Z-scores: The most versatile statistic for media research

z-scores

While there are dozens of statistical methods available to analyze data, the simple z-score is probably the most versatile.  The following information is from Roger Wimmer's The Research Doctor Archive: 

What are z-scores?

Whenever we conduct a test of any kind (e.g., tests in school, music tests, personality ratings, or even Arbitron ratings), we collect some type of scores. The next logical step is to give meaning to these numbers by comparing them to one another. For example, how does a vocabulary test score of 95 compare to a score 84? In a music test, how does a song score of 82 compare to a score of 72? And so on. Without these comparisons, the scores have no meaning.

Although there are different ways to compare scores to one another (e.g., percentile ranks), the best way is to determine each score's standard deviation (or "average difference") above or below the mean of the total group of scores. This placement in standard deviation units above or below the mean is called a z-score, or standard score.

But wait. What is standard deviation? To understand that, you need to understand another term—variance. In simple terms, variance indicates the amount of difference that exists in a set of scores or data—how much the data vary.  If the variance, which differs from one test to another, is large, it means that the respondents did not agree very much on whatever it was they were rating or scoring.  Obviously, if the variance is small, the respondents agreed (were similar) in their ratings or scores.

The standard deviation is the square root of the variance. The advantage of the standard deviation is that it is expressed in the same units as the original scores. For example, if you test something using a 10-point scale, the standard deviation will exist somewhere between 1 and 10; a 65-point test will have a standard deviation between 1 and 65. And so on.

The standard deviation (SD) is important because it is used in calculating z-scores. What we need are symbols for Score (X), Mean (M), and standard deviation (SD).  With those symbols, the z-score formula is X - M / SD, or subtract the Mean of the group of scores from each individual score, and divide by the standard deviation.  The typical way the z-score formula is shown is:

Z =	X- M
	SD

All z-scores have a mean of zero and standard deviation of 1, and ranges (roughly) between –3.00 and +3.00. A z-score of "0" is average; a positive z-score is above average; a negative z-score is below average.  Here is a picture of a normal curve, showing that about 68% of a sample falls between -1 and +1 standard deviations, and about 98% fall between -2 and +2 standard deviations.

z-scores relate to the normal curve (the bell curve you may remember from when your teacher said that he/she was going to "curve" the test scores). Because of this, we know where things "stand" when using z-scores. For example, about 68% off your scores for a test will fall between –1.00 and +1.00 z-scores (standard deviations) from the mean.

z-scores allow you to compare "apples to oranges." For example, when you conduct a music test, you can't compare the raw scores of the males to the raw scores of the females, or one age cell to another. But you can withz-scores. In addition, if you compute z-scores for a music test, you can compare the scores from one market to scores in another market. You can't do this with the raw scores (regardless of the rating scale you use).

By the way, z-scores are known as standard scores because they are computed by transforming your scores into another metric by using the standard deviation. Get it? standard-ized scores. This procedure is known as a monotonic transformation . . . that is, all of your scores were transformed to another form using the same (monotonic) z-score formula.