1. 12b. Regression Analysis, Part 2 CSCI N207 Data Analysis Using Spreadsheet Lingma Acheson linglu@iupui.edu Department of Computer and Information Science, IUPUI

2. Fitting the Data

3. If there are more than two data points, chances are they don’t all fit in one straight line. We need to find the equation for a straight line that does the “best job” of reproducing the data. About half of the data points should fall above our line (“positive residual”) and about half should fall below (“negative residual”).

4. Residual Difference between the measured and the calculated Y-values:

5. Finding the Slope ( m ) of an Estimated Line The slope of the estimated line is given by the ratio of the covariance between the X and Y data sets and of the variance of the X data set:

6. Finding the y-Intercept ( b ) of an Estimated Line Once we’ve found the slope, we can find the Y- intercept using the standard equation for a line, with one exception: we must use the means of the X and Y data sets as our coordinates (since the actual data points are unlikely to be on the estimated line): Excel functions: –m: SLOPE(..,..) –b: INTERCEPT(..,..)

7. Practice Find an equation for the trendline of the following data set and predict the reading hours when aptitude is 25, 33 or 45. Student Reading Aptitude Reading Hours 1205 251 352 4357 5308 6358 7103 852 9155 10409

8. Predicting Values Once we get the slope ( m ) and the y-intercept ( b ) of the estimated line, we have a mathematical relation that ties the X variable to the Y variable. Once we have this relation, we can use it to predict X- and Y- coordinates that are not part of the data sets. E.g. What is the estimated reading hours if two new students coming in, one has a reading aptitude of 25 and another one 46? y = 0.2029x + 0.9429 x = 25, y = 0.2029*25 + 0.9429 = 6.0154 x = 46, y = 0.2029*46 + 0.9429 = 10.2763

9. Interpolation Interpolation is the process by which we use the formula for estimated line to predict a value of Y for a given value of X that is not included in the data set, but is within the range of the data set. The given value of X and the predicted Y -value will be on the estimated line.

10. Extrapolation Extrapolation is the process by which we use the formula for estimated line to predict a value of Y for a given value of X that is not included in the data set AND is not within the range of the data set. The given value of X and the predicted Y -value will be on the estimated line, but outside of the range of the data set.

11. R 2 Value How good is the line? How confident is the prediction? R : C orrelation Coefficient, -1 ≤ R≤ 1 R 2 :Coefficient of Determination, 0 ≤ R 2 ≤ 1 The Coefficient of Determination is used to measure the certainty of making predictions from a graph. It represents the percent of data closest to the trendline. The closer it is to 1, the more confident the prediction is. - From "Correlation Coefficient" (http://mathbits.com/MathBits/TISection/Statistics2/correlation.htm)

12. Excel Functions TREND() - Returns predicted Y values in a linear trend when passed X data. Add Trendline (from the Chart menu) Returns the trendline, equation, and correlation coefficient for a set of X,Y data.