# Z Scores and Percentiles

Z Scores and Percentiles

Mary is a social worker who leads a treatment group of young adults diagnosed with chronic anxiety. Group members are selected for treatment based on their scores on a particular screening device, Anxiety Scale A. This instrument has a mean of 60 and a standard deviation of 12. Those scoring over 72 on Anxiety Scale A are eligible to join the treatment group.

Suppose there is a vacancy in the group that must be filled. Mary refers to her files and finds that she has a client named Jessie who qualifies for the treatment because her score on Anxiety Scale A was 76. However, there is a new client, Marvin, who was referred to the treatment center from another clinic. His records indicate that at his previous treatment center Marvin scored a 68 on Anxiety Scale B, another valid and reliable measuring instrument. Anxiety Scale B has a mean of 52 and a standard deviation of 10.

Mary is forming a new treatment group of young adults with very high anxiety levels. She will only admit those in the 95th percentile of above on Anxiety Scale A.

Question is : What is the lowest raw score that a client must have on Anxiety Scale A in order to be admitted into this new group? Also assuming some clinics use Anxiety Scale B and often refer their clients to Mary, what is the cutoff point (lowest raw score) for these clients to be admitted into the new group?

I also have a chart that is too lengthy to include with this question.. It is areas under the Standard Normal Curve (the z table) it has z scores listed and the percnet that the z equals. If I need to look up certain z scores to give you the percent please email me back with those numbers.. Thank you !!!!!

#### Solution Preview

The same conent appears here in plain text as well as the MS word document, less some formatting.

<br>

<br>First, a quick review on normal distributions so that I know we're speaking the same language:

<br>

<br>We can describe some property (such as anxiety levels) of a population as having a mean m, and standard deviation s. Any given person in this population can have any value x for this property, however it is most likely that they will have a score close to the population mean m. Exactly how close to m is given by the standard deviation s - 68% of the population will have scores within one standard deviation of the mean.

<br>

<br>What if we wanted to compare different things, or different ways of measuring the same thing, such as the two tests for anxiety in this problem? If you measured heights in meters, you'd find that mine was 1.73 m, a mere .05 m below the average. But if you measure in inches I'm 68 inches tall, a whole 2 inches below the average. The different size of the units makes it difficult to tell exactly where I am relative to the rest of the population. What's handy is to convert everything to a normalized system of measurement, so we use the standard deviation.

<br>

<br>Let's say the standard deviation of heights is 10 cm or 4 inches. Then in both cases I would be .5 standard deviations below average, which isn't too bad at all.

<br>

<br>This is the essence of the z-score. You simply find the difference between your particular subject's value and the mean ...