Ch 4 : More on Two-Variable Data 4.1 – Transforming Relationships

Is there a linear relationship?

4.1 Nonlinear Data Linear Data: Exponential Data: A variable grows linearly over time if it adds a fixed increment in each equal time period Exponential Data: A variable grows exponentially if it is multiplied by a fixed number greater than 1 in each equal time period Exponential decay occurs when the factor is less than 1

4.1 Nonlinear Data How do we know if data is exponential? Data graphically appears exponential A common ratio exists: Ratio between subsequent points is roughly the same number if data is exponential

4.1 Nonlinear Data Perform linear transformations to “linearize” the data This will allow us to find the exponential equation and look at statistics such as correlation For exponential data log only the y-values to perform a linear transformation (i.e. linearize the data) A scatterplot of the x-values vs logy-values should reveal linear data We can now find the correlation and look at residuals from this “linearized data” To find the exponential equation of the original data, you must perform an inverse transformation on this linear equation (x-values vs logy-values)

4.1 Nonlinear Data Residuals of Exponential Data: Residuals are based off linear data, therefore look at the residuals of the transformed equation ( ) plot the “logy-values” based on “x-values”

4.1 Nonlinear Data Power Regression: One quantity is proportional to a second quantity raised to a power These always pass through the origin Examples:

4.1 Nonlinear Data Linear Transformation of a Power Regression: log the x-values and log the y-values to perform a linear transformation (i.e. linearize the data) A scatterplot of the logx-values vs logy-values should reveal linear data We can now find the correlation and look at residuals from this “linearized data” To find the power regression equation of the original data, you must perform an inverse transformation on this linear equation (logx-values vs logy-values)