# Biostatistics for the Clinician ## 1.6 Standard Scores

### 1.6.1 Why Important?

Standard scores or z-scores appear frequently in the medical literature. It turns out that it's a natural question to ask for some value, "How many standard deviations is it from the mean?". The z-score is the answer to the question.

The z-score is particularly important because it tells you not only something about the value itself, but also where the value lies in the distribution. Typically, for example, if the value is 3 standard deviations above the mean you know it's three times the average distance above the mean and represents one of the higher scores in the sample. On the other hand, if the value is one standard deviation below the mean then you typically know it is on the low end of the midrange of the values from the sample. But, there is much more that is important about z-scores. In fact, the z-score opens the door to doing statistical inference for quantitative variables.

Standard Scores
Practice
Exercise 1:
The z-score tells you the distance the value is above or below the mean in:

No Response
Raw score units
Mean units
Standard deviation units
Interquartile range units

### 1.6.2 Z-Scores

For every value from a sample, a corresonding z-score can be computed. The z-score is simply the signed distance the sample value is from the mean in standard deviations. This statement defining a z-score is represented concisely in the simple formula for computing z-scores you see below (see Figure). In the formula x represents the sample value, the greek letter mu represents the mean, and the greek letter sigma represents the standard deviation.

Z-Score Formula Given z-scores, now you can take a whole bunch of data like life expectancies and instantly find values for people that express where they rank with respect to others. In other words, the z-score formula gives you a way of normalizing or collapsing the data to a common standard based on how many standard deviations values lie from the mean. To put it another way. Subtracting the value of the mean from each one of the values and dividing each of these differences by its standard deviation parametizes the original distribution so that it has a mean of 0 all the time and a standard deviation of 1. So, given the shape of the distribution, you can build one table for it. In other words, no matter what your data looks like, no matter what the mean value is, you can reduce it to one standard table by reformulating your data using the z-score formula. You then can take all kinds of experiments and build tables for them because you can normalize it or reduce it by doing things like forming a z-value.

Another way to illustrate this is to present the following problem, "Suppose you have two people. One has an IQ of 130 on the WAIS IQ test which has a mean of 100 and a standard deviation of approximately 10. The other has an IQ of 145 on the Stanford Binet IQ test which also has a mean of 100, but has a standard deviation of approximately 15. According to the IQ tests, who is the smartest?" Given no knowledge of statistics the answer is far from obvious. On the other hand, with z-scores you can quickly calculate that each person has an IQ 3 standard deviations above the mean. In other words, you can quickly use z-scores to find that both have approximately the same intelligence.

Standard Scores
Practice
Exercise 2:
Z-scores provide a common standard for comparison of different measures.

No Response
True
False

### 1.6.3 General Z-Score Properties

Because every sample value has a correponding z-score it is possible then to graph the distribution of z-scores for every sample. The z-score distributions share a number of common properties that it is worthwhile to know. These are summarized below.

Properties of Z-Scores
• The mean of the z-scores is always 0.
• The standard deviation of the z-scores is always 1.
• The graph of the z-score distribution always has the same shape as the original distribution of sample values.
• The sum of the squared z-scores is always equal to the number of z-score values.
• Z-scores above 0 represent sample values above the mean, while z-scores below 0 represent sample values below the mean.

Standard Scores
Practice
Exercise 3:
The mean of the z-scores is equal to:

No Response
0
1
100
68

### 1.6.4 Gaussian Z-Score Properties

Given the z-score properties above, it is obvious that if the sample values have a Gaussian (normal) distribution then the z-scores will also have a Gaussian distribution. The distribution of z-scores having a Gaussian distribution has a special name because of its fundamental importance in statistics. It is called the standard normal distribution. All Gaussian or normal distributions can be transformed using the z-score formula to the standard normal distribution.

Statisticians know a great deal about the standard normal distribution. Consequently, they also know a great deal about the entire family of Gaussian distributions. All of the previous properties of z-score distributions hold for the standard normal distribution. But, in addition, probability values for all sample values are known and tabled. So, for example, it is known then that for any normal distribution, approximately 68% of values lie within one standard deviation of the mean. Approximately 95% of values lie with 2 standard deviations of the mean. Approximately 2.1% of values lie below 2 standard deviations below the mean. Approximately 2.1% of values lie above 2 standard deviations above the mean. In general, all probabilities associated with the normal distribution have already been computed and are tabled (see Figure below).

Standard Normal, Gaussian, or Bell Curve Standard Scores
Practice
Exercise 4:
The percentage of values that lie within one standard deviation of the mean in a Gaussian distribution is approximately:

No Response
2.1%
50%
68%
75%
95%

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