A normal distribution is a specific mathematical distribution that is rarely observed in real data, but yet it has tremendously important statistical applications. As we will see, even when a real population of data does not have a normal distribution, the sampling distribution of means may be very close to a normal distribution.

When we have a normal distribution and we know the mean and the standard deviation of that distribution, we know everything there is to know about the shape of that distribution. We can calculate the probability of observing a score within any specific range.

Solve this yourself now. _____ The solution follows.

A score of 600 is one standard deviation above the mean. Using a formula to calculate the z value, we find z = (x - m )/s = (600 - 500)/100 = +1.00; z is the number of standard deviations that the score of interest differs from the mean. Now we can use a z table of probabilities (areas under the standard normal distribution) to find the desired probability. From the z table (available in your introductory statistics book) we find that the probability that a randomly selected z score in a normal distribution will exceed z=1.00 is .1587 or about 16%. This is equivalent to the probability that a randomly selected individual from this population will have an SAT score over 600 is about 16%. The probability that a randomly selected individual from this population will have an SAT score at or below 600 is 100% - 16% = 84%.

Exercise: What is the percentile score for a person in this population who has an SAT score of 650? _____ Solve this yourself now. The solution follows.

A score of 650 corresponds to a z score of (650 - 500)/100 = 150/100 = 1.50. An SAT score at or above 650 in this normally distributed population is as likely as a z score at or above 1.50 in a standardized normal distribution. We can consult a z table to find this probability, which is .0668. This tells us that 93% of the distribution is at or below this z value. Thus, an SAT score of 650 is at the 93^rd percentile in this population.

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

• Go to the Sampling Distributions applet
• An overview of the applet
• Exercise 1: Accuracy of sample means
• Introduction to the Applet
• Review of z-scores
• Review of Central Limit Theorem
• Answers for Exercise 1 hand calculation problems
• Follow up questions: Questions to assess knowledge
• Download the exercise in Rich Text Format or as an Adobe Acrobat pdf

Questions, comments, difficulties? Please contact Dale Berger.