How to Calculate Z Score & Probability
- 1). Measure the feature of interest in the population which you are studying. As an example, consider a class of 100 students (the population) who have written a test. You would collect the marks (the feature of interest) of all 100 students on that test.
- 2). Calculate the average value (or "mean") of the feature of interest for your population. This is done by adding together all measured values and dividing by the number of values. An average test score of 73.8 percent will be used for the example.
- 3). Calculate the standard deviation - sigma - for the population. This can be done by hand but it is much easier to find sigma using a standard scientific calculator or a software program such as Excel. The standard deviation is a measure of the amount of variation, or spread, in the population values. The sigma value which will be used for the test scores example is 8.6 percent.
- 4). Determine the value of the population feature for which you want to calculate a probability. The Z test will give you the probability of finding a feature in the population which is either higher or lower than your chosen value. In the case of the test scores, assume that you wish to know the probability that a student scored 90 percent or higher on the test, so the value you choose would be 90.
- 5). Calculate the Z score for your chosen value. This is done by subtracting your chosen value from the population average, then dividing that difference by the sigma for your population. So the Z score for a mark of 90 percent would be equal to (90 - 73.8)/8.6 = 1.88.
- 6). Look up your calculated Z score on a table of standard Z values. To do so, look down the leftmost column of the table until you find a value equal to the first two digits of your Z score (1.8 in the example). This locates the row you will use. Look along the uppermost row until you find a value equal to the second decimal place digit of your Z score (0.08 in the example). This locates the column you will use. Where your column and row intersect, read off the number value in that location. In the test example, a Z score of 1.88 corresponds to a reading of 0.4699. This number is the probability (in decimal form) of finding a student having a test score between the average score and your chosen value of 90 percent.
- 7). Subtract the probability just determined from 0.50. This is necessary because the Z table always gives the probability of finding a value between the mean and your chosen value, and you instead want to know the probability of exceeding your chosen value. The reason for subtracting from 0.50 is that the total probability of any score being on one side of the mean is 50 percent. For the example, the value you would calculate would be 0.50 - 0.4699 = 0.0301.
- 8). Multiply the number you just calculated by 100. This is the percent probability of finding a value in the population at or above your chosen value. So the probability that any given student scored at or above 90% is 3.01 percent. The probability of a mark below 90% would simply be 100 minus this value, or 96.99 percent.
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