Statistics Dictionary

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Linear Transformation

A linear transformation is a change to a variable characterized by one or more of the following operations: adding a constant to the variable, subtracting a constant from the variable, multiplying the variable by a constant, and/or dividing the variable by a constant.

When a linear transformation is applied to a random variable , a new random variable is created. To illustrate, let X be a random variable, and let m and b be constants. Each of the following examples show how a linear transformation of X defines a new random variable Y.

  • Adding a constant: Y = X + b
  • Subtracting a constant: Y = X - b
  • Multiplying by a constant: Y = mX
  • Dividing by a constant: Y = X/m
  • Multiplying by a constant and adding a constant: Y = mX + b
  • Dividing by a constant and subtracting a constant: Y = X/m - b

Note: Suppose X and Z are variables, and the correlation between X and Z is equal to r. If a new variable Y is created by applying a linear transformation to X, then the correlation between Y and Z will also equal r.

See also:   Linear Transformations of Random Variables | AP Statistics Tutorial: Transformations to Achieve Linearity