Review - Normal Distribution: SAT scores example
A normal distribution is a specific mathematical distribution that is rarely observed in real data, 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.
Exercise: Assume that SAT scores among U.S. college students are normally distributed with 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. _____ The solution follows.
A score of 600 is 100 points, or one standard deviation above the mean. We can use a formula to calculate the z value:
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, 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 probability that a sample of 9 randomly selected people have an average SAT score at or below 600? _____ Solve this yourself now. The solution follows.
The standard error (standard deviation of the distribution of possible sample
means) is 
.
Thus , for sample means the z formula is:
A mean of 600 for a sample of 9 people corresponds to a z score of 3.0. Finding a sample mean at or above 600 in this normally distributed population is as likely as a z score at or above 3.0 in a standardized normal distribution. We can consult a z table to find this probability, which is .001. This tells us that over 99% of the distribution is at or below this z value.
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.