This tutorial will cover z-scores. You’ll learn about:
Z-scores are values that allow you to make one-to-one comparisons between a couple of different distributions.
Standard Scores/z-scores
A value that explains how many standard deviations away from the mean an observation is. It can be positive (if the value is above the mean) or negative (if the value is below the mean).
Oftentimes, you want to compare two things, but it's not really fair to compare them directly. Z-scores help with this.
Imagine you are taking a math class.
1. For the first exam of the year, the class mean was 88 points, and standard deviation was 5. You scored a 92, so you did better than the class average.
2. The second test was much harder. You scored an 80, which is a lot worse than you did the first time, but the class mean was 74. So though your score went down, the class average also went down. The standard deviation, this time, was 4 points.
Did you do better on the first or the second test, relative to your classmates?
It's obvious that you scored higher on the first test, but relative to your classmates, did you do better on the first test or the second test?
Just based on the scores, it's not fair to say that you did on the first test. You want to see how you did relative to your classmates. So z-scores are going to allow you to make this comparison.
Z-scores are sometimes called standardized scores because they're measuring how many standard deviations away from the mean your observation is. In the previous example, for the first test of the year, the standard deviation was 5 for the first exam, and you scored 4 points higher than average, a 92 compared to the average score of 88.
This means that you z-score is less than one: 4 points higher, with a standard deviation of 1 point. So you're less than one standard deviation above the mean.
In fact, more specifically, your z-score is positive 4/5: positive 4 means above the mean, and divided by 5.
How do you get this calculation? You use the formula:
Symbolically, this is written:
The raw score you can call x. And so you can write symbolically z, z for z-score, is equal to x, the raw score, minus mu, the mean, divided by standard deviation, sigma.
Compare the standardized scores by comparing the z-score from the first test to the z-score from the second. You already determined that the first z-score was positive 0.8. So how did you do on the second test?What you should have come up with is that your second z-score is 80:
This is your score on the second test (80), minus the mean for the class average for the second test (74), divided by the standard deviation of the class for the second test (4). This gives you positive 1.5, or 1.5 standard deviations above the mean.
What does this mean? Because positive 1.5 is larger than positive 0.8, your score on the second test was, in fact, better relative to the rest of her class.
The other thing that's worth noting is that a z-score can be negative. If you're subtracting a bigger number from a smaller number, i.e. if the raw score is a smaller number than the mean, then you'll end up with a value that's negative.
If the raw score is below the mean, the z-score will be negative. If the raw score is above the mean, the z-score is positive. If the raw score and the mean are the same, the z-score is 0.
How are z-scores used in standard distributions?
Men's heights follow a normal distribution, with a mean of 68 and a standard deviation of 3. Here is a graph with standard deviations marked:
Imagine, then, converting the 59, 62, 65, etc., into z-scores.
If the standard deviations were converted to z-scores, would that normal distribution look like then?Because the noted numbers are integers of standard deviations away from the mean, and that's what z-scores measure, each can be represented by z-scores.
This normal distribution of z-scores is called the standard normal distribution. Standard, because it's the normal distribution of standardized scores.
Thank you and good luck!
Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS
A value that explains how many standard deviations away from the mean an observation is. It can be positive (if the value is above the mean) or negative (if the value is below the mean).