Unit 1 AP Statistics

Linear Transformations

Changing the unit of measurement

The same variable can be recorded in different units of measurement. Americans commonly record distances in miles and temperatures in degrees Fahrenheit. Most of the rest of the world measures distances in kilometers and temperatures in degrees Celsius. Fortunately, it is easy to convert from one unit of measurement to another. In doing so, we perform a linear transformation.

LINEAR TRANSFORMATION

A linear transformation changes the original variable x into the new variable xnew given by an equation of the form

xnew = a + bx

Adding the constant a shifts all values of x upward or downward by the same amount.

Multiplying by the positive constant b changes the size of the unit of measurement.

Effects of Linear Transformations on Mean, Median, Standard Deviation, and Variance

Assume that xnew is a linear transformation of x. Then,

xnew = a + bx

What is the relationship between the mean, median, standard deviation, and variance of x and the mean, median, standard deviation, and variance of xnew?

Mean of xnew = a + b(mean of x)
Median of xnew = a + b(median of x)
Standard Deviation of xnew = b(sd of x)
Variance of xnew = b² (variance of x)

Let's consider the following data set:

{3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 19, 22, 23}
= 12 median = 11 sd = 6.41 variance = 41.08

Let's investigate what happens when 2 is added to value in the data set. 2 will be substituted into the transformation equation for a. The constant b will have the value of 1.

It is easy to see that the mean will now be 14. xnew = 2 +1(12)

What about the median? The median will be 13 an increase of 2. xnew = 2 + 1(11)

The standard deviation will not change. Even though the center of the distribution moved the distance of each point from the mean didn't change. The same will be true for the variance as well. Adding a constant amount to each observation does not change the spread.

Now let's investigate the transformation of increasing each term in the data set by 10%. a will be 0 and b = 1.10.

The mean will be xnew = 0 + 1.1(12) = 13.2

The median will be xnew = 0 + 1.1(11) = 12.1

The standard deviation will be xnew = 1.1(6.41) = 7.05

The variance will be xnew = 1.21(41.08) = 45.19

This time the center and the spread moved by 10%.

Linear transformations do not change the shape of a distribution. As you saw in the previous example, changing the units of measurement can affect the center and spread of the distribution. Fortunately, the effects of such changes follow a simple pattern.

EFFECT OF A LINEAR TRANSFORMATION

To see the effect of a linear transformation on measures of center and spread, apply these rules:

• Multiplying each observation by a positive number b multiplies both measures of center (mean and median) and measure of spread (standard deviation ) by b. The variance will be multiplied by b².

• Adding the same number a (either positive or negative) to each observation adds a to measures of center and to quartiles but does not change measures of spread (the standard deviation or the IQR).

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