Review of Z Scores

The normal distribution describes a theoretical distribution of values that follow a specific mathematical formula. Although normal distributions may have different means and standard deviations, all normal distributions are "bell-curve" shaped, symmetrical with a peak at the mean (see Figure 1 for examples). Tails of a normal distribution are asymptotic, indefinitely decreasing but never touching the x-axis. The total area under the standardized normal curve sums to 1.00 (i.e., 100%).

Figure 1. Three normal distributions whose means and standard deviations vary

Some measurements in the natural world may approximate normal distributions (e.g., perhaps the weights of adult hippopotamuses, heights of palm trees, students' IQs, and people's happiness). The normal distribution may characterize distributions of individual data points in some populations of scores, a large sample drawn from such a population, or the theoretical distribution of sample statistics such as the mean. For more information on the normal distribution and its history, see this article from Wikipedia.

• The normal distribution is important in inferential statistics because certain theoretical distributions, such as the distribution of possible means, can be very close to normal even when the population distributions are not normal.
• By using the areas underneath normal distributions, we can calculate probabilities of different outcomes, including how likely it is to obtain a mean within a certain range.

Important note: Before we use the normal distribution to compute probabilities, we must verify that the shape of the distribution of interest is very close to normal.

Standard Normal Distributions and Z Scores

A normal distribution that is standardized (so that it has a mean of 0 and a SD of 1) is called the standard normal distribution, or the normal distribution of z-scores. If we know the mean m ("mu"), and standard deviation s ("sigma") of a set of scores which are normally distributed, we can standardize each "raw" score, x, by converting it into a z score by using the following formula on each individual score:

A z score reflects how many standard deviations above or below the population mean a raw score is. For instance, on a scale that has a mean of 500 and a standard deviation of 100, a score of 450 would equal a z score of (450-500)/100 = -50/100 = -0.50, which indicates that the score is half a standard deviation below the mean.

• Note that converting x scores to z scores does NOT change the shape of the distribution. The distribution of z scores is normal if and only if the distribution of x is normal.

By converting normally distributed scores into z scores, we can ascertain the probabilities of obtaining specific ranges of scores using either a table for the standard normal distribution (i.e., a z table) or a calculator like the WISE p-z converter.

Caution: It is not appropriate to use the z table to find probabilities unless you are confident that the shape of your distribution of interest is very close to the normal distribution!

Calculating a Z Score and Probability

Q1: Suppose that SAT scores among U.S. college students are normally distributed with a mean of 500 and a standard deviation of 100. What is the probability that a randomly selected individual from this population has an SAT score at or below 600?

Solve this yourself now. You may use either a z table of probabilities or the p-z converter to find the desired probability.

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