 Welcome! We're going to learn about the difference between the two different types of standard normal tables that you might encounter. You're going to see two different types of z-tables. Remember, the z-table is the normal distribution, special case of it, with a mean of zero and a standard deviation of one. Well, they're two kinds of z-tables. Some text book, she's one of them, and some use the other. It's good to know how to use both of them. This is the table we've been using. We might call it the zero to z normal distribution table. It shows you the area under the standard normal curve of the z-distribution between zero, the mean, and a particular z-value. And we've been using it in two ways. You have a z-value, you look up the area under the curve, that's the shaded blue area in the picture, or if you have the probability, the area under the curve, you will go into the table with that and you look up the closest z-value. For example, if I tell you the weight of adult men is normally distributed, and these are the symbols of normal distribution. If I tell you that it's normally distributed with a mean of 150 pounds and a signal of 10 pounds, what's the probability that a randomly selected male will weigh between 140 and 155 pounds? Again, we've never asked you what is the probability that an adult man weighs 150 pounds. Technically, the answer is zero. Nobody in the world, nobody on the planet earth, weighs exactly 150 pounds. This is a continuous measurement. 150 pounds means 150, but 8 million zeroes after it. Nobody weighs back. So technically, even though you think you weigh 150 pounds, that's rounded. So we never asked that kind of question, but we might ask between 140 and 155. Remember that we're not asked. That's an interval. Well, to get the answer, we're going to use the normal distribution, z, the standard normal distribution, and notice asking the question between 140 and 155. We can't do that directly. So we'll do 150 to 155 and 140 to 150. Now, of course, we don't have a table that has pounds and those kind of numbers. We have to convert everything into the z-score. So we're going to talk about x-scores. That's going to be in the original units, like pounds, and the z-score, z-value, which is never in units, pure number. So you don't talk about pounds or dollars. It's a pure number. So we're going to convert the 140 into a z-score. Well, using the formula that z equals x minus mu over sigma, so z of 140 becomes 140 minus 150 over 10, and those pounds over the pounds give a pure number. It's minus one. You've got a z-score of minus one, or in other words, saying this. You're minus one standard deviation from the mean. Well, minus one is the same as plus one. It's symmetric. And if you look at your table from zero to one, you'll find you have .3413. That's the amount of area that comes out of the z-table. What about 155? Well, 155 to 150, if you convert that into a z-score, 155 minus 150 over 10, that's plus .50. Half a standard deviation from the mean. Looking at the table for .50, you find the area under the curve between zero and .5 is .1915. Now, we're going to add this up. It's like little pieces. You'll have to add up because we know the area under the entire curve is one. That makes it a probability distribution. So we add .3413 to .1915. We add .5328. Roughly 53.28% of adult men will weigh between 140 and 155 pounds. We're in another way of doing this. If I ask you, what is the probability that a randomly selected adult man will weigh between 140 and 155? Well, that's .5328. We have the other type of standard normal table. It's called the cumulative standard normal table. It gives you areas under the z-distribution, under the standard normal curve, between negative infinity to a z-value, to a particular z-value. So the one on the left gives you the area under the curve up until a z-value that's negative. And the table on the right gives you the area under the curve from negative infinity up until a z-value on the positive side of the distribution. Okay, as we explained, the cumulative z-table provides the area underneath the curve from minus infinity to a specific z-value. The other z-table that we used, that shows you the area between zero and z. Now, a z-score of zero always puts you at the 50th percentile, right? And a positive z-score means you're above the 50th percentile. A negative z-score means you're below the 50th percentile. And again, as explained before, there are two cumulative standard normal distribution tables. One for negative z-scores and one for positive z-scores. Well, I will show you how to solve a problem through either table. For example, see, one of the areas between minus infinity and minus 170. Well, you can find it directly in the cumulative z-table. If you go to the cumulative z-table, the one that shows the negative values of z, you'll see directly that if you go to minus 1.70, the answer is 0.0446. On the other hand, alternatively, to use the zero to z-table, then what you'd have to do is you'd have to go to... You have to understand how the zero to z works. You get the area between zero and the z-value. And you'll note the area between zero and minus 170. Well, you know, it's the same as the area between zero and plus 170. It's 0.4554. So if you went over here on the left tail, you subtract 0.50. That's half the distribution that goes from minus infinity to zero. And you subtract. You do 0.500 minus 0.4554, and you get the same answer of 0446. Obviously, it's a lot easier to use the cumulative z-table for this kind of problem. As we can see, the problems in normal distribution probabilities involving percentiles, and you've done a few of them by now, are truly easier to do with the cumulative z-distribution table because it's already in the form that you want. It's in the form of percentiles, negative infinity to z-value, as opposed to having to do subtraction, subtracting from 0.5 and getting the tail probability. Here's an example. Suppose that you take a standardized exam, and the scores of this exam follow a normal distribution. Whatever grade you got, when you convert it to a z-score to standardize it, you get a z-score of negative 0.53. From the table, trying to figure out the percentile that you're at, you see that the probability of getting a score between negative infinity and negative 0.53, the z-value, the standardized value of your score, is 0.2981, that's the direct table value. So what I tell you is approximately 29.81% of the scores are lower than yours, and you're at the almost 30th percentile. Here are some more problems for you to do on your own using the cumulative z-table.