Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination

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1 Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination

2 What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used for the x variable and the vertical axis is used for the y variable. How is Scatter Plot helpful? The pattern of the plotted points in a Scatter Plot will enable us to see whether there is a relationship between the two variables.

3 Example: The study time and midterm exam score for a random sample of 10 students in an elementary statistics course are shown in the following table. Student A B C D E F G H I J Study Time; x (Hours) Midterm Score; y Draw the scatter plot.

4 Solution: We plot the ordered-pair (3, 57) for student A, ordered-pair (4, 65) for student B, and so on to draw the Scatter Plot.

5 What is a Regression Equation? The Regression Equation algebraically describes the best linear relationship between two variables x and y. The Regression Equation is usually written in the following form. ŷ = a + bx What is the Regression Line? The Regression Line is the graph of the Regression Equation.

6 How do we compute a and b? Compute x, y, and xy. Compute x 2, and y 2. Now we use the formulas below. b = n( xy) ( x)( y) n( x 2 ) ( x) 2 and a = ( y)( x 2 ) ( x)( xy) n( x 2 ) ( x) 2

7 Example: The study time and midterm exam score for a random sample of 10 students in an elementary statistics course are shown in the following table. Student A B C D E F G H I J Study Time; x (Hours) Midterm Score; y Find the equation of the regression line in which the x variable is the study time, and y variable is the midterm score. Draw the regression line and scatter plot in the same coordinate system.

8 Solution: We first find and verify that x = 64, y = 782, xy = 5277, x 2 = 464, y 2 = 62632, and then we apply these values in the the formula b = n( xy) ( x)( y) n( x 2 ) ( x) 2 = 10(5277) (64)(782) 10(464) (64) 2 =

9 Solution Continued: and a = ( y)( x 2 ) ( x)( xy) n( x 2 ) ( x) 2 = (782)(464) (64)(5277) 10(464) (64) 2 = So the equation of the regression line is ŷ = a + bx = x

10 Solution Continued: Here is the graph of the regression line as well as the scatter plot.

11 What is a Correlation? A Correlation between two variables is when there is an apparent association between the values of one variable with the corresponding values from the other variable. What is the Linear Correlation Coefficient? The Linear Correlation Coefficient is a numerical value that measures the strength of the linear correlation between the paired x and y for all values in the sample. We denote this value by r.

12 What are the properties of r? 1 r 1 It is not designed to measure the strength of a nonlinear relationship. It is very sensitive and changes value if the sample contains any outliers. The Linear Correlation Coefficient is considered significant when r is fairly close to 1.

13 How do we compute r? Compute x, y, and xy. Compute x 2, and y 2. Now we use the formula r = n( xy) ( x)( y) n( x 2 ) ( x) 2 n( y 2 ) ( y) 2 It is worth noting that r is usually calculated with a computer software or a calculator.

14 Example: The study time and midterm exam score for a random sample of 10 students in an elementary statistics course are shown in the following table. Student A B C D E F G H I J Study Time; x (Hours) Midterm Score; y Find the value of linear correlation coefficient r.

15 Solution: We first find and verify that x = 64, y = 782, xy = 5277, x 2 = 464, y 2 = 62632, and then we apply these values in the the formula r = = = n( xy) ( x)( y) n( x 2 ) ( x) 2 n( y 2 ) ( y) 2 10(5277) (64)(782) 10(464) (64) 2 10(62632) (782)

16 What is the Coefficient of Determination? The Coefficient of Determination is a numerical value usually provided in percentage that indicates what percentage of the dependent variable y is explained by the independent variable x. We denote this value by r 2. How do we compute r 2? We simply square the value of r and then convert it to a percentage by moving the decimal point two places to the right.

17 Example: The study time and midterm exam score for a random sample of 10 students in an elementary statistics course are shown in the following table. Student A B C D E F G H I J Study Time; x (Hours) Midterm Score; y Find the value of coefficient of determination r 2 and explain what this number describes in the context of this example.

18 Solution: We have already used this example and found the value of the linear correlation coefficient r. We got r Now we square this number to get the coefficient of determination. r 2 = (0.959) 2 = % So 92.0% of the midterm scores are explained by the study time.

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