5.1 – Calculating Z-Scores for Individual Scores

The z-score is a statistic that measures how many standard deviations an individual’s score is away from the mean. It is essentially a deviation (which tells you how far above or below the mean) that is put on the scale of the standard deviation. The formulas are as follows:

So let’s go back to our example where the mean on the test is μ = 75, and the standard deviation is σ = 20 (we are assuming at this point that this group of scores is a population), you can then calculate the z-score for your test score of 65:

[latex]z=\frac{X-\mu}{\sigma}=\frac{65-75}{20}=\frac{-10}{20}=-0.50[/latex]

This z-score of -0.50 tells you that your test score of 65 was half (0.50) a standard deviation below (-) the mean. In other words, the number tells you how far a score is from the mean in terms of a standard deviation, and the sign (+/-) tells you whether the score was above (+) or below (-) the mean.

If we calculate the z-score for the alternate exam version where the standard deviation instead was σ = 5, we get the following:

[latex]z=\frac{X-\mu}{\sigma}=\frac{65-75}{5}=\frac{-10}{5}=-2.00[/latex]

This z-score of -2.00 tells you that your test score of 65 was two (2.00) standard deviations below (-) the mean.

Using a z-score to Determine the Individual’s Score (X)

Sometimes, however, we might want to calculate an individual’s score from a given z-score. As long as we know the mean and the standard deviation, we can do this using one of these formulas:

So, let’s say that an individual took a test that had a mean of μ = 50 and a standard deviation of σ = 10, and we were told that their z-score was z = +1.00.

We can use some simple reasoning to figure out their actual score (X). Based on the z-score of z = +1.00, we know that this person’s “score is one standard deviation above the mean.” We were told that the standard deviation is σ = 10, so we now know that the person’s score should be 10 points above the mean. If the mean is μ = 50, then their score is 50 + 10 = 60.

However, we can simply plug the numbers into our formula:

[latex]X=\mu + z\sigma=50 + (+1.00)(10)=50 + 10 = 60[/latex]

So, let’s say that an individual took a test that had a mean of μ = 50 and a standard deviation of σ = 10, and we know that their z-score was z = -1.50.

We can use some simple reasoning to figure out their actual score (X). Based on the z-score of z = -1.50, we know that this person’s “score is 1.50 standard deviations below the mean.” If the standard deviation is σ = 10, we can then calculate that one and one-half (1.50) of a standard deviation of 10 is 1.50 x 10 = 15. This then tells us that the person’s score should be 15 points below the mean. If the mean is 50, then their score is the mean, μ = 50, minus 15, or 35.

Again, we could also simply plug the numbers into our formula:

[latex]X=\mu + z\sigma=50 + (-1.50)(10)=50 - 15 = 35[/latex]

Again, we determine that the person’s score on the test was X = 35. And we also know that the score of  35 is one and one-half standard deviations below the mean.

Populations vs. Samples

You may note that all of our examples above used the population formulas for z-scores.

Thankfully, the z-score formula doesn’t differ between populations and samples (except for the symbols), so the calculations are the same.