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

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 0.3Copyright 2013, JJBarnette2

Unit 0, Session 0.3Copyright 2013, JJBarnette3 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

Unit 0, Session 0.3Copyright 2013, JJBarnette4 1. Cartesian Coordinate System We have a two-dimensional space One dimension is horizontal, we label this X One dimension is vertical, we label this Y The 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 

Unit 0, Session 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 Negative Y value is Positive X value is Positive Y value is Negative Copyright 2013, JJBarnette

Unit 0, Session 0.3Copyright 2013, JJBarnette6 1. Cartesian Coordinate System We 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

Unit 0, Session 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 Copyright 2013, JJBarnette

Unit 0, Session 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 Copyright 2013, JJBarnette

Unit 0, Session 0.3Copyright 2013, JJBarnette9 0,0 Using Part of the Cartesian Coordinate System for Frequency Histogram/Polygon Score or Score Category, Low to High F P r e e r q c u or e e n n t c y

Unit 0, Session 0.3Copyright 2013, JJBarnette10 1. Cartesian Coordinate System We 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 lines We often look to see if there is a straight line, what we will refer to as a linear relationship

Unit 0, Session 0.3Copyright 2013, JJBarnette11 +Y -Y -X+X X=0, Y=0 Any of these could represent areas that could be used to plot matched pairs of scores. The area just needs to include the full range of X and Y scores

Unit 0, Session 0.3Copyright 2013, JJBarnette12 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

Unit 0, Session 0.3Copyright 2013, JJBarnette13 * 12, 30 * 85, 40

Unit 0, Session 0.3Copyright 2013, JJBarnette14 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

Unit 0, Session 0.3Copyright 2013, JJBarnette15 3,10 6,16 9,20

Unit 0, Session 0.3Copyright 2013, JJBarnette16 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 b 1, 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 b 0, is the value on Y when X= 0, where it intersects the Y axis

Unit 0, Session 0.3Copyright 2013, JJBarnette17 4. Characteristics of a Straight Line Every straight line is defined as: Y= b 0 + b 1 X 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

Unit 0, Session 0.3Copyright 2013, JJBarnette18 4. Characteristics of a Straight Line 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: b 0 = 5.44 (Y-Intercept, where Y axis crossed) b 1 = 1.22 (change in Y as X increases 1 unit) So, our equation is: Y= X We can plot our line on our scatterplot

Unit 0, Session 0.3Copyright 2013, JJBarnette19

Unit 0, Session 0.3Copyright 2013, JJBarnette20 4. Characteristics of a Straight Line 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

Unit 0, Session 0.3Copyright 2013, JJBarnette21 Conclusion I 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