Z-scores and Probability

Two facts from the Z-distribution demonstration will make it easy to calculate the probability of a certain value in a distribution. First, we can express any value in a distribution as a Z-score, a number that indicates how many standard deviations from the mean a score lies. Second, we know the probability of observing a value based on how many deviation from the mean a score lies. Thus, knowing the Z-score will allow us to calculate the probability of observing that value. If a score is within one standard deviation away from the mean (Z=-1 or +1), then we know that it is a fairly likely score. We know this because over 65% of our distribution will fall within one standard deviation of the mean. We can make more precise probability statements if we refer to a table of values that represents the probability of observing a value based on the Z-score. The Z-table we use shows the area of a distribution we would expect to observe between the mean and any Z-score.

In the following examples, the mean time it takes expectant mothers to locate a baby face in a crowd is 77 milliseconds. There is a standard deviation of 10 milliseconds for the recognition of the babies faces. What proportion of expectant mothers took an average of 90 (X=90) milliseconds or less to recognize the babies faces?

The previous example demonstrated finding the probability of observing a known X value. We started with an X value found the corresponding Z-score and used the Z-table to find the probability. We can also find raw scores for certain percentages of the population (working the problem in reverse order). Perhaps we know the mean and standard deviation of a population, for example, and want to find the raw score that corresponds to a certain percentage of the distribution. Now we start with the probability, find the Z-value and solve for X.

In the following example the average miles per gallon (MPG) a Ford motor car gets is 23 with a standard deviation of 5. How many miles to the gallon does the top 10% of Ford cars get?