RelativeResourceManager-3

# RelativeResourceManager-3 - Notes PSY 2801 Summer 2009...

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PSY 2801: Summer 2009 z-scores/Normal Distribution Jeff Jones University of Minnesota Jeff Jones Ch 6 Data Transformations Often, your data will make more sense in a different form. For instance, if you have interval data, then the 0 point doesn’t make sense, so it might be more practical to transform your data in order to make the 0 point equal to the mean . Jeff Jones Ch 6 Data Transformations Data Transformations : Any mathematical operation changing the original metric of your data. This mathematical operation is performed to every data point . For instance: y i = f ( x i ) is a transformation, with x i the old data point for person i, y i the new data point for person i, and f ( ) the function performed on every data point . Jeff Jones Ch 6 Notes Notes Notes

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Data Transformations Examples of Data Transformations: y i = 2 x i y i = ln ( x i ) + 4 y i = x i 44 y i = sin ( x i ) + cos ( x 2 i ) y i = x i - 20 2 y i = x 3 i The left column contains linear transformations while the right column contains non-linear transformations . Jeff Jones Ch 6 Linear Transformations Linear Transformations : Any transformation that can be written in the form y i = ax i + b 1 The mean of a linear transformation: ¯ y = a ¯ x + b 2 The variance of a linear transformation: s 2 y = a 2 s 2 x 3 The shape does not change Linear transformations change the mean and the variance of the data. However, the shape doesn’t change it is as though we are drawing the same picture with a different center and on a different scale. Jeff Jones Ch 6 Linear Transformations This is what happens if we transform data by an arbitrary linear function: -2 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Plot of Distribution K n = 200 K Density 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 Density Plot of K Transformed by (K + 3) * pi K Transformed Linearly Density Jeff Jones Ch 6 Notes Notes Notes
Non-Linear Transformations Non-Linear Transformations : Anything that cannot be written in the form of a linear transformation. It could be any function. Linear functions are a class of function (they’re special), so statisticians make note of them and regard anything else as non-linear . Non-Linear Transformations often change the mean, the variance and the shape of the distribution. Jeff Jones Ch 6 Non-Linear Transformations This is what happens if we transform data by an arbitrary non-linear function: -2 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Plot of Distribution K n = 200 K Density 0 20 40 60 80 120 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Density Plot of K Transformed by K^2 K Transformed Non-Linearly Density Jeff Jones Ch 6 Non-Linear Transformations This is a different non-linear transformation: -2 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Plot of Distribution K n = 200 K Density -4 -2 0 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Density Plot of K Transformed by log(K) K Transformed Non-Linearly Density Jeff Jones Ch 6 Notes Notes Notes

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