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Statistics: Correlation and Linear Regression

Study of correlation and linear regression in statistical analysis.

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    Page 1 of 7 S tatistics Dr S. Therianos Correlation and Linear Regression Correlation: A t echnique used to decide whether there is a statistically significant linear relationship between the values of the dependent and independent variables. Regression: A statistical method used to describe the nature of the relationship between the depen dent and independent variables with a linear equation , called the Regression Line. Association: A vague term used to describe the relationship between two variables. We shall investigate relationships between variables. If only two variables are involved the relationship is simple . For example, is there a relationship between a person’s age and his or her blood pressure? If more than two variables are studied we have a multiple relationship . For example a student’s success in college depends on many factors such as the hours devoted to studying, the GPA, high school background, social and economic conditions. In this course we study only simple relationships . There are many types of relationships: A relationship is positive if both variables increase or decrease at the same time. For example there is a positive relationship between a person’s height and weight. A relationship is negative if one variable increases and the other decreases, or vice versa. For example, for older people, as age increases, in general, agility decreases. Also, a relationship can be linear or nonlinear . Our purpose is to answer the following questions statistically: 1. Is there an association between the variables? Basically this means a re the two variables x and y somehow related? 2. If the variables are related, what is the type or shape of the relationship?

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    Page 2 of 7 This means is the relationship linear or non - linear? Is it positive or negative? To answer these two questions we plot the points (x, y). The resulting graph is called a scatter plot , or scatter diagram or scatter - gram . 3. If the scatter plot shows a linear relationship or correlation how do we measure it? The strength of the linear relationship is measured by a number called the linear correlation coefficient r. 4. What kind of predictions can be made from the relationship? Predictions are made all the time and in all areas, including weather forecasting, stock market, sales predictions, crop predictions, sports, energy, global warming, health etc. Some predictions are more accurate than others depending on the strength of the relationship between the variables and the honesty of the researche r . You probably read about the “scientists” that conveniently omitted certain data that did not support their predictions about global warming. Scatter Plots: The existence of a relationship between two variables can be visually determined by a scatter plot or scatter diagram . Def : A Scatter Plot is a graph of ordered pairs of sample data. The points are plotted but they are not joined together. The Independent variable is always on the Horizontal axis. The Dependent variable is always on the Vertical axis. The purpose of a s catter d iagram is to give us a visual representation of the pairs of data and help us determine if there is any “pattern ”. A s the x - values increase do the y - values inc r ease, or decreas e, or do the y - values seem completely unaffected. To put it in another way a scatter diagram shows if there is any correlation between the values of the dependent and the in dependent variables. Consider the following scatter diagram. There appears to be a “ linear pattern ” or “ linear trend ". It seems that we can draw a straight line that approximates all the points. This line is called the Regression line . We shall compute its equation later . Ex. 1: Construct a scatter diagram for the following pairs of points: x 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 y 0.6 1.7 2.4 2.3 3.3 3.2 4.5 4.0 4.8 6.4

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