7.1 – Applying Z-scores and Probability to Groups of Scores

At the end of the previous chapter, we looked at how we can use z-scores and probability to explore answers to research questions by examining the probability of an individual having a particular range of scores on any measurement. To do this, we would follow this process:

1. Convert the score to a z-score.
2. Draw the distribution of scores and shade the necessary part of the distribution to determine the type of proportion shaded (body, tail, or slice).
3. Look up the correct proportion (which is the same as the probability) in the Unit Normal Table.

In order to follow this process, we needed to know 3 things:

• The scores are normally distributed (this allows us to use the Unit Normal Table)
• The mean score for the group to which the individual is compared
• The standard deviation for the group to which the individual is compared

This process is an incredibly useful tool to help answer research questions. However, the vast majority of research is not performed on a sample of one individual but is rather performed on a sample of a group of individuals.

Remember that research is usually performed on a sample, rather than the entire population, and the ultimate goal is to be able to take whatever happens with the sample and try to generalize it to population.  Research on one individual increases the likelihood that the results are ultimately due to the individual participant and not the treatment. Showing that one person responded well to the treatment provides insufficient evidence that everyone else in the population would respond in the same way (this is called replication). A big problem with a small sample (the smallest sample being one individual) is that the sample is not going to be representative of the population. By sampling more than one person, we increase the odds that our sample is representative.

However, if we sample a group of individuals for a research study, the process we learned in the last chapter won’t work because it is intended for looking at whether individual scores are unlikely. Thus, we will need to adjust our z-score formulas a bit to handle samples that involve more than one individual.

Let’s return to the research study on the impact of the newly developed “math pill” that we explored at the end of the last chapter.

Let’s say that this time the researcher wants to perform a more rigorous research study and thus they decide to sample n = 9 participants for the study. (While having n = 9 people in the sample is not that much larger than a sample of n = 1, for now, it will help us to understand how to use samples of a group without having too many numbers with which to work.)

As with the previous version of the “math pill” study, the researcher will give each of the n = 9 participants the pill and then have them each take a standardized math achievement test with the following characteristics:

• Scores are normally distributed
• Have a mean of μ = 100
• Have a standard deviation of σ = 15

Once the testing is completed, the n = 9 participants have the following scores:

At this point, it appears that many of the n = 9 participants have somewhat higher than average scores on the math test (remember that the mean for the test is μ = 100), which seems to support the researcher’s belief that the “math pill” improve math performance.

But if the researcher tries to apply the process from the last chapter to explore the probability of getting these scores by chance they run into an obstacle. There isn’t just one individual score; there are nine! And our goal with this version of the research study is not to determine if an individual’s score is extreme, but instead to determine if the group of n = 9 participants’ scores are extreme, as a group.

This is where statisticians had to rework the process of using z-scores and probability to help answer the research questions like this one where you sample a group of people to participate in a research study. In order to accomplish this, we will need to explore two new topics:

• Distribution of Sample Means
• Central Limit Theorem

We will cover those topics in the next two sections.