Standard Normal Distribution: Videos & Practice Problems

Understanding the standard normal distribution and z-scores is crucial for calculating probabilities. To find probabilities associated with z-scores, utilize a z-table, which provides cumulative probabilities to the left of a given z-score. For areas between two z-scores, subtract the smaller cumulative probability from the larger. Conversely, when given a probability, reverse the process to find the corresponding z-score. This foundational knowledge is essential for statistical analysis, including hypothesis testing and confidence intervals.

Finding Standard Normal Probabilities using z-Table

Welcome back, everyone. So in the last few videos, we learned about continuous random variables, and we were introduced to the standard normal distribution, a bell curve that looks like this with some special properties. We saw that on the horizontal axis, we had numbers, which we called z-scores, which basically measure how far away from the center you are in terms of standard deviations. One of the most fundamental skills that you'll absolutely have to know and use for the rest of the course is being able to find probabilities when given z-scores. I know that sounds kind of complicated, but I'm going to walk you through it step by step.

We're actually just going to use a table to do most of this. We're going to look up some values and then write down some numbers while drawing some graphs. I'm going to break this down for you, and we're going to do a ton of examples. Let's go ahead and get started here. We're going to be looking at how to find standard normal probabilities using something called a table, a z-table.

Basically, what happens is when we looked at continuous random variables, we always saw that probabilities were essentially areas under the density curve. We saw how to do this with uniform distribution. We had these rectangles. Finding areas of rectangles is easy because you just do width times height. But for a standard normal distribution, finding areas is less straightforward.

How do you find an area of this sort of weird wedge shape like this? You can't use it doing doing formulas, and instead we find these areas using a table. We call this table a z-table, and it's going to be in your books. It's going to be in your courses, and we actually are going to include one for you in your worksheets. So go ahead and have it handy.

It basically just looks like this with some graphs and a bunch of columns and cells with rows and numbers. I'm going to walk you through how to use it in just a second here. Alright? So basically, what happens in all of these problems is you're going to be asked to find a probability p of z being either less than or greater than, in other words, to the left or to the right, of some given z score. Alright?

So how do we actually go ahead and do that? Well, the first thing you're going to do is you're actually just going to go ahead and draw a graph. The more you draw these graphs and visualize what the problem is asking you, the less likely you are to make a mistake. So I always want you to draw these graphs. We're going to sketch the normal curve and basically look at what the z scores are and then highlight or shade a specific area.

So let's take a look at our first problem over here. We're going to sketch a graph to represent each probability and then find it using a z-table, and we're going to see how to do that in just a second. So first, let's go ahead and sketch the graph. I've already got most of it drawn for you. We basically have a standard normal graph that looks like this.

The midline, you know, where it is, is basically zero, and we're interested in a z-score of negative 0.64. We're asked to find the probability that z is less than that number. So essentially what happens here is negative z-scores are basically going to be to the left of that middle. And, we have these z-scores that could be decimals, and usually they're expressed to the hundredths place. So I've got a z-score that I'm interested in, which is negative 0.64.

And I'm asked to find the probability that I have a number that's to the left of that or less than that. And essentially what that means here is that I'm looking at the area to the left of that z-score. So basically, this line here is going to be the boundary of where I shade this region, and this shading over here essentially is going to translate to an area or a probability. Basically, what this problem is asking is what is the probability, in other words, what's the area of this little piece of the normal curve? How do I find that?

I have to look this up using a table. So once you've drawn out the graph and you've sketched the normal curve, you're going to look up a z-score that you're interested in, and that z-score is going to be just the one that we're given, negative 0.64. So we're going to look up the z-score in a table, and I'm going to show you how to do that in just a second here. Alright? So let's go ahead and look at that table here.

And basically, the way this works is you're going to have a bunch of rows and columns where the z-scores are shown over here in the first column and first row. The tenths place is always going to be going down, and the hundredths place will always be going across. And basically, these on the left will be negative z-scores, and these on the right will be positive z-scores. So, for example, if we're looking at a z-score of negative 0.64.

The way you use this is you basically just go down the rows until you find the tenths place that you're interested in. In this case, we're going to go down the rows until we find negative 0.6, and then we're going to go across the columns until we find the right hundredths place, which is going to be 0.04. Those two things where they intersect, that's going to be the number that you're interested in. When they intersect right over here, which is a probability of 0.2611, These things will always be represented in terms of four decimals, and they're always going to be numbers that are between zero and one, as we can see if you look just across this table over here. Alright?

Essentially, what this variable or sorry. This number represents is the probability or area that is contained within this little piece of the graph up until that z-score. What's really important when you look at this table and you use it is that these values will always be cumulative areas or probabilities from the left. In other words, when you look up a z-score, these yellow numbers correspond to areas, and by default, they're always to the left of that z-score that you're interested in. Alright?

That's going to be really, really important. In fact, most books and courses will be consistent with this as well. If your professor has their own method, you should go ahead and stick to that, but this is always how we're going to do it. Okay? So let's go back to our page over here because we actually found our answer.

We found that the probability of the area to the left of negative 0.64 is this number over here, which is 0.2611. Alright. What happens is these areas are always going to be shown to the left of that score, and that's really important there. Make sure you write that down. And our probability that we just got from the table was 0.2611.

So is that the right answer to our problem? Actually, it definitely is. So in other words, the probability or the area that z is to the left of that negative 0.64 is 0.2611, and we're done here. So essentially, this piece of the normal curve has an area of 0.2611. Alright?

So basically what happens is whenever you're asked to find where, a probability of z being to the left or less than some number, you're always just going to use the exact same number that you're just reading off the z-table. Alright? Now unfortunately, what happens is that won't always happen. And, in some cases, you'll be asked to find the probability that z is greater than some number. And essentially what that means is we're going to be looking at an area to the right.

And I'm going to show you how to do that. So let's just go ahead and start off with our normal curve. Just do the best you can at drawing a little sketch. It's going to be something like this. It doesn't have to be perfect.

Let me try that again. Yeah. It doesn't have to be perfect, but this is going to look something like that. Alright. So here we have our normal curve.

What I'm going to do here is I'm just going to write where does that z-score end up being? It's a positive number, so it's going to be to the right of that midline. And if you don't exactly know where to draw the line, you can kind of just take a best guess. You can look at your z-table to give you a rough idea of where these things should be. Two points zero is going to be somewhere over here, so it doesn't leave a whole lot of area underneath that curve when you get all the way out there to those high twos and three numbers.

So that's what I'm going to do. I'm going to go ahead and over here, I'm just going to draw this as, like, my best guess. It doesn't have to be perfect. Z equals 2.27. Alright?

Now essentially what I'm trying to do here is I'm trying to find a probability that z is greater than that number. In other words, what I'm going to do here is draw that line. And because it's greater than, this is going to be an area to the right of that number. So this is actually the area that I'm interested in.

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

Answer: (B)

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

Answer: (B)

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, where you 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 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 z-score. 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 -0.64, we just looked up a z-score after drawing our curve and shading the area. When we found that area, which was 0.2611, we had 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 had to look up a z-score of 2.27. We used our z table to find the probability 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 one 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 -0.64 over here, slightly to the left of zero, and you've got 2.27 all the way over here. And now 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, -0.64 and 2.27.

And you're going to look up those probabilities in a 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

Answer: (D)

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.

They'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 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 gauge where the z-value is going to be. Remember, the total area under this entire standard normal curve is one. So if we get values of areas that are very small closer to zero, 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 one, like 0.99, then we know that 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 kind of 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 around one 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 are basically two 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 0.8051, or is it something else? Well, all you have to do is just remember how the values in the z-table are organized.

Remember that you have your z-scores in the columns and the rows, and these values over here in yellow are those 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, 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 to do here is when you're given an area to the left, you're just going to look up the same exact probability.

0.8051, that's the value you're going to look up in your table. Okay? So let's actually go ahead and do that. So now what we're going to look for is we're going to look for this value over here, 0.8051. And remember, what happens is I'm going to be looking at the right side of the graph because I should expect to get a z-score that's slightly positive.

So let's go over here and see how this works. So what I'm going to do is I'm going to try to look for this number, and this is a little bit sort of more tedious because you have to look through a bunch of numbers here. Basically, I kind of just, like, try to find the zero point eights, and then just kind of start from there. We actually can see that the point sevens start right, end in this row, and then you start transitioning to the point eights. And lo and behold, our answer is right over here.

So we've got our probability, our area of 0.8051. Now we just have to find out what z-score that corresponds to. So again, I'm just going to draw the rectangle that goes across that row and column, and basically that's going to tell me the z-score. It's going to be these two numbers, this is the tenths place, and this is the hundredths place. Alright?

So in other words, this area corresponds to a z-score that equals 0.86, and that's our answer. Alright? So in other words, the probability that we looked up in our table was basically just the same probability that was given to us in our problem because it's an area from the left of that to the left of that z-score. And what we found is that the z-score associated with that is 0.86. Alright?

And that's it. That's really all you have to do with this. So basically what happens is whenever you are working with areas to the left, whether it's z-scores to probabilities or probabilities to z-scores, either one, these are always the simplest problems because you have to you don't have to do anything with the values that you're given in the problem, in the table. You can kind of just blindly trust them. So let's take a look now at our second problem, which is a probability to the right, so an area to the right.

So in this case, what we have is that we have p of some value is greater than some unknown z-score, and that equals 0.67. So, again, this is an area to the right. Let's see how this works. Right? So the first thing we want to do is just sketch the region, that's associated with this problem.

So we have an area of 0.67. What does that represent? That's an area to the right of some unknown z-score. And, again, we can use this number to figure out where exactly we're going to draw this. Right?

If this was a value that was really, really close to one, remember, because this is an area to the right, this would actually have a lot of area under the normal curve. Because this is an area that's slightly above 0.5, then that means that the area to the right of this line has to be slightly more than half the curve. So basically, I know to draw this line somewhere over here because when I go to fill in this area, it's going to be slightly more than half the graph. Alright? So notice how what I'm doing here is whenever I'm shading areas to the right, I like to do this in blue, and we'll see why in just a second.

Alright? So let's go on to our values here. What's the probability that's actually given to us in the problem? Well, the probability that's given to us is just 0.67. Alright?

So that's just, you know, straight up the number that we were given from this problem here, that's this area. So is that the value that we go look up in our table? And again, we just have to ask ourselves, is the shaded area the same as the types of shaded areas that we see in our z-table? And it's actually not. So what happens here is if you're ever given an area to the right of a z-score, you must first solve for the area to the left.

That's always what you have to do first before you can go over and start using your z-table. Alright? So you always solve for the area to the left before you move on to that that second step of using your z-table. So let's go ahead and do this first. The probability we're actually going to look up is not going to be 0.6776 seven zero zero.

It's going to be one minus that. So we're just going to look up this value 0.33. So basically what's happening here is in order to find this z-score that we're interested in, we're going to have to when we use our z-table, we're going to have to look up the area that is to the left of that z-score in order to find out what this number is. Alright? So that's basically this yellow region over here.

So what we should expect to find is we should expect to find some small negative number. Alright? So this is the area so this is the probability that we're actually going to look up. So let's go over to our z-table and see if we can actually find that area. Alright?

So we know we're going to be looking at the negatives, so let's go over here and see if we can find 0.33. So 0.33. If you start going down the list over here, you're going to go all the way down, and you start hitting the the threes somewhere around, over here in this in this region. Alright? So we can see here that at 0.33 actually ends up being over here, and that's our probability.

So now we just go ahead and find out what the, tenths and hundredths place is. Alright? So with the tenths and hundredths place are just going to be these columns, and if you go up, you're going to see that this is 0.4, and then this is going to be 0.04. So if you put those two numbers together, the z-score here ends up being z equals negative 0.44. Alright.

Let's go back to our problem here and see if that makes sense. We got a number that is small and slightly negative, so that perfectly does make sense, and that's our z-score. So this is going to be zero point, 44, but it's going to be negative. Alright, folks? So that's all there is to it, really.

So, again, with the areas to the left, you kind of just can go and use the z-table as it is, and you don't have to manipulate any of those numbers. Whenever you're working with areas to the right, there's always one step that you have to do, which is you have to subtract the area that's given to you to find the area that you're supposed to look up in that z-table, and then you can go do that as, as usual. Alright, folks. That's it for this one. Let me know if you have any questions.

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

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

Answer: (A)

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

Answer: (A)

Here’s what students ask on this topic:

What is a standard normal distribution and how is it used in statistics?

The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. It is represented by a bell-shaped curve, known as the Gaussian curve. In statistics, it is used to find probabilities and z-scores, which measure how many standard deviations an element is from the mean. The standard normal distribution is crucial for hypothesis testing and confidence interval estimation. By converting data to z-scores, we can use the standard normal distribution to determine the likelihood of a data point occurring within a given range. This is done using a z-table, which provides cumulative probabilities for z-scores.

How do you find probabilities using a z-table?

To find probabilities using a z-table, first convert your data point to a z-score using the formula: $\frac{x-\mu }{\sigma }$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. Next, use the z-table to find the cumulative probability associated with the z-score. The z-table provides the area under the curve to the left of the z-score. For probabilities to the right, subtract the table value from 1. For probabilities between two z-scores, subtract the smaller area from the larger. This process helps in determining the likelihood of a data point falling within a specific range.

How do you calculate the probability between two z-scores?

To calculate the probability between two z-scores, first find the cumulative probability for each z-score using a z-table. The z-table provides the area under the standard normal curve to the left of a given z-score. Once you have these probabilities, subtract the smaller probability from the larger one. This difference gives you the probability of a data point falling between the two z-scores. Mathematically, if $P(z<a)$ and $P(z<b)$ are the cumulative probabilities for z-scores $a$ and $b$, the probability between them is $P(a<z<b)=P(z<b)-P(z<a)$.

How do you find a z-score from a given probability?

To find a z-score from a given probability, first determine whether the probability is to the left or right of the z-score. If it is to the left, directly use the z-table to find the corresponding z-score. If the probability is to the right, subtract it from 1 to find the left-side probability, then use the z-table. The z-table lists cumulative probabilities from the left, so match the given probability with a table value to find the z-score. This process is essential for interpreting probabilities in the context of the standard normal distribution and is widely used in statistical analysis.

What is the importance of understanding z-scores in business statistics?

Understanding z-scores in business statistics is crucial as they allow for the standardization of data, making it easier to compare different datasets. Z-scores indicate how many standard deviations a data point is from the mean, providing insight into its relative position within a distribution. This is particularly useful for identifying outliers, assessing risk, and making data-driven decisions. In hypothesis testing, z-scores help determine the significance of results. By converting raw data into z-scores, businesses can apply statistical methods to analyze trends, forecast outcomes, and optimize strategies, ultimately leading to more informed and effective decision-making.