Effects of Linear Transformations on Mean, Median, sd, and Variance
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
Mean of Y = b(mean of X) +A
Median of Y = b(median of X) + A
sd of Y = b(sd of X)
Variance of Y = b² (variance of X)
Note in calculation block value of X should be respective mean, median sd or variance X give as given in the above formula.
