About the Two Different Standard Normal (Z) Tables Each slide has its own narration in an audio file. For the explanation of any slide click on the audio icon to start it. Professor Friedman's Statistics Course by H & L Friedman is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 

Two Tables, same answer. You will see two different types of Z Tables. Some textbooks use one, some use the other, some include both for reference and give you a choice. It is best to become familiar enough to use both types of Z tables. This will help you to better understand using normal probabilities and making inferences using the standard normal distribution. The Two Z Tables

The “0 to z” Standard Normal Distribution Table Areas Between 0 and a Z-value The Two Z Tables

Normal Distribution Source: Levine et al, Business Statistics, Pearson. The Two Z Tables

Example: Weight If the weight of males is N.D. with μ=150 and σ=10, what is the probability that a randomly selected male will weigh between 140 lbs and 155 lbs?   [Important Note: Always remember that the probability that X is equal to any one particular value is zero, P(X=value) =0, since the normal distribution is continuous.]  The Two Z Tables

Example: Weight Solution: Z = (140 – 150)/ 10 = -1.00 s.d. from mean   Z = (140 – 150)/ 10 = -1.00 s.d. from mean Area under the curve = .3413 (from Z table) Z = (155 – 150) / 10 =+.50 s.d. from mean Area under the curve = .1915 (from Z table) Answer: .3413 + .1915 = .5328 The Two Z Tables

The Cumulative Standard Normal Distribution Table Areas Between -∞ and a Z-value The Two Z Tables

Cumulative Standard Normal Table The Two Z Tables

Cumulative Z-Table The cumulative Z-Table provides the area underneath the curve from -∞ to a specific Z value. The other Z-table we used, shows areas between 0 and Z. A Z-score of 0.00 will place you at the 50th percentile. A positive Z-score means you are above the 50th percentile and a negative Z-score means you are below the 50th percentile. Note that there are two Cumulative Standard Normal Distribution tables, one for negative Z scores and one for positive Z scores. In either case, you are finding the area between -∞ and a specific Z score. The Two Z Tables

Using the Cumulative Z-Table For example, if we want to know the area between -∞ and -1.70, we can find that directly from the Cumulative Z Table, which tells us that it is .0446. Alternatively, since the “0 to Z” Table shows the area between 0 and a z value, we note that the area between 0 and -1.70 (same as the area between 0 and +1.70) is .4554. Thus, the area under the Z distribution between -∞ and -1.70 is .5000 - .4554 = .0446. The Two Z Tables

Cumulative Z-Table: Percentiles You can easily use the Cumulative Z Table to determine percentiles of the normal distribution. For example, Suppose that your grade on a standardized test (whose scores follows a normal distribution) converts to a Z-score of -0.53. From the table, the probability of getting a score between -∞ and -0.53 is .2981. Approximately 29.81% of the scores are lower than yours. In effect, you are almost at the 30th percentile. Using the 0-to-Z table, the area between 0 and +0.53 (which is the same as the area between 0 and -0.53) is .2019. For the percentile, .5000 - .2019 = .2981. Say your Z-score on the standardized test is +1.75. This converts to a cumulative normal probability of .9599. This means that the area between -∞ and +1.75 is about 96% and your score puts you at approximately the 96th percentile. Using the 0-to-Z table, the area between 0 and +1.75 is .4599; add the area between -∞ and 0.000 which is .5000 and you get .9599. The Two Z Tables

Cumulative Z-Table: Examples Example 1: If your score on a standardized test that follows a normal distribution is +2.17, you are at which percentile? Answer: .985 or 98.5%. Only 1.5% of people taking the test get better scores. Example 2: If your score on a standardized test that follows a normal distribution is -1.34, you are at which percentile? Answer: .0901 or 9.01%. 91.99% of people taking the test get better scores. The Two Z Tables