Statistics Simplified

1. Statistics Simplified Prabhakar Jain

2. Buying A Raffle Ticket What is the Value of the Ticket (not cost)? What are my Chances of Winning ? Is it a Good thing to DO?

3. Process � To Answer some of the Questions raised in our Previous slide, we need to have little more Information What is the Selling Price of the Ticket ($1)? How many tickets will be sold(1000)? How many Items will be drawn to Give Away (1 Gift Certificate to a local Restaurant)? What is the Total value of give away item(s) ($50)? How many Tickets I am Buying (1)?

4. Process � Money expected to be collected $1,000 Ticket Selling Price * No. of Tickets to be sold. Money Spent for Give Away Item(s) $50. No. of Item(s) Given Away * Cost of Item(s) Do this for Each Unique Item Given Add up Costs for Each Item

5. Answers Value of the Ticket Cost of Give Away Item(s) / Total Money Collected $50 / 1000 = $0.05 (Nickle) Chances of Winning No. of tickets you Bought / Total No. of Tickets Sold 1 / 1000 = 0.001 or 0.1% (1 tenth of 1 Percent). If you feel good about the Cause, and you can afford it, You should do it.

8. Population is Described by Parameters Sample is Described by Statistics

9. Graphical Representation of Data Dot Plots Histogram Frequency Polygon Stems and Leaves Time Series Plot Pareto Chart

10. Dot Plot

11. Histogram

12. Box Plot

13. Time Series Plot

14. The Mean Is also known as Average Sum of all Values / Number of Values Also called a Measure of Central Tendency

15. Median Next Measure of Central Tendency Central Value After arranging the data in ascending order, When Number of Data points is Odd, it is the Middle Value When Number of Data points is Even, it is the average value of the two middle numbers.

16. Mode It is the Most Frequently Occurring Value in the data set

17. Normal Distribution When Mean, Median, and Mode values of a data set are same. Has a Bell Shaped Curve Both Sides of the Bell Shaped Curve are Symmetrical

18. Empirical Rule If you have Normal Distribution, You can expect the following 68% of the points will be within 1 Sigma 95% of the Points will be within 2 Sigma 99.7% of the Points will be within 3 Sigma

19. Other Distributions Binomial Poisson Uniform Exponential Weibull Geometric

20. Uniform Distribution Rolling of a Dice. Possible Outcomes are 61 = 6. Probability of rolling 1 is 1/6. Probability of rolling any of the other numbers 2, 3, 4, 5, and 6 is also 1/6. Having a constant Probability for all possible outcomes, makes it a Uniform Distribution.

21. Let us Review rolling 2 Dice Possible Outcomes from Rolling 2 Dice are 62 = 36. Probability of rolling; 1 is 0 (We are adding the numbers rolled). 2 or 12 is 1/36 (1 or 6 on each dice). 3 or 11 is 2/36 (1 on 1st and 2 on the 2nd or 2 on the 1st and 1 on the 2nd). 4 or 10 is 3/36 (1st dice could have 1, 2, or 3 and 2nd dice needs to the 3, 2, or 1 respectively). 5 or 9 is 4/36 (1st dice could have 1, 2, 3, or 4 and 2nd dice needs to the 4, 3, 2, or 1 respectively). 6 or 8 is 5/36 (1st dice could have 1, 2, 3, 4, or 5 and 2nd dice needs to the 5, 4, 3, 2, or 1 respectively). 7 is 6/36 (1st dice could have 1, 2, 3, 4, 5, or 6 and 2nd dice needs to the 6, 5, 4, 3, 2, or 1 respectively).

22. Results from rolling 2 Dice Probabilities for numbers are not Uniform They vary 1/36, 2/36, 3/36, 4/36, 5/36, 6/36 If we layout the numbers and their respective Probabilities, It will appear very much like a Normal Distribution It will also follow the Empirical Rule

23. Other Distributions By Applying some Transformation to the Data (Averages) and if subgroup is large enough, then the Distribution of the averages will tend to be Normally Distributed. This Implies that X-Bar chart will follow the Rules for the Normal Distribution, and not the underlying distribution of the original Data. So for Control Charts for Variable Data, We follow rules, applicable for the Normal Distribution. Control Limits at +/- 3 Sigma would mean 99.7% within control limits, and If you see a plot point beyond 3 Sigma Control Limits, There is 99.7% Chance, that you will find a Assignable Cause, and there is only 0.3%, That it happened by Chance.

24. Control Charts Variables Data X-Bar and R X-Bar and S IX&MR Attribute Data p or np Chart c or u Chart

25. Process Capability In Order to see if Process is in Control We need to verify that process is Stable Western Electric Rules is one way to Check for stability. If you have frequent out of Control Conditions, We should not be calculating the Process Capability. Because, if Process is not Stable, It is not Predictable.

26. Western Electric Rules 1 Point Beyond 3 Sigma Limits 7 Points in a row on Same Side of Center Line (Target) 7 Points in a row increasing or decreasing (Trend) 14 Points in a row alternating up and down (Cycle) 2 out of 3 Points > 2 Sigma from Center Line - same side (Target) 4 out of 5 Points > 1 Sigma from Center Line - same side (Target) 15 Points in a row within 1 Sigma of Center Line � either side (Reduced Variation) 8 Points in a row > 1 Sigma from Center Line � either side (Spread)

27. Process Capability � Cp = (Total Tolerance) / 6s. (Does not take Process Target into consideration) Cpl (lower) = (Process Mean � Lower Spec.) / 3s. Cpu (upper) = (Upper Spec. � Process Mean) / 3s. Cpk = Minimum of (Cpu, Cpl). A negative Cpk means Process is Targeted outside the Specification Limits. Cpk = 0 means Process is targeted exactly at either Lower or Upper Specification Limit.

28. Process Capability - Cpk Cpk is looking for No. of 3s�s from the Center of the Process to any given point X (Specification Limit may be one Example) Z Score is looking for No. of s�s from the Center of the Process to any given point X (Specification Limit may be one Example) By these two Definitions Cpk = Z / 3. Z = Cpk * 3.

29. Z - Table

30. Process Capability � Once you have evaluated the Process, you know if your Process is Capable of producing either Customer�s or Yours established Requirements Cpk (Minimum of 1.67 for Automotive) PPM or PPB Yields (May be set by the Upper Management) Throughput (May be established by Plant Manager level)

31. Process Capability � If you are not happy with the outcome of your Process, you need to take some action. Normally, things do not change on its own. SPC may identify, which area need improvement Other Process Capability Metrics Customer Complaints Regulatory / Government Organization

32. Process Capability � To improve the Process Capability, various tools are available Value Stream Mapping Six Sigma Lean Kaizen Theory of Constraints (Bottlenecks) TPM

33. Statistics Most of the Improvement tools utilize use of Statistics to analyze. You do not have to be a Statistician to make use of these tools. Many Statistical Software, available in the market, will make this task easier. You need to have statistical thinking. You should be able to ask the Right Questions. You should be able to understand the process of getting to the end result.

34. Regression Analysis: Weight versus D2H The regression equation is Weight = - 0.1081 + 0.01019 D2H S = 0.346437 R-Sq = 95.9% R-Sq(adj) = 95.9% Analysis of Variance Source DF SS MS F P Regression 1 832.207 832.207 6933.98 0.000 Error 293 35.165 0.120 Total 294 867.372

36. Questions ?