7.4 – Z-scores for Sample Means

In this section, we are on the path to learning how to find probabilities from a sample mean. We will take the skills that we learned in the previous section, “Finding Proportions or Probabilities from an Individual Score (X),” and now adapt them so that they work with a sample mean (M). 

The main thing that we need to adapt is the z-score calculations. We learned the following formulas for an individual score (X):

[latex]z = \frac{X - \mu}{\sigma}[/latex]

[latex]X = \mu + z\sigma[/latex]

It might help to remember that the numerator (X – μ) measures how much the score (in this case the individual’s score, X) differs from what we expect it to be (which is the population mean, μ). We call this a “deviation.” The denominator is the average amount of difference we can expect an individual’s score to deviate from the mean (which we call the standard deviation, or “average” deviation from the mean, σ).

Suppose we want to look at how a randomly selected student scored on our last exam, and I told you that the exam had a mean of μ = 80 with a standard deviation of σ = 10. You can see that because our research question makes the students in our class the population of interest, we depict the mean and standard deviation as population parameters (using Greek symbols). Now, if I were to randomly select an individual from our class, I can make an educated guess because I know that students score 80 on average (which would be our best guess at this point), plus or minus about 10 points. Using the 68-95-99 rule, I know that about 68% of students scored somewhere between 80 ± 10, or between 70 and 90, while 95% scored somewhere between 80 ± 20, or between 60 and 100.

If I randomly selected a student with a score of 94, I can now calculate the z-score for that student by using the z-score formula that solves for z:

[latex]z = \frac{X - \mu}{\sigma} = \frac{94 - 80}{10} = {+14}{10} = +1.40[/latex]

We can see that the numerator of +14 means that the individual’s score of 94 is 14 points above the mean of 80, which is a deviation of +14. The denominator tells us that individuals in the class as a whole deviate from the mean by an average of 10. That means that some deviate more than 10 from the mean, and some deviate less than 10 from the mean. It also should be noted that the deviations from the mean can be either positive or negative (some scores are above the mean and some are below the mean).

What about sample means?

If we want to calculate z-scores for sample means, we are going to use formulas that are structured in the same way as the z-score formulas for individual scores:

Individual (X)

[latex]z = \frac{X - \mu}{\sigma}[/latex]

Sample Mean (M)

[latex]z = \frac{M - \mu_{_{\!M}}}{\sigma_{_{\!M}}}[/latex]

When looking at the new z-score formula for sample means, hopefully, you can see that it’s not that different from the z-score formula for individual scores. As we discussed above, the numerator ([latex]M - \mu_{_{\!M}}[/latex]) measures how much the score (in this case the sample mean, M) differs from what we expect it to be (which is a thing we call the “expected value of the means,” [latex]\mu_{_{\!M}}[/latex]). Again, we can technically call this a “deviation.” However, it is also a measure of how much sampling error is found in the sample mean. Remember that any difference between a sample mean and the population mean is considered sampling error. The denominator is the average amount of difference we can expect an individual’s score to deviate from the mean (which we will call the "standard error", or the average amount of sampling error we can expect for any sample mean, [latex]\sigma_{_{\!M}}[/latex]).

We can also apply what we learned about the central limit theorem to calculate some of the components in our formula and also simplify the formula. Here, again, is what the central limit theorem tells us:

1. The shape of the distribution of sample means will be approximately normal, as n approaches infinity.
2. The mean of all these sample means will equal the population mean, [latex]\mu_{_{\!M}} = \mu[/latex].
3. The standard deviation of all these sample means can be calculated as the standard error which is equal to: [latex]\sigma_{\!{_{\!M}}} = \frac{\sigma}{\sqrt n}[/latex].

This provides us with the formula to calculate the standard error:

[latex]\sigma_{\!{_{\!M}}} = \frac{\sigma}{\sqrt n}[/latex]

And it also tells us that mean of all the sample means, also known as the “expected value of the mean” ([latex]\mu_{_{\!M}}[/latex] is the same as the population mean for individual scores ([latex]\mu[/latex]): 

[latex]\mu_{_{\!M}} = \mu[/latex]

This last part helps us simplify the z-score formula for sample means to the following:

[latex]z = \frac{M - \mu}{\sigma_{_{\!M}}}[/latex]

And now we have all the components we need to apply z-scores and probability to sample means to answer questions from the real world of research and practice.