1. Green Belt – SIX SIGMA OPERATIONAL Central Limit Theorem

2. Flextronics Corporate Presentation Module Objectives At the end of this module the student will be able to: Describe the basic meaning of the Central Limit Theorem Explain (mathematically) how the means of samples from various distributions are similar to and different from their parent distribution

3. Flextronics Corporate Presentation Inferential Statistics - 2 Major Topics Parameter Estimation e.g. Engineer interested in estimating the parameter mean tensile strength of steel Scope : - Point Estimation Central Limit Theorem -Interval Estimation Confidence Intervals Hypothesis Testing e.g. Engineer interested in concluding if temperature t 1 resulting in higher yield than temperature t 2 Scope: Hypothesis formulation Discrete Data Testing Continuous Data Testing ANOVA Regression

4. Flextronics Corporate Presentation Central Limit Theorem The Central Limit Theorem and Confidence Intervals are the fundamental tools used in making (inferential) statistical decisions. The CLT is the basic concept of inferential statistics. It allows us to make inferences about a population characteristics based upon sample data. Confidence Intervals are derived from the Central Limit Theorem.

5. Flextronics Corporate Presentation Introductory Exercise Follow the instructions listed at the bottom of the slide to... Generate 100 rows by 10 columns of random data; mean = 0, std dev = 1. Plot the I-chart with the data Plot the Xbar chart with the means of the rows Compare the control limits of the charts Plot & compare the histograms for the data and the means of the rows Use Calc > Random Data > Normal Distribution to generate the data Use Calc>Row Statistics (select mean) to create C11 as Xbar Use Data > Stack to stack C1-C10 data in Column C12, label as Stack Use Stat > Control Charts > Individuals to obtain the I-chart Use Stat > Control Charts > Individuals to obtain the Xbar chart

6. Flextronics Corporate Presentation Individual vs Xbar Charts Why the difference? After all, it’s the same data ? How do the control limits compare ?

7. Flextronics Corporate Presentation Original Histogram vs Sampling Histogram The new histogram shows the sampling distribution of the mean Note the difference between the two histograms (Use Graph > Histogram to plot Stack and Xbar data)

8. Flextronics Corporate Presentation - Generate 100 x 10 random Binomial data, with Trials=20, p=0.1 - Plot the histograms (original and means of subgroup 10) What About Sampling on Binomial Distribution?

9. Flextronics Corporate Presentation Definition of the Central Limit Theorem: Part 1 If random samples of size n are taken from a distribution with mean  and standard deviation , then the sample means will form a distribution with the same mean but with a smaller standard deviation given by Population DistributionSampling Distribution x-  -  - - n  * The concept holds for both normally and non-normally distributed data x  xxmean )( n x       x n x x   

10. Flextronics Corporate Presentation Definition of The Standard Error of the Mean Standard error of the mean (SE Mean ) is the standard deviation of the distribution of means = Standard Error of the Mean  = Standard Deviation for the Individual Scores n= Sample Size for the Mean This formula shows that a “sample mean” is less variable than a single individual observation by a factor of the square-root of the sample size (n) The SE Mean tells us that the distribution of sample means has less variance than the original population for all n > 1 n x   Population of Individuals Population of Sample Average

11. Flextronics Corporate Presentation Note: If  is unknown, the sample standard deviation s may be substituted into the previous equation. Then the estimated standard error of is: n s ˆ x  

12. Flextronics Corporate Presentation Practical Application We usually rely on one reading per part from a measurement system (MS, which is used to estimate the “true” quality of our characteristic We can reduce measurement system error using the CLT by taking averages of two or more readings* on the same part *Repeated measurements should reflect the entire measurement system. The precision of our measurement system then is increased by a factor of the square-root of the sample size (number of repeat measurements) This is NOT an excuse to avoid fixing the gage! n likewise n call MS meanMSx    )( :Re  222 MSPartsTotal    2 2 )( meanMS % Contribution of MS =

13. Flextronics Corporate Presentation Standard Error and Sample Size

14. Flextronics Corporate Presentation Central Limit Theorem: Part 2 As n increases, the distribution of means becomes more normally distributed for any distribution.

15. Flextronics Corporate Presentation Sampling Distribution of for Various Sample Sizes X n = 1 n = 5 n = 30 n = 2

16. Flextronics Corporate Presentation Summary The central limit theorem allows us to assume that the distribution of sample means from any distribution will approximate the normal distribution if “n” is sufficiently high (n > 30 for unknown distributions). For a normal distribution, the central limit theorem also allows us to assume that the distributions of sample means are themselves normal, regardless of sample size. The standard error of the mean shows that as sample size increases, the standard deviation of the sample means decreases. The standard error will help us calculate confidence intervals.

17. Flextronics Corporate Presentation Module Review The student is now able to: Describe the basic meaning of the Central Limit Theorem Explain (mathematically) how the means of samples from various distributions are similar to and different from their parent distribution