1. Graphical Analysis

2. Why Graph Data? Graphical methods Require very little training Easy to use Massive amounts of data can be presented more readily Can provide an understanding of the distribution of the data May be easier to interpret for individuals with less mathematical background than engineers

3. Graphical methods Quantitative data (numerical data) Cost of a computer (continuous) Number of production defects (discrete) Weight of a person (continuous) Parts produced this month (discrete) Temperature of etch bath (continuous) Graphical tools Line charts Histograms Scatter charts

4. Graphical methods Qualitative data (categorical and attribute) Type of equipment (Manual, automated, semi- automated) Operator (Tom, Nina, Jose) Graphical tools Bar charts Pie charts Pareto charts

5. Getting Started Classify data Quantitative vs. Qualitative Continuous or discrete (quantitative) Chose the right graphical tool Chose axes and scales to provide best “view” of data Label graphs to eliminate ambiguity

6. Graphical Analysis Examples

7. Bar or Column Graph Displays frequency of observations that fall into nominal categories

8. Line Chart Shows trends in data at equal intervals

9. Graphical methods Acceptable graph

10. Graphical methods Better graph

11. Graphical Analysis Details Always label axis with titles and units Always use chart titles Use scales that are appropriate to the range of data being plotted Use legends only when they add value Use both points and lines on line graphs only if it is appropriate – don’t use if the data is discrete

12. Histograms Histograms are pictorial representations of the distribution of a measured quantity or of counted items. It is a quick tool to use to display the average and the amount of variation present.

13. Histogram example

14. The Pareto principle Dr. Joseph Juran (of total quality management fame) formulated the Pareto Principle after expanding on the work of Wilfredo Pareto, a nineteenth century economist and sociologist. The Pareto Principle states that a small number of causes is responsible for a large percentage of the effect--usually a 20-percent to 80-percent ratio.

15. Pareto example

16. Histogram Example in Excel

17. ENGR 112 Fitting Equations to Data

18. Introduction Engineers frequently collect paired data in order to understand Characteristics of an object Behavior of a system Relationships between paired data is often developed graphically Mathematical relationships between paired data can provide additional insight

19. Regression Analysis Regression analysis is a mathematical analysis technique used to determine something about the relationship between random variables.

20. Regression Analysis Goal To develop a statistical model that can be used to predict the value of a variable based on the value of another

21. Regression Analysis Regression models are used primarily for the purpose of prediction Regression models typically involve A dependent or response variable Represented as  y One or more independent or explanatory variables Represented as  x 1, x 2, …,x n

22. Regression Analysis Our focus? Models with only one explanatory variable These models are called simple linear regression models

23. Regression Analysis A scatter diagram is used to plot an independent variable vs. a dependent variable

24. Regression Analysis Remember!! Relationships between variables can take many forms Selection of the proper mathematical model is influenced by the distribution of the X and Y values on the scatter diagram

25. Regression Analysis X Y X Y X Y X Y

26. Regression Analysis Model SIMPLE LINEAR REGRESSION MODEL However, both  0 and  1 are population parameters  i  Represents the random error in Y for each observation i that occurs Y i =  0 +  1 X i +  i

27. Regression Analysis Model Since we will be working with samples, the previous model becomes Where b 0 = Y intercept (estimate of  0 ) Value of Y when X = 0 b 1 = Slope (estimate of  1 ) Expected change in Y per unit change in X Y i = Predicted (estimated) value of Y Y i = b 0 + b 1 X i ^ ^

28. Regression Analysis Model What happened with the error term? Unfortunately, it is not gone. We still have errors in the estimated values

29. Regression Analysis Find the straight line That BEST fits the data

30. Regression Analysis X Y 0 0 Positive Straight-Line Relationship e1e1 e2e2 e3e3 e4e4 e5e5 Y i = b 0 + b 1 X i b0b0 b1b1

31. Least Squares Method Mathematical technique that determines the values of b 0 and b 1 It does so by minimizing the following expression Min

32. Least Squares Method Resulting equations Equations (1) and (2) are called the “normal equations” (1) (2)

33. Least Squares Method Assume the following values Resulting equations

34. Assessing Fit How do we know how good a regression model is? Sum of squares of errors (SSE) Good if we have additional models to compare against Coefficient of determination  r 2 A value close to 1 suggests a good fit Where do we get these values?