability (continuous data). Section 49-2
LCL
Lower Control Limit (This is not the specifcation limit)
Upper Specification Limit
Lower Specification
LSL
Lower Specification Limit
Idea: Make a Form of a Part Itself
Mark on the Form Where the part was found to be defective. Great Graphical Idea.
Poisson Probability Distribution: Conditions for Use
Must know population mean. Must know occurences per unit. For Approx Binomial Distribution, sample size must be ≥ 16, -or- sample size times probability < 7 AND sample size < 10% of population size.
Poisson Probability Distribution: Definition
Named after S. Poisson, used to model situations which occur randomly over time, distances, volumes, rates etc. Useful when it is difficult or impossible to count the NON-occurences, such as how many NON-dots are there per square inch when you are looking at dots per square inch. Used to approximate binomial distributions.
Student t-Distribution: Definition
Named after W.S. Gosset whose pseudonym was Student. Used to test hypotheses about means, regression coefficients, and other stats. Draw conclusions about populations less than 30. As n (sample) approaches 30, t-distribution approaches normal distribution.
Z-Score Table Decimal Places
On a Z score table, the left side will show the main portion of the s number from top to bottom. From there, move to the right until the further decimal is reached to read the Z value.
3 Medium Common Distributions
Poisson Distribution Student t-Distribution Paired t-Distribution
Mu: μ
Population Mean (n * p, sample size x probability) Section 50-3
Relating Ppk to Z-Score
Ppk = Z-Score / 3 Section 47-6
Pp
Process Potential
Weibull Distribution: Results that can be obtained
Reliability for a given period of time. Probability of failure at a point in time. Mean life. Failure rate. Warranty time. B(X) life. Can be used regardless of the underlining distribution to obtain life data. Section 52-5
Analytical (Inferential) Statistics
Sample data is used to estimate something about the population. xbar (sample mean) and infer μ (population mean) s (sample stdev) and infer σ (population stdev) n (sample size and infer N (population size) Section 55-3
Tally
Simply a check sheet, where we tally the reason for something, the frequency of something, comments. Section 54-4
Concept of Probability
Sum of probabilities = 1. Σ(P) =1 Reliability = 1 - Unreliability Stated in % Section 52-2
Paired t-Test: Definition
Test hypotheses about two population means from two small sets of sample data (n <30) by pairing (n1 = n2). Use when comparing two sample groups. Can be used for before and after situations. Must compare the same things at different times. Section 50-9
Cpk Example
The Average % defective is 0.1%, Evaluation of the control chart shows that the process is in control. Working the Z-Score table backwards from 0.00135, we find that Z = 3 (exact is 3.09). Cpk will be Z/3 or 1.03. Section 47-8
Z Table Shading
The Z Score table will give you the value for the area under the shaded area shown before the table.
Rbar
The average of the ranges.
Chi-Square Distribution (x^2): Definition
The distribution of the sample variances tells whether two factors are related or are dependent on one another. Section 53-2
Binomial Distribution: Definition
The probability that an output of a process contains x DEFECTIVES (really, both good or bad characteristics can be measured this way. Defective is just the term used with Binomial Distributions. Section 48-2
Chi-Square Distribution: Alternative Hypothesis
The two factors are dependent. Choice of one factor DOES matter in outcome of the other. Section 53-4
Chi-Square Distribution: Assertion (Null Hypothesis)
The two factors are independent. Choice of one factor does not matter in the outcome of the other. Section 53-4
Significance Level (Alpha α)
These are the points at the end or ends of the normal distribution. While Confidence Level is in the middle (the comfortable area in which we are confident). Section 55-8
Chi-Square Distribution: Purpose
To compare the % of items distributed among several categories. To compare the categories within two factors. Section 53-4
F Distribution: When used
To test hypotheses about equality of population variances based on sample variances. Assumes that both populations are normal. Infer information about populations from sample data. Section 48-10
UCL
Upper Control Limit. (This is NOT the specifcation limit)
Exponential Distribution: Definition
Used frequently in reliability work to find the time between outcomes, especially when outcomes are consistent thru time. Can be used to predict the length of time equipment will operate. Used when the number of occurences follows a Poisson Distribution. No need to use tables, formula only. Section 49-4,5
Weibull Distribution: Definition
Used in reliability. Life data analysis of a product. Times to failure. Makes predictions about the life of all products based on a sample of the population. Quick analysis of reliability. Parts tested to failure. Exact time to failure unknown. Has a wide range of other applications. Section 52-3
Interval Estimate: Definition
Useds with inferential statistics to develop a confidence interval. The interval within which it is believed (with a certain degree of confidence) the population parameter lies. Any parameter estimate based on sample statistic has some amount of sampling error.
Weibull Plot
Uses log scale X Axis: Life Values in miles, time, cycles, etc. Y Axis: Accumulated % of failure up to 100% Usually the plot is a straight line. We can find X by bouncing off the intersection of the plot to the Y, and vice versa. Mean time to failure MTTF is always at 63.2% failure point. Section 52-7
Process Potential: 4 Numbers Required
Xbar (mean, central tendancy) s (stdev, fatness, dispersion of the bell) USL (upper specification limit) LSL (lower specification limit)
Calculating Upper Limit of Process Variation
Xbar + (Stdev * 3), 3 sigma to the right
Calculating Lower Limit of Process Variation
Xbar - (Stdev * 3), 3 sigma to the left
Z-Score Example 2
Z-Score <-1.45 P(Z<-1.45) What % of parts are OK? P(Z≤ 1.45) = 0.0735 -or- 1-0.0735 = 0.9265 or 92.65% will be OK. Section 47-5
Z Score Example 1
Z-score = 2.62. P(Z≤2.62) What % of parts will be OK? 1-Z = % -or- 1-.0044 = .9956 or 99.56% will be OK. Section 47-4
Degrees of Freedom (df)
n (sample -1). One less than the total sample size. For example, if you have a sample size of 4, and the mean is 2.75, and the first 3 terms are 1,1,1, then the 4th term MUST be 8 to make the mean 2.75 (11/4). No other term will work. Only the first 3 terms are free to be anything. Section 50-7
Binomial Variable Terms, Most Common
p = probability of occurrence on each trial r = number of occurences of defective n = number of trials Section 48-6
Chi-Square Distribution: Formula
x^2 = Σ (Observed - Expected)^2/ Expected
