# Finding the Probability of Z-Value Ranges

The current page provides a guide on finding the probability associated with certain z-value ranges when assuming a normal distribution.  This guide requires the use of a z-value table.  These can be found in the back of most introductory statistics textbooks or performing a simple google search for a “z-value table.”  If you have any questions or comments about this guide, please email me at MHoward@SouthAlabama.edu.

Sometime researchers and practitioners want to identify the probability of obtaining a certain value while assuming a normal distribution in their data.  For example, someone may want to find the likelihood of obtaining a value that is one standard deviation less than the mean.  Or, they might want to find the likelihood of obtaining a value that is two standard deviations above the mean.  Or, it is even possible to identify the likelihood of obtaining a value between one standard deviation below and two standard deviations above the mean.

To find these probabilities, it is possible to use a z-score table.  A z-score table identifies the probability of obtaining a value LESS THAN a certain z-score, or to the left of the value when visually observing a normal distribution.  So, if we want to find the probability of obtaining a value below a certain z-score, we just find the associated z-score in the table.

However, we can also use the table to identify the probability of obtaining a value above the z-score as well as between two z-scores.  To find the probability of obtaining a value above a z-score, we simply invoke the complement rule (the probability of something not occurring) regarding the probability of obtaining the value below a z-score (the value found in the table).  So, we would just take the number 1 and subtract the probability found in the table, which provides the probability of obtaining a value above the specified z-score.

Lastly, to find the probability of finding a value between two z-scores, we take the probability of finding a value below the upper range and subtract the probability of finding a value below the lower range.  This is because we do not want to include the probability below the lower range.  If this is confusing, that is okay.  We are going to review three examples.

To begin, we identify the range of values that we are interested in.  Is it less than a certain z-value?  Is it greater than a certain z-value?  Is it between two z-values?

Let’s say that we are interested in the probability of obtaining a z-value less than -1.0 (negative one standard deviation from the mean).  To find this probability, we would go to our table and start with the left-hand column that includes numbers to the tenth place (e.g. one decimal point).  We would find the value that corresponds to our value of interest down to the tenth place.  In this example, that would be -1.0.  Then, we would go over to the column that includes the value of interest in the hundredth place (e.g. two decimal points) at the top of the column.  In this example, that would be .00 (-1.0 and .00 combine to -1.00).  Lastly, we find the number that corresponds to the intersection of the -1.0 row and the .00 column, which is .1587.  Therefore, the probability of obtaining a z-value less than -1.0 is about 16 percent.

For the next example, let’s say that we are interested in the probability of obtaining a z-value greater than 1.0 (one standard deviation above the mean).  We would first go to our table and again start with the left-hand column that includes numbers in the tenth place.  We would find the value that corresponds to our value of interest down to the tenth place.  In this example, that would be 1.0.  Then, we would go over to the column that includes the value of interest in the hundredth place at the top of the column.  In this example, that would be .00.  We then find the number that corresponds to the intersection of the 1.0 row and the .00 column, which is .8413.  Lastly, because we are interested in the probability of obtaining a z-value greater than 1.0, we find the inverse of this probability by invoking the complement rule.  We take 1 and subtract .8413 to obtain .1587.  Therefore, the probability of obtaining a z-value more than 1.0 is about 16 percent.

For the last example, let’s say that we are interested in the probability of obtaining a z-value between -1.0 and 1.0 (between one standard deviation below and above the mean).  We would first find the upper bound of this range, and go to our table to find the value that corresponds to our value of interest down to the tenth place in the left-hand column.  In this example, that would be 1.0.  We then go over to the column that includes the value of interest in the hundredth place at the top of the column.  In this example, that would be .00.  We then find the number that corresponds to the intersection of the 1.0 row and the .00 column, which is .8413.  However, we now need to find the lower bound of our range.  To do so, we follow the same steps and find the intersection of -1.0 and .00 in our table, which would provide a value of .1587.  Lastly, we take the probability associated with our upper range and subtract the probability associated with our lower range, which would be .8413 + .1587 = .6826.  Or, about 68 percent.

That’s all for finding the probabilities associated with obtaining a certain z-value.  IF you are still confused, feel free to email me at MHoward@SouthAlabama.edu!