Linear Transformations

Notes

If \(x\) represents a variable, then a linear transformation is an equation of the form:

\[y = a + b x\]

where \(a\), \(b\) are some numbers. For instance \(y = x + 10\), or \(y = 2x - 1\).

We think of \(y\) here as a new variable, and the equation tells us how to convert values of the one variable into values of the other variable.

Examples:

\(x\) is a student’s height measured in “inches from 5 feet”. \(y\) is their actual height in inches. Then: \[y = x + 60\]
\(x\) is a student’s height in inches, \(y\) is their height in meters. 1 meter is \(39.37\) inches. So: \[y = x / 39.37 = \frac{1}{39.37} x\]
\(x\) is a room’s temperatures measured in F, \(y\) is the same temperatures in \(C\). The relation between Fahrenheit and Celsius degrees is: \[F = 1.8 C + 32\] So: \[y = \frac{x - 32}{1.8} = \frac{x}{1.8} - \frac{32}{1.8} = \frac{1}{1.8}x -\frac{32}{1.8}\]

A linear transformation between two variables tells us how individual values transform to each other.

So for instance if we had the temperature in Fahrenheit, \(x=56\), then we can find the corresponding temperature in Celsius: \[y=\frac{1}{1.8}\times 56 - \frac{32}{1.8} = 13.333\]

But how measures of center or spread behave requires more thinking!

Behavior of variables under linear transformation

Assume \(y = a + bx\). Then we can observe the following relation in the properties between \(x\) and \(y\).

shape

stays the same (modes, skewness, outliers)

center

Follows the same transformation (mean, median do that)

e.g. \(\bar y = a + b\bar x\)

spread

Only follows the multiplier (std. dev., IQR do that)

e.g. \(s_y = b s_x\).

aspect	multiplication by \(b\)	addition of \(a\)
shape center spread	ignores/unaffected affected affected	ignores/unaffected affected ignores/unaffected

Practice: If some temperatures have a mean of \(67\) degrees F, and standard deviation of \(5\) degrees F, how would the corresponding temperatures in C behave?

A special case: Standardized scores

Standardized scores, also called \(z\)-scores, are given by the following linear transformation:

\[z = \frac{x-\bar x}{s_x}\]

Alternatively, they relate to \(x\) via:

\[x = \bar x + s_x z\]

Key properties:

Think of \(z\)-score as saying “how many standard deviations away from the mean you are”.
\(z\)-scores are unitless.
\(z\)-scores always have a mean of 0 and a standard deviation of 1.
They are great for comparing quantities that are measured in different scales.

Practice

In the Behavioral Survey data we have examined, let us consider as \(x\) the height variable. The variable has a mean \(\bar x=67.18\) and a standard deviation \(s_x = 4.126\).

The maximum height was \(93\). What \(z\)-score corresponds to that?
We would consider a typical range of heights anything with a \(z\)-score between \(-2\) and \(2\). What heights would that cover? Convert the \(z\)s to \(x\)s to answer this.