Standard Normal Distribution - Online Tutor, Practice Problems & Exam Prep

Finding Standard Normal Probabilities using z-Table

Welcome back, everyone. So in a previous video, we saw how to find the variance and standard deviation of a dataset, which just basically represents how spread out the data values are. But now we've been talking about discrete random variables and probability distributions, but you may have to find the variance and standard deviation for those. Conceptually, it means the same exact thing. Basically, how spread out are the numbers, the outcomes of your experiments.

But because we're dealing with probabilities, the equation is a little bit different. I'm going to show you how to calculate these in this video, and, basically, it's just going to be this equation over here. It looks a little scary at first, but I'm going to show you that all that happens is that when we're given a discrete random variable, we're going to make a table with columns for \(x \times p(x)\), then \(x^2 \times p(x)\). Alright? Let's go ahead and just jump right into this example, and I'll show you how this works.

Alright? So again, these equations look kinda scary at first, but basically, all that this equation is saying is you're going to take each value, each \(x\), square it, multiply it by its associated probability, add all those things up, and then we're going to subtract the mean squared. Alright? This bottom equation is something that you may see in a book. It's a little bit more conceptually easy to understand.

You take the difference between each value and the mean, then square that, add all those things up, and multiply by each probability. Alright? So it's a little bit different, but, ultimately, you'll get the same exact answer. Personally, I find this first equation is easier to use. That's the one we're going to use in this problem.

Alright? So let's just jump right into our problem. These are numbers we've already seen before. This table here shows the probability distribution for a number of kids that you would find in a random household in a random town. We're going to find the variance and standard deviation of this distribution here.

Okay? So again, when we have these data values that are given to us horizontally like this, first thing you want to do is you want to build a table with the values listed vertically. It's much easier to do it this way. We've already seen actually all of these numbers before, so we're not going to recalculate them. But, basically, if you were to just say each \(x \times p(x)\), you would get this column over here, and you would already even find the mean, which was the expected value, which is 1.1.

Alright? We actually already know what the mean of this distribution is. The mean is just 1.1 or 1.10, which means that we can also find the mean squared, because we just square that number. So, basically, this is just going to be 1.21 if you calculate that. Alright?

So if you look at this equation here, this giant mess over here, we actually already know what this number is. We already know what the mu squared. So, really, all that happens here is we just have to figure out what is the \(x^2 \times p(x)\). And, again, that's why you want to build a little column here for \(x^2 \times p(x)\). Okay?

So I'm just going to take each number and square it and then multiply it by its probability. So we've got \(0^2 \times 0.15\), which is still just 0. Then I've got \(1^2 \times 0.60\), which is still 0.60. Then I've got \(2^2 \times 0.30\), and that's going to equal, oh, I'm sorry. This is supposed to be 25 over here, which is going to equal, and that's going to equal 1.

Great. So if you add up all of these things here inside of this column, what you should get is you should get 1.60. So, basically, what happens is this represents the sum of \(x^2 \times p(x)\). Right? That's basically what this number represents.

That's actually what goes right here in this column. So you should make these little rows here. It's going to help you organize this information. So this is going to be 1.60. Okay?

So now how do you find the variance? Well, basically, what happens is you're going to take this thing over here and you're just going to subtract this thing over here. So we're going to take 1.60 like this, and we're going to subtract the previous number of 1.21. That's going to give us a variance of 0.39. Right?

This is basically this number minus this number. That's how you find the variance. Alright? So that's our first answer. We got 0.39.

And the last question we have to answer is what's the standard deviation? And remember, whenever you have one of these things, you should always figure out the other because they're just related by squares and square roots. The variance is just the standard deviation squared. So we just basically take the square root of this number we just found over here. What you should get is you should get a standard deviation of 0.62.

You may see this rounded to the first decimal place just because all of our data values are integers. So you may see this rounded to 0.6. And, basically, what happens is this has units, the standard deviation over here. The standard deviation would be 0.6, you know, kids per household or something like that. Okay?

So these are your two answers. You've got a variance of 0.39 or 0.4, and then we've got 0.6 kids. Alright? So, basically, what happens is, again, this represents the spread of the outcomes of this experiment. We've got a mean of 1.1, that's the expected value, with a standard deviation of about 0.6 kids.

Alright? That's it for this one. Now that we understand a little bit more about this, let's go ahead and take a look at some practice. Thanks for watching.

Find the area under the standard normal distribution to the left of a z-score of $1.21$.

• A. 0.1131
• B. 0.8869
• C. 0.8849
• D. 0.1151

Find the area under the standard normal distribution to the right of a z-score of $-0.44$.

• A. 0.3300
• B. 0.6700
• C. 0.6654
• D. 0.3346

Probabilities Between Two Values

Welcome back, everyone. So in the last couple of videos, we saw how to find probabilities using given z scores in a z table. Sometimes we were asked for an area to the left and sometimes an area to the right. But there's actually a third type of problem that's very commonly asked. We actually have to calculate a probability that is between two z scores using a z table, something like between negative 0.64 and 2.27.

I want to show you how to do that in this video, and, basically, we're going to use pretty much the same operation that we did, over here. We're going to draw out some of our curves. We're going to look up some z scores, and then we're going to have to do some stuff with the probabilities. And, basically, what that operation is is we're just going to subtract the area of the smaller z score from the larger. That's all we have to do.

So let's just jump right into our problems here and kind of recap what we've learned so far. Let's get started. So with an area to the left, something like p where your z is less than negative 0.64, we just looked up a z score after drawing our curve and shading the area. And when we found that area, which was 2.2611, we didn't have to do anything else. We could just read that right off of the z table, and that was our answer.

So our answer here was 0.2611. Now with an area to the right, we basically did a similar procedure. We have to look up a z score of 2.27. We use our z table to find the probability it was 0.9884, but that wasn't our answer because that corresponds to an area that's from the left all the way up until that score of 2.27. To get the complement, we just take 1 minus that number, and then this was our answer.

So now what we want to do is we want to find an area not to the left or to the right of some number, but actually in between two numbers. So what does that look like on a graph? Always draw the graph first. It's basically going to look something like this. You have negative 0.64 over here, slightly to the left of 0, and you've got 2.27 all the way over here.

And then what you want to do is you want to basically draw all of the area that is contained between those two numbers. So basically, the area that we're interested in with the probability is this area that's shaded in yellow over here. This is what you're looking for. How do I actually find that using a z table? You're just going to use a very similar procedure.

You're just going to have to identify the two z scores you have to look up, which in this case are just our two numbers, negative 0.64 and 2.27, and you're going to look up those probabilities in the z table. Now I'm going to skip that step because we've already done that over here. We already know what those numbers are. But if you didn't know what these numbers are, you would just go into the z table and find those. Alright?

So the two numbers we're going to use are 0.2611. I'm going to go ahead and color this in green, and then the 0.9884, I'm going to go cover this in blue. Alright? So how do I get this area over here that's between the two numbers? Well, let's think about this.

Right? This 0.9884 is basically if I take all of the cumulative area starting from the left all the way up until that number. So in other words, it's basically all of this area over here. But I don't want all of that. I want basically only this piece in between the two z scores.

So how do I subtract this little piece over here? Well, that's basically the probability that is the 0.2611 that we get when we calculate the probability for this z score over here. So, essentially, what happens is if I just take the larger blue area and then subtract the smaller green area, then basically all I'm left with is just the area between the two z scores. That's what's going on here. Alright.

So we just subtract the larger from the smaller. Okay. So what I'm going to do here is we're just going to take, we've got our two numbers here. I'm going to take 0.9884, and we're going to subtract the 0.2611 for a grand total of 0.7273. So that over here is going to represent the area or the probability in between those two z scores.

Alright? So it's a pretty straightforward procedure. Let me know if you have any questions. Let's go ahead and take a look at some practice.

Find the area of the shaded region under the standard normal distribution.

• A. 0.7881
• B. 0.7119
• C. 0.2119
• D. 0.2881

Z-Scores from Probabilities

Welcome back, everyone. So in the last couple of videos, I showed you how to find a probability from a given z score using the z table. We would always draw a sketch of the problem, and then we would look up a given z score and find the probability. Unfortunately, some problems aren't as straightforward. The examples we're going to work out here actually will ask you to do the opposite.

It'll give you areas or probabilities under the standard normal curve and ask you to find the z score. Now, if this sounds kind of complicated, don't sweat it because, really, we're just going to use the exact same things. We're always just going to draw a sketch, and we're going to use our z table. But, basically, we're just going to reverse the order of the values we're looking up and writing down. So let's just jump right into our problems here so I can show you how this works.

Let's get started. So we're going to find the z scores from each one of these probabilities over here, and we're always just going to sketch the region under the normal curve if it's not already given to us. You should always be doing this. Remember, you should always draw the graph. It's going to help you minimize the chance that you'll make a mistake.

So let's take a look at our first problem here. We're going to find the z score from this given area. Notice how this problem is a little bit different. Instead of a z score that's already known to us and then us having to shade the area to the left, now that's already given to us. We know that this area is 0.8051, and we're trying to find the z score that is associated with that probability.

Alright? So how do we do this? Well, one of the things I want you to notice here is that we can basically use this number, this area, to try and gauge where the z value is going to be. Remember, the total area under this entire standard normal curve is 1. So if we get values of areas that are very small closer to 0, that means that our z value is going to be somewhere over here to the left because it's going to represent a very, very small sliver of the graph.

If we get values that are very close to 1, like 0.99, then we know to draw the line somewhere over here because it's going to encompass all of this area to the left of that z score. So what happens is in this case, we have 0.8, which is, you know, somewhere between half and one, and it's about in the middle. So, basically, you can use this to gauge where your z value is going to be, which in this case is right over here. So what that means is that when we go and find our z score using the table, we should expect to get a positive number that is, you know, around 1 or something like that. That's where the z score looks to be.

So let's go and see if that's actually true. So how do you actually solve these problems? Well, there's basically 2 questions you have to answer. What is the probability that's given to you? And that's actually pretty straightforward because you just copy it straight from the problem.

The probability that's given is 0.8051. The second question is, what's the probability that I have to look up in the table? Is it just the same value? Is this just the point 8051, or is it something else? Well, all you have to do is just think about remember how are the tables how are the values in the z table actually organized.

Remember that you have your z scores in the columns and the rows, and these values over here in yellow are assuming that they're basically those are the values that are cumulative probabilities from the left of the graph. So all you have to do is just compare the shaded parts that you've drawn in the problem to what's going on here in the z table, and they're actually the same. So these values are going to be here to the left, and that's exactly what you're trying to find. You're trying to find an area or the z score with an area to the left. So, basically, all you have

Find the z-score such that $P$(Z < z)$=0.6331$.

• A. 0.34
• B. 0.75
• C. 0.62
• D. 0.66

Find the z-score corresponding to the probability/area shown under the standard normal curve below.

• A. -1.52
• B. 1.52
• C. 0.82
• D. -0.82