The z-score is the number of standard deviations away from the mean. It allows for standardizing scores so that they can be compared across samples or tests.
Introduction to Z-Scores
Chris took the SATs and got a 2100 out of 2400. Pat took the ACTs and got a 32 out of 36. Who has the higher score?
The SATs and ACTs are measured on entirely different scales, so it's impossible to tell from the raw scores who did better. But, thanks to statistics, with just a little bit of knowledge about the distribution of test scores across the population, we can standardize Chris and Pat's scores by putting them on the same scale. If we know the mean and standard deviation of both tests, we can compare the two scores with respect to how many standard deviations they are above or below the mean. We call this the z-score, the number of standard deviations a given score is from the mean.
Calculating The Z-Score
Let's say the SATs have a mean of 1500, and a standard deviation of 200. That means Chris's score is 600 points higher than the mean, or 3 standard deviations above the mean. Thus, Chris's z-score on the SATs is 3.
Let's say the average ACT score is 24, and the standard deviation is 4. That means Pat's score is 8 points above the mean, or 2 standard deviations above the mean. Thus, Pat's z-score on the ACT is 2.
What if Jean had scored a 20 on the ACT? In that case, Jean's score would be 4 points below the mean, or 1 standard deviation below the mean. Thus, Jean's z-score on the ACT would be -1.
The Formula for the Z-Score
In all three cases, the z-score was found by subtracting the mean from the raw score and then dividing by the standard deviation.
This is read as x minus mu, divided by sigma, where:
x = sample score mu = population mean sigma = population standard deviation
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Returning to the original example, we now know that Chris scored 3 standard deviations above the mean, while Pat scored only 2 standard deviations above the mean. We can therefore say that Chris did better, relative to the population, than Pat. Of course, before we can call that our final conclusion, we have to consider a few caveats.
Whenever we compare two z-scores, we also need to make sure that assumptions underlying the comparisons are valid. First, we are assuming that the population means and standard deviations are coming from a normal distribution, or something close to it. Z-scores are valid only when we are dealing with normal distributions. Second, we are assuming that the measures that we are comparing are testing the same thing. As long as the SAT and ACT measure the same construct, and both are drawn from a normal distribution, we can feel confident saying that Chris did better than Pat on a college admissions test. Otherwise, something is not normal, and/or we are comparing apples and oranges.
The z-score is the number of standard deviations away from the mean, and can be used to compare different scores when the mean and standard deviation of the population are known. These comparisons are based on the assumptions that the distributions of the population is normal, and that the two scores are drawn from tests that measure the same constructs.
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