How to Convert Z Score to Probability

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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

  • Calculator
  • 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.

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