Once upon a time, I graded a test on a curve. Now, I don’t mean that I added extra points to everyone’s scores. Instead, I calculated how each student’s test score compared to the mean test score for all the students. For that particular test, an 85% was the average grade, so it was a C instead of the usual B. The students never asked me to grade on a curve again. In statistics, comparing scores to the mean is both useful and easy. All you have to do is calculate a z-score
How to Find the z-Score
The z-score basically converts an individual raw score into a new score that shows how it compares to the mean. Once we have a z-score for one value, it becomes easier to compare it to other values. Let’s take a look at the formula
The numerator essentially tells us how much the raw score differs from the mean. This is sometimes called the deviant score; it is most often used for standard deviation calculations.
The standard deviation is in the denominator. This indicates that you are converting the deviant score into units of standard deviations. Now, we can compare raw scores based on how many standard deviations they are from the mean instead of the actual value itself.
Any distribution of raw scores can be converted to a distribution of z-scores. This means that instead of the raw score units, like test percentage points, along the x-axis, you would have units of standard deviations. Which is mighty useful for measuring.
Measuring with the z-Score
In my state, the ACT is used as the primary college entrance exam. In a neighboring state, colleges use the SAT. When one of my students wants to go out of state for college, I usually get a question like, “Dr. Clark! I got a 1290 on the SAT, but a 26 on the ACT. Which score is better?” Since we are familiar with the ACT’s 1-36 scale, I have no idea. But that’s where z-scores come into play.
Let’s calculate the z-score of my student’s SAT score. The national average SAT score is 1002 with a standard deviation of 194 points. Using the formula from earlier, the z-score is +1.48. This means that the student’s score is 1.48 standard deviations above the mean. Since the ACT is on a different scale, we will need its z-score in order to compare the two.
But what is the z-score of my student’s ACT score? The national average ACT score is 20.8 with a standard deviation of 5.6. When we change this to a z-score, we get +0.92. This means that my student’s scored less than 1 standard deviation above the mean. The student’s SAT score is the better of the two scores.
But, wait, what did you just say?
A z-score just measures how much a score deviates from the mean in terms of standard deviations. This means that we can compare two raw scores by putting them both in terms of standard deviations. An SAT score that is 1.48 standard deviations above the mean is higher scoring (compared to its mean) than an ACT score that is 0.92 standard deviations above its mean.
So a z-score allows you to compare raw scores, even from different distributions. This is because they are in terms of standard deviations instead of other units. All in all, z-scores are a strong statistical tool worth having and using. happy statistics!