Source: Normal Distribution Graph created from public domain http://commons.wikimedia.org/wiki/File:Z_cumulative.svg All other tables and graphs created by Author
We're going to take an opportunity now to review the standard normal table. This is the table that we use when you have a normal distribution and you want to find probabilities or percent. So the table can be used actually to do four things.
The table value itself gives you the percent of observations below a particular z-score. You can also find the percent above a particular z-score by subtracting the table value from 100%, because the table value always gives the area to the left. You can also find the percent of observations between two z-scores by subtracting the table values. Or you can find the percent of values outside of two z-scores by finding both the percent above the higher number and the percent below the lower number, which is sort of a combo of these other options.
So let's take a look. Men's heights are normally distributed with a mean of 68 inches and a standard deviation of 3 inches. The example, what percent of men are over six feet tall? So here's the normal distribution regular. 72 inches is our cutoff value. And we want the percent of men that are taller than that.
We're going to take this normal distribution, centered at 68 with a standard deviation of 3, and convert it into the standard deviation of 1 and mean of zero. This is called the standard normal curve. The 72 standardizes to positive 1.33 for a z-score.
This is the standard normal table in which the area to the left counts as the probability that an observation falls at or below a particular z-score. So our z-score was positive 1.33. We're going to look in the positive z-score table.
Positive z-scores deal with the tenths place and the hundredths place. Because our z-score was positive 1.33, we're going to go in the 1.3 row and the 0.3 column. We're going to find 0.9082, which is the area to the left of 1.33. But the question was asking the area above. So simply subtract from 100% and you get 9.18% of adult men have heights over 72 inches.
We can do a different type of problem by asking, what percent of men are shorter than 63.5 inches? It's the same type of question, except this is what it looks like now. 63.5 falls right between 62 and 65. The z-score ends up being negative 1.5 It's 1.5 standard deviations below the mean of 68.
We can use the negative z-score table, and go to the negative 1.5 row and the zero hundredths column, and find that the probability is 0.0668. So about 7% of men are shorter than that.
We can do another type of problem, which is finding the area between two values, between something like 5'6" and 5'9". So something like this is a little trickier. When we standardize the values of 66, which is 5'6", and 69, which is 5'9", we end up with these two z-scores.
The area corresponding to the z-score of positive 0.33, we look in the positive z-score table at 0.3 row and 0.03 column to find that the area below that is 0.6293. But when you look at the negative z-score table for the negative 0.67 z-score, you find that its probability in the negative 0.6 row and the 0.07 column is 0.2514. The area between is the area below the 0.33 z-score but not below the negative 0.67 z-score. So subtract the table values, and obtain the answer of 0.3779, or about 38% of men are between those two heights.
Lastly, we can find the area outside of a particular region. So what percent of men are not within 2.5 inches of the mean? Well, what does this one look like? It looks like this, where this grey area is the particular area that we want.
So here, all we do is add the two heights. Now, because of the symmetry of the normal curve, we can actually just find one of these two areas and double it. In general, we wouldn't be able to do that.
But what we're going to do is convert both of these to z-scores of negative 0.83 and positive 0.83. We would find the area below the negative 0.83 z-score, which is 0.2033. And normally we would find the area above the positive 0.83 z-score. But we don't have to do that, because it's the same as the area below the negative 0.83 z-score. So we can just use the symmetry and double it to obtain about 41% of men being outside that range.
So to recap, it's possible to use the standard normal table to find the percent of values above or below a particular value, or between two values, or even outside two values. And that's not listed here because it's not done horribly often. And we use z-scores on the normal distribution to do that.
The normal probability table, also called the z-table or the standard normal table, can find these percents by finding the percent of values below. And it always gives you the area below a certain z-score. And all we have to do is subtract as necessary, maybe from one, or maybe from some other probability.
And so that is the standard normal table. And that was a quick review. But we've seen it before. And we're going to use it again. Good luck. And we'll see you next time.
(0:00-0:57) Uses of the Standard Normal Table
(0:58-1:44) Example 1: Men taller than 72 inches
(1:45-2:43) Using the Standard Normal Table
(2:44-3:26) Example 2: Men shorter than 63.5 inches
(3:27-4:52) Example 3: Heights between 66 and 69 inches
(4:53-6:12) Example 4: Heights not within 2.5 inches of 68.
The table that allows us to find the percent of observations below a particular z-score in the normal distribution.