In statistics, a collection of numbers that has a bell shaped distribution, symmetrical around an average value, is said to be "normal." You can change any value in such a data set to a z score, which is a measure of how many standard deviations the value is from the average of the set. You can then use this z score to calculate the probability that a randomly chosen number out of your data set is either below or above the value you converted to z. Once you have a z score, you can convert it to a probability by finding your value of z on a table of standard values.
Things You'll Need
- Z table
Determine what type of probability you want to find from your z score. You can find the probability that a random number in your data set is either above or below the value you used to calculate the z score. For example, if you have a data set consisting of the heights of a group of people, you may want to find the probability that a given person in that group is above 6 feet tall.
Find the probability value associated with your z score on a standard table of z values. To do this, first look along the leftmost column of the table until you find the first two digits of your z score. This will align you with the table row you need. For example, if you calculated a z score of 2.15 for 6 feet from your height data, you would find the digits "2.1" along the leftmost column and determine that this aligns with the twenty second row.
Find the third digit of your z score in the topmost row of the table -- this will identify your required column within the table. In the case of the height example, the third digit of the z score is 0.05, so you would look for this along the uppermost row and find that it aligns with the sixth column.
Look for the spot within the body of the table where the row and column you have identified meet. Here you will find the probability value related to your z score. In the case of the example, you would look for the intersection of the twenty second row and the sixth column and find the probability there is 0.4842.
Subtract the probability you just determined from 0.5, if you wish to calculate the probability of finding a number in the data set which is greater than your value. The reason for this step is that the z table actually gives the probability of finding a value between the average and your z score. The probability in the case of the example would be calculated as 0.5 - 0.4842 = 0.0158.
Multiply the result of your last calculation by 100 to change it into percentage form. You now have the probability that a randomly chosen number in your normally distributed data set will exceed the value you originally converted to a z score. For the example, the probability that a person chosen at random from your group has a height that exceeds 6 feet is 1.58 percent.
Subtract the percentage you just found from 100 to find the probability that a randomly chosen number will be below the value you converted to a z score. In the example, you would calculate 100 minus 1.58 and therefore there is a 98.42 percent probability that a random person in the group is below 6 feet.
Tips & Warnings
- The z score method of finding probability only works properly if your data set has a normal distribution.