# 4.1 - Standard Scores

4.1 - Standard Scores

## Standardized Scores (also called "standard scores" or "z-scores")

In Lesson 3 we learned that the standard deviation provides a measure of variability about the mean. Generally, most observations are within one standard deviation of the mean, and observations more than three standard deviations away from the mean are very rare. Thus, the standard deviation provides a natural yardstick by which to gauge where an observation stands relative to others. If you are five standard deviations above the mean then you know you are at the top of the list; one standard deviation below the mean and you know you are on the low side but not too far down. The standardized score is a measure of relative standing on a list, it is just the number of standard deviations above (+) or below (-) the mean you are. To compute the standardized score of a value, you take

Standardized Score (Z-score) formula
$$z = \text{standardized score} = \dfrac{\text{(value - mean)}}{\text{standard deviation}}$$

These numbers are called "standardized" because the list of standardized scores itself always has a mean of 0 and a standard deviation of 1.0. That's because subtracting the mean from every value makes the new mean equal zero and dividing every value by the standard deviation makes the new standard deviation equal to 1.

## Example 4.1

According to EPA data, the gas mileage for compact SUVs in the 2013 model year has a mean of approximately 22 mpg and a standard deviation of about 3 mpg. One SUV gets 25 mpg. Thus, its standardized score is z = (25 - 22)/3 = 1. It is one standard deviation above the mean. Another SUV gets 20.5 mpg. Thus, its standardized score is z = (20.5 - 22)/3 = -0.5. It is one-half of a standard deviation below the mean.

The standardized scores give you a way to compare relative standing of values on different lists where the distributions might have roughly similar shapes.

## Example 4.2

According to EPA data, 4-cylinder 2013 model year cars have CO2 emissions that average 333 ppm (parts per million) with a standard deviation of 51 ppm; while 6-cylinder cars made that year average 431 ppm with a standard deviation of 44 ppm. Which vehicle has higher CO2 emissions relative to other cars with the same number of cylinders; the 4-cylinder Honda Civic that emits 284 ppm or the 6-cylinder Toyota Camry that emits 358 ppm?

The Honda Civic would have a z-score of $$\dfrac{(284-333)}{51}=-0.96$$ while the Toyota Camry would have a z-score of $$\dfrac{(358-431)}{44}=-1.66$$. The Honda Civic has a higher relative CO2 emissions.

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