How to Calculate Z Scores

A Z score allows you to take any given sample within a set of data and to determine how many standard deviations above or below the mean it is. To find the Z score of a sample, you'll need to find the standard deviation and mean of a set of data, find the difference between a value in the sample and the mean, and divide it by the standard deviation. If you follow these steps, you'll see that calculating the Z score isn't as tricky as it sounds.

STEPS

1. 
   Collect a meaningful amount of samples of your variable of interest. It's important to gather a large number of samples of the variable of interest to make sure that all reasonable variations from the average are covered. Samples should be randomly chosen. Let's say your sample of interest is the height of palm trees in Florida, and that you've collected the measurements of 25 different trees. The different trees measure 7 feet, 8 feet, 8 feet, 7.5 feet, and so on.
   • Keep in mind that if the sample of interest is the height of palm trees, measuring only palm trees in Florida will give an answer that is meaningful to Florida palm trees only. Palm trees selected randomly around the world should be chosen to arrive at an answer that is meaningful for palm trees as a flora.
   • Make sure to pick an appropriate sample size. The sample must be large enough to give a meaningful answer, but this does not mean that every palm tree in the world must be measured as a sample. The need for the most accurate answer possible must be weighed against the enormous mathematical task of taking all possible samples in to account. Measuring just 2 or 3 palm trees won't give you a meaningful result; however, measuring 2,000 or 3,000 palm trees may be difficult.
   • There is no hard and fast answer for the sample size you need to use; it is basically dependent on how accurate the answer needs to be. Consult a statistical textbook or online university presentation to get an idea of what sample size is needed to get the desired accuracy.
2. 
   
3. 
   Find the sample mean. Add together the values of all of the samples. Divide this sum by the number of samples used. This number is the average, or mean, value. Just add up all of the different values of the 25 heights of palm trees and divide this result by the amount of palm trees. The mean is often represented by the symbol x̅. Let's say that you got 199 when you added up the values of all of the samples, and that there are 25 of them. Just divide 199 by 25 to get 7.96, the sample mean.
   • Picture a bell curve with the sample mean, 7.96, right in the center of the curve.
4. Calculate the standard deviation of the sample. This represents how tightly or loosely the values are grouped around the mean. In this example, the standard deviation of the set of data is 0.888819442, which rounds to 0.89. Here's how you should calculate the standard deviation:^[1]
   • Subtract the sample mean from each sample. So, for example, if you subtract the sample mean, 7.96, from the sample 7, you get 7-7.96, or -.96. If you subtract the sample mean, 7.96, from the sample 8, then you get 8-7.96, or .04, and so on. Continue doing this until you've found the difference between each sample and the sample mean. You should have 25 new data points. They will be as follows: -.96, .04, .04, -.46, 1.04, .54, .04, -.96, 1.54, .04, 1.04, -1.46, .54, 1.04, .04, -.96, .04, 1.04, .54, -.96, -1.46, -.96, .54, -.96, and 1.04.
   • Find the variance. The variance is the average of the squared differences from the mean. So, now that you have your 25 differences, you should square them all and then add up the result. You would add up (-.96)² + (.04)² + (.04)² + (-.46)² + (1.04)² + (.54)² + (.04)² + (-.96)² + (1.54)² + (.04)² + (1.04)² + (-1.46)² + (.54)² + (1.04)² + (.04)² + (-.96)² + (.04)² + (1.04)² + (.54)² + (-.96)² + (-1.46)² + (-.96)² + (.54)² + (-.96)² + (1.04)². Once you get the result, then simply divide it by 24 (or N-1, where N = the number of data points in your sample) to obtain the variance of .79.
   • Find the square root of the variance. To get the standard deviation, simply find the square root of .79. √(.79) = 0.888819442, rounding to 0.89.
5. Calculate the Z score. To calculate the Z score of any sample, you just have to plug it in to a simple formula: Z score = (sample value - sample mean)/standard deviation. The result of that division is the Z score of the chosen sample, indicating how many standard deviations away from the mean the chosen sample lies. If your Z score is positive, then it tells you how many standard deviations above the mean the sample lies; if it is negative, then it tells you how many standard deviations below the mean the sample lies. Here is how to calculate the Z score for the sample 6.5 feet:
   • Subtract 7.96, the mean, from 6.5, the sample value. 6.5 - 7.96 = -1.46
   • Divide -1.46 by .89, the standard deviation. -1.46/.89 = -1.64
   • This means that the sample 6.5 falls -1.64 standard deviations below the mean.
6. Consider using a Z table to convert the Z score to a percentage. You can easily find a table in a textbook or online. This is another way of interpreting the data. If you'd like to take this a step further, check out the value of -1.64 on a Z table to see what percentage of the samples fall below 6.5. Look it up under the row of -1.6 and column of .04 and see that you get .0505 as the "p value." This gives you a 5% chance that a sample will fall below or on 6.5. You can see that this is not a high percentage because all but 2 of the samples were higher than 6.5.
   • This can be useful if you want to compare your test results to a larger population. Let's say you calculated the standard deviation of your result on the Math component of the SAT to have a standard deviation of .736. Look it up on the Z Table and get .7673, or roughly .77. This means that if you look up your score on the bell curve, that 77% of people would have scored on or above it.