We're going to take an opportunity now to review the standard normal table. Specifically we are going to focus on:
This is the table that we use when you have a normal distribution and you want to find probabilities or percent. The table can be used actually to do four things.
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. What percent of men are over six feet tall? Here's the normal distribution regular. 72 inches is our cutoff value. You want the percent of men that are taller than that.
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.
The image above is called the standard normal curve. The 72 standardizes to positive 1.33 for a z-score.
Below, are standard normal tables in which the area to the left counts as the probability that an observation falls at or below a particular z-score. There is a negative and positive curve and table.
NEGATIVE Below is the negative probability curve and and negative z-table (table attached as pdf below the video for your convenience):
POSITIVE: Below is a positive probability curve and positive z-table, which looks like this (table attached as pdf below the video for your convenience):
Our z-score was positive 1.33, so you will look in the positive z-score table. Positive z-scores deal with the tenths place and the hundredths place.
Because your z-score was positive 1.33, you will go to the 1.3 row (tenths) and the 0.3 column (hundredths). At that intersection, you will find 0.9082, which is the areas to the left of 1.33.
But the question was asking the area above. So now you simply subtract from 100% and you get 9.18% of adult men have heights over 72 inches.
You 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.
You 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. About 7% of men are shorter than that.
You can do another type of problem, which is finding the area between two values, between something like 5'6" and 5'9".
Something like this is a little trickier. When you standardize the values of 66, which is 5'6", and 69, which is 5'9", you end up with these two z-scores.
The area corresponding to the z-score of positive 0.33, 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, you 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.
All you do is add the two probabilities. Now, because of the symmetry of the normal curve, you can actually just find one of these two areas and double it. In general, you wouldn't be able to do that.
But what you're going to do is convert both of these to z-scores of negative 0.83 and positive 0.83.
What percent of men are not within 2.5 inches of the mean?
You would find the area below the negative 0.83 z-score, which is 0.2033. Normally you would find the area above the positive 0.83 z-score. But you don't have to do that, because it's the same as the area below the negative 0.83 z-score. Just use the symmetry and double it to obtain about 41% of men being outside that range.
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. Z-scores are used 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. All you have to do is subtract as necessary, maybe from one, or maybe from some other probability.
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.
Source: This work adapted from Sophia Author Jonathan Osters. Image of z-score table CC https://creativecommons.org/licenses/by-sa/4.0/
Source: Adapted from Sophia author Jonathan Osters.
The table that allows us to find the percent of observations below a particular z-score in the normal distribution.