Graphical Analysis

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Graphical Analysis
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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

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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

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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

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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

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Graphical Analysis

• Examples

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Bar or Column Graph

• Displays frequency of observations that fall into
  nominal categories

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Line Chart

• Shows trends in data at equal intervals

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Graphical methods

• Acceptable graph

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Graphical methods

• Better graph

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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 dont use if the data is
  discrete

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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.

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Histogram example
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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.

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Pareto example
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Histogram Example in Excel
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ENGR 112

• Fitting Equations to Data

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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

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Regression Analysis

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

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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

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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 ? x1, x2, ,xn

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Regression Analysis

• Our focus?
• Models with only one explanatory variable
• These models are called simple linear regression
  models

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Regression Analysis

• A scatter diagram is used to plot an independent
  variable vs. a dependent variable

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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

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Regression Analysis
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Regression Analysis Model

• SIMPLE LINEAR REGRESSION MODEL
• However, both b0 and b1 are population parameters
• ei ? Represents the random error in Y for each
  observation i that occurs

Yi b0 b1Xi ei
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Regression Analysis Model

• Since we will be working with samples, the
  previous model becomes
• Where
• b0 Y intercept (estimate of b0)
• Value of Y when X 0
• b1 Slope (estimate of b1)
• Expected change in Y per unit change in X
• Yi Predicted (estimated) value of Y

Yi b0 b1Xi

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Regression Analysis Model

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

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Regression Analysis

• Find the straight line
• That BEST fits the data

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Regression Analysis

• Positive Straight-Line Relationship

Yi b0 b1Xi
Y
b1
e4
e2
b0
e5
e3
e1
0
X
0
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Least Squares Method

• Mathematical technique that determines the values
  of b0 and b1
• It does so by minimizing the following expression

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Least Squares Method

• Resulting equations
• Equations (1) and (2) are called the normal
  equations

(1)
(2)
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Least Squares Method

• Assume the following values
• Resulting equations

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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 ? r2
• A value close to 1 suggests a good fit

Where do we get these values?