Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

6.6 Interpreting and Using ZScores

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Digging Deeper into Group Models

segmentChapter 9  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 10  The Logic of Inference

segmentChapter 11  Model Comparison with F

segmentChapter 12  Parameter Estimation and Confidence Intervals

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
6.6 Interpreting and Using ZScores
How ZScores Are Different From Standard Deviation
Standard deviation (SD) is roughly the average deviation of all scores from the mean. It can be seen as an indicator of the spread of the distribution. A zscore uses SD as a sort of ruler for measuring how far an individual score is above or below the mean.
A zscore tells you how many standard deviations a score is from the mean of its distribution, but doesn’t tell you what the standard deviation is (or what the mean is). Another way to think about it is that a zscore is a way of comparing a deviation of a score (the numerator) to the standard deviation of the distribution (the denominator).
Let’s use zscores to help us make sense of our Thumb
data. Calculate the zscore for a 65.1 mm thumb.
require(coursekata)
# this saves the mean and standard deviation of Thumb
mean < mean(Fingers$Thumb)
sd < sd(Fingers$Thumb)
# write code to calculate the zscore for a 65.1 mm Thumb
mean < mean(Fingers$Thumb)
sd < sd(Fingers$Thumb)
(65.1  mean) / sd
ex() %>% {
check_output_expr(., "(65.1  mean) / sd")
}
0.572534942855165
A single zscore tells us how many standard deviations away this particular 65.1 mm thumb is from the mean. Because the standard deviation is roughly the average distance of all scores from the mean, it is likely that most scores are clustered between one standard deviation above and one standard deviation below the mean. It is less likely to find scores that are two or three standard deviations away from the mean. Zscores give us a way to characterize scores in a bit finer way than just bigger or smaller than the mean.
Using ZScores to Compare Scores From Different Distributions
One more use for the zscore is to compare scores that come from different distributions, even if the variables are measured on different scales.
Here’s the distribution of scores for all players of the video game Kargle again. We know that the distribution is roughly normal, the mean score is 35,000, and the standard deviation is 5,000.
Her zscore is +2. Wow, two standard deviations from the mean! Not a lot of scores are way up there.
Now let’s say you have another friend who doesn’t play Kargle at all. She plays a similar game, though—Spargle! Spargle may be similar, but it has a completely different scoring system. Although the scores on Spargle are roughly normally distributed, their mean is 50, and the standard deviation is 5. This other friend has a high score of 65 on Spargle.
Now: what if we want to know which friend, in general, is a better gamer? The one who plays Kargle, or the one who plays Spargle? This is a hard question, and there are lots of ways to answer it. The zscore provides one way.
We’ve summarized the zscores for your two friends in the table below.
Player  Player Score  Game Mean  Game SD  Player ZScore 
Kargle Player  45,000  35,000  5,000  +2.0 
Spargle Player  65  50  5  +3.0 
Looking at the zscores helps us to compare the abilities of these two players, even though they play games with different scoring systems. Based on the zscores, we could say that the Spargle player is a better gamer, because she scored three standard deviations above the mean, compared with only two standard deviations above the mean for the Kargle player.
Of course, nothing is really definite with such comparisons. Someone might argue that Spargle is a much easier game, and so the people who play it tend to be novices. Maybe the Kargle player is better, because even though her zscore is lower, she is being compared to a more awesome group of gamers!