AppliedStatsCourse

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. $s_y = b s_x$.

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aspect multiplication by $b$ addition of $a$ ——- ———————- ——————– shape ignores/unaffected ignores/unaffected center affected affected spread 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 $s_x = 4.126$.

1. The maximum height was $93$. What $z$-score corresponds to that?
2. 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.