Unit 0, Pre-Course Math Review Session 0.3, Graphing J. Jackson Barnette Professor of Biostatistics.

Unit 0, Pre-Course Math ReviewSession 0.3, Graphing

J. Jackson Barnette

Professor of Biostatistics

Copyright 2013, JJBarnette 2

Session 0.3 Graphing

Topics in the session include:

1. The Cartesian coordinate graphing system

2. Plotting the points

3. The scatterplot

4. Characteristics of a straight (linear) line

Unit 0, Session 0.3

Unit 0, Session 0.3 Copyright 2013, JJBarnette 3

1. Cartesian Coordinate Graphing System, Equation for Line

We do a reasonable amount of graphing in the course and we use the Cartesian Coordinate Graphing System for several types of graphs

For some types of graphs, we are also interested in plotting a straight line that best represents a set of matched data points

For this, we need to find the characteristics of a straight line, the slope and Y-intercept


1. Cartesian Coordinate System

We have a two-dimensional spaceOne dimension is horizontal, we label this XOne dimension is vertical, we label this YThe X and Y axes cross at the point where X= 0 and Y= 0

In theory, the X axis goes from - to +and the Y axis goes from - to +Our values will tend to be pretty small

compared with

Copyright 2013, JJBarnetteUnit 0, Session 0.3 5

+Y

-Y

-X +X

X=0, Y=0

Values are Positive for both X and Y

Values are Negative for both X and Y

X value is NegativeY value is Positive

X value is PositiveY value is Negative


1. Cartesian Coordinate SystemWe will use only parts of the possible graphing

system in such a way that we can benefit from the visual nature of the data presented

We first look at the part of the system we will use to graph a frequency/percent histogram (bar graph) or polygon

We have frequency or percent plotted on Y, beginning at 0 and score or score category on the X axis, beginning at any point on X, negative or positive, plotted with lower scores on left and higher scores to the right


+Y

-Y

-X +X

We might use this part of the system to graph data such as a frequency histogram or polygon

Score or Score Category will be plotted on X Low to High

Frequency or percent Will be plotted on Y 0


+Y

-Y

-X +X

We might use this part of the system to graph data such as a frequency histogram or polygon

Score or Score Category will be plotted on X Low to High

High

0


0,0

Using Part of the Cartesian Coordinate System for Frequency Histogram/Polygon

Score or Score Category, Low to High

F Pr ee rq cu or ee n n tcy


1. Cartesian Coordinate SystemWe also use this graphing system to plot

matched scores on two variables, labeled X and Y, to examine the possible relationship between these two variables

We plot the matched pairs of data and then we look to see if there is a pattern to the scores

Patterns could be straight or curved linesWe often look to see if there is a straight line,

what we will refer to as a linear relationship


+Y

-Y

-X +X

X=0, Y=0

Any of these could representareas that could be used to plot matched pairs of scores.The area just needs to include the full range of X and Y scores


2. Plotting the Points

We would have a set of matched pairs of X and Y scores

Each matched pair is plotted as a single point

The next slide shows two points being plotted

There will be many more than just these two plotted, but this just illustrates how the points are plotted


Scatterplot Structure with Two Pairs of X and Y Values PlottedX= 12 and Y= 30 and X= 85 and Y= 40

Value onY Variable

LOW Y VALUE

HIGH YVALUE

LOW XVALUE

Value onX Variable

HIGH X VALUE

X=12

Y= 30

Y= 40

X= 85

*12, 30

*85, 40


3. The Scatterplot

A scatterplot is used to graphically display a set of matched data on two variables X and Y

The next slide shows an example of such a scatterplot with ten matched pairs of scores


Scatterplot of Two Metric Variables

4

6

8

10

12

14

16

18

20

22

2 3 4 5 6 7 8 9 10 11

X Variable Value

Y V

ari

ab

le V

alu

e

3,10

6,16

9,20


4. Characteristics of a Straight Line

We will want to find the straight line that best fits this set of points (a linear pattern)

Every straight line has two characteristics:

1. The slope, which we will symbolize as b1, is the change in the value of variable Y when X increases by ONE (1) unit

2. The Y-intercept (also called the constant), which we will symbolize as b0, is the value on Y when X= 0, where it intersects the Y axis



Every straight line is defined as:

Y= b0 + b1X

Y= slope times a value of X + the Y-Intercept

Sometimes this is labeled as: Y= bX + a

This is just different symbols for the same things



In the course, we will look at ways we find these two values

Let’s assume that for these data points, we get the two characteristics of:

b0= 5.44 (Y-Intercept, where Y axis crossed)

b1= 1.22 (change in Y as X increases 1 unit)

So, our equation is: Y= 5.44 + 1.22X

We can plot our line on our scatterplot


Scatterplot of Two Metric Variableswith Line of Best Fit, Y= 1.22X + 5.44

4

6

8

10

12

14

16

18

20

22

0 1 2 3 4 5 6 7 8 9 10 11

X Variable Value

Y V

ari

ab

le V

alu

e

b 0, Y-Intercept= 5.44

X Increases 1

Y Increases 1.22b 1, the Slope



Every straight line is defined by only two values, the slope and the Y-Intercept

We will find these values and use them to plot the line that best fits the set of points

We will then use this relationship to make predictions on the Y variable when we have known values on the X variable


ConclusionI hope this review has been helpful in

reacquainting you or introducing you to many of the mathematics operations used in introductory statistics

You may want to copy these slides as pdf file handouts from the Course Website and refer to them occasionally as we go through the course

If you find errors, feel other topics should be included, or if some of these presentations are not clear, please contact Dr. Barnette

Unit 0, Pre-Course Math Review Session 0.3, Graphing J. Jackson Barnette Professor of Biostatistics.

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