Understanding Z-Scores and Their Applications

Okay, so welcome to my video on z-scores. Z-scores are just another way of expressing average and standard deviation. If you have a z-score of 1, that just means that you're one standard deviation above the average. Or if you have a z-score of negative 1, that means that you are one standard deviation below the average. So let me draw this for you on the normal curve and I think it will make a little bit more sense. If you remember from my last video, we always put the average in the middle of the normal curve. And this Greek letter mu represents the average. And the average always has a z-score of 0. So if I draw the z-score on the normal curve, I'm going to put the number 0 where the average is supposed to be, right in the middle. And the distance of the average... to one standard deviation above the average is going to have a z-score of positive one. So I'm going to put a positive one, one standard deviation above the average. And the distance of two standard deviations above the average is going to have a z-score of two. So I'm going to draw a positive two, which represents two standard deviations above the average. And the same concept applies to the left, except we use negative numbers. So one standard deviation to the left of the average is going to be a negative 1 z-score. And two standard deviations below the average is going to have a negative 2 z-score. And if you remember from my previous video, we know from the empirical rule that 68% of the data lies within one standard deviation. And the same concept applies for z-scores. We know that 68% of... of the data or 68% of the area under the curve is in between the z-scores negative 1 and positive 1. The empirical rule also tells us that 95% of the area under the curve is in between two standard deviations and once again the same thing applies for z-scores. 95% of the data or the area under the curve is in between the z-scores negative 2 and positive 2. So many of you have probably noticed that z-scores are just another way of expressing standard deviation and are wondering why do we even need to use z-scores in the first place. And z-scores are particularly useful because you can find the area for any number under the curve. Let's say that we're somewhere between 1 and 2. Let's say that we're 1.5 standard deviations away from the average. So we draw a line. And this number is 1.5. We are 1.5 standard deviations away from the average. And we can't use the empirical rule to find the area to the right or to the left of 1.5. So let's say we needed to find the area above 1.5. This is when z-scores are particularly useful because we can use a z-table to find this area. Hopefully this will make a little bit more sense once we get started with an example. So let's get started right away. Okay, so here we have an example that is somewhat similar to the example from my last video. And in this video we're going to solve this using z-scores. And it says that the test scores for a class are normally distributed. Okay, so that means that we could use our normal curve. And it gives us that the average... which is the Greek letter mu, is equal to 75, and the standard deviation, which is the Greek letter sigma, is equal to 10. So the first thing I'm going to do is place my average on the normal curve. The average, once again, always goes directly in the middle of the normal curve. Okay, so now that we've placed our average of 75 directly in the middle of the curve, let's see what this problem is asking us. It says, what is the probability that a student scored above a 60? So we need to find out what's the probability that a student scored above a 60. So let's place this value of 60 on our normal curve. We know that our value of 60 is less than 75. So that means that we have to place that value of 60 below the value of 75. So we can place our value of 60. anywhere below the value of 75. So now if we want to find the probability of a student scoring above a 60, the only thing we have to do is find the area above the value of 60. But the question is, how exactly do we find that area? And this is why we need to use our z-scores. So the formula that we need to use to find the z-value for any point on the curve is equal to any value x minus... the average, which is the Greek letter mu, all divided by the standard deviation. So this is the formula that you should memorize. So write this down. This is the formula to find the z-score for any value on the curve. So let's find the z-score for our value of 60. The z-score is going to be equal to x, which is our value of 60, minus the average which is 75 divided by our standard deviation which is equal to 10. So if we simplify this we have 60 minus 75 in the numerator which is negative 15 divided by 10 negative 15 divided by 10 is negative 1.5 so our z-score for the value of 60 is equal to negative 1.5 so now I'm going to draw the z-value in blue below the value of 60. This value of 60 has a z-score of negative 1.5. So what that means is that this value of 60 is 1.5 standard deviations below our average of 75. So now let's find the z-score or the z-value for the average of 75. The average always has a z-score of 0 every single time. So right in the middle below our average we're going to write our z-score of 0. So now in order to solve this problem we need to solve these areas under the curve. Notice how we have three separate sections. We have this area to the left of 60 which I'll call section 1. We have this area in between 60 and 75 which I'll call section 2 or area 2. And we have this this area to the right. of 75 which I'll call section 3 or area 3. So now we're going to find the area of these three sections and we're going to use our Z values to help us with that. So the first thing we're going to do is we're going to find the area to the left of our Z value of negative 1.5. So in order to do this we need to use our Z table. So let me show you what a Z table looks like. Here we have a Z table, and you've got to be careful which type of Z table you use, because they're all a little bit different. In this particular one, all the areas given to you are to the left of the value of Z. Notice how in this picture, it shows you the area is to the left of our Z value. So make sure that you know which type of Z table that you're using. So we need to find the area to the left of our Z value of negative 1.5. So if we go on the left side of our table, here we have a value of negative 1.5. This is the area of negative 1.50. If you go to the right, this is the area of negative 1.51, and so on and so forth. This is the area to the left of negative 1.52. So we need to find the area to the left of negative 1.50, which is this number right here. 0.0668. So we know that the area to the left of negative 1.5 is equal to 0.0668 or you could say that this area is 6.68% of the area under the curve. Alright, so now let's take a look at section 3. You can probably figure this out just by looking at it. We know that this is half of the curve. because the average of 75 is directly in the middle. So we know that to the right of this 75 is half the curve, so we know that this area of section 3 is equal to 50% of the curve. Alright, so now let me erase some stuff on the right just to give myself a little bit more space. Alright, so what do we know so far? We know that the area of section 1 is 6. point six 8 percent of the total area under the curve. We know that the area of section 3 is equal to 50 percent of the total area under the curve and we don't know what what the area of section 2 is at this point. But we do know that the total area under the curve must equal 100 percent. So if we add the area of section 1 plus the area of section 2 and the area of section 3. If we add all those together it has to equal 100%. So we can solve this equation for the area of section 2 just by subtracting area 1 and area 3. So the area of section 2 is equal to 100% minus the area of 1 which is 6.68% minus the area of section 3 which is 50%. So we know that the area of section 2 is equal to 100 minus 6.68 minus 50 which is equal to 43.32%. So now let's label this on our curve. We know that this area of section 2 is 43.32%. of the area under the curve. So once again I'm running out of space so I'm going to erase some stuff here on the right. Okay so now we have everything that we need to know in order to solve this problem. The problem is asking us what is the probability that a student scored above a 60. So we need to find the area above our value of 60. So the probability that a random student X scored greater than a 60 is equal to the area of section 2 plus the area of section 3. Those are the areas above 60. Or another way we could say this is the probability that our Z value is greater than negative 1.5 is equal to the area of section 2 plus the area of section 3. These are two ways of saying the probability that a student scored above a 60. Alright, so let's plug in our numbers. Our area of section 2 is 43.32% plus our area of section 3, which is 50%. So the probability that a student scored above a 60 is 43.32 plus 50, which is equal to 93. 0.32 percent. So I hope this gave you a better idea of z-scores and how to use them to find probability. In my next video I'm going to talk about t-scores. It's not always possible to use this method of z-scores which I just taught you. So sometimes we need to use another method which we use t-scores instead. So stay tuned for my next video. I really hope you're enjoying these and I will see you later.

Or if you have a z-score of negative 1, that means that you are one standard deviation below the average. So let me draw this for you on the normal curve and I think it will make a little bit more sense. If you remember from my last video, we always put the average in the middle of the normal curve.

And this Greek letter mu represents the average. And the average always has a z-score of 0. So if I draw the z-score on the normal curve, I'm going to put the number 0 where the average is supposed to be, right in the middle. And the distance of the average...

to one standard deviation above the average is going to have a z-score of positive one. So I'm going to put a positive one, one standard deviation above the average. And the distance of two standard deviations above the average is going to have a z-score of two. So I'm going to draw a positive two, which represents two standard deviations above the average. And the same concept applies to the left, except we use negative numbers.

So one standard deviation to the left of the average is going to be a negative 1 z-score. And two standard deviations below the average is going to have a negative 2 z-score. And if you remember from my previous video, we know from the empirical rule that 68% of the data lies within one standard deviation.

And the same concept applies for z-scores. We know that 68% of... of the data or 68% of the area under the curve is in between the z-scores negative 1 and positive 1. The empirical rule also tells us that 95% of the area under the curve is in between two standard deviations and once again the same thing applies for z-scores. 95% of the data or the area under the curve is in between the z-scores negative 2 and positive 2. So many of you have probably noticed that z-scores are just another way of expressing standard deviation and are wondering why do we even need to use z-scores in the first place. And z-scores are particularly useful because you can find the area for any number under the curve.

Let's say that we're somewhere between 1 and 2. Let's say that we're 1.5 standard deviations away from the average. So we draw a line. And this number is 1.5.

We are 1.5 standard deviations away from the average. And we can't use the empirical rule to find the area to the right or to the left of 1.5. So let's say we needed to find the area above 1.5. This is when z-scores are particularly useful because we can use a z-table to find this area. Hopefully this will make a little bit more sense once we get started with an example.

So let's get started right away. Okay, so here we have an example that is somewhat similar to the example from my last video. And in this video we're going to solve this using z-scores. And it says that the test scores for a class are normally distributed.

Okay, so that means that we could use our normal curve. And it gives us that the average... which is the Greek letter mu, is equal to 75, and the standard deviation, which is the Greek letter sigma, is equal to 10. So the first thing I'm going to do is place my average on the normal curve.

The average, once again, always goes directly in the middle of the normal curve. Okay, so now that we've placed our average of 75 directly in the middle of the curve, let's see what this problem is asking us. It says, what is the probability that a student scored above a 60? So we need to find out what's the probability that a student scored above a 60. So let's place this value of 60 on our normal curve. We know that our value of 60 is less than 75. So that means that we have to place that value of 60 below the value of 75. So we can place our value of 60. anywhere below the value of 75. So now if we want to find the probability of a student scoring above a 60, the only thing we have to do is find the area above the value of 60. But the question is, how exactly do we find that area?

And this is why we need to use our z-scores. So the formula that we need to use to find the z-value for any point on the curve is equal to any value x minus... the average, which is the Greek letter mu, all divided by the standard deviation. So this is the formula that you should memorize. So write this down.

This is the formula to find the z-score for any value on the curve. So let's find the z-score for our value of 60. The z-score is going to be equal to x, which is our value of 60, minus the average which is 75 divided by our standard deviation which is equal to 10. So if we simplify this we have 60 minus 75 in the numerator which is negative 15 divided by 10 negative 15 divided by 10 is negative 1.5 so our z-score for the value of 60 is equal to negative 1.5 so now I'm going to draw the z-value in blue below the value of 60. This value of 60 has a z-score of negative 1.5. So what that means is that this value of 60 is 1.5 standard deviations below our average of 75. So now let's find the z-score or the z-value for the average of 75. The average always has a z-score of 0 every single time.

So right in the middle below our average we're going to write our z-score of 0. So now in order to solve this problem we need to solve these areas under the curve. Notice how we have three separate sections. We have this area to the left of 60 which I'll call section 1. We have this area in between 60 and 75 which I'll call section 2 or area 2. And we have this this area to the right. of 75 which I'll call section 3 or area 3. So now we're going to find the area of these three sections and we're going to use our Z values to help us with that.

So the first thing we're going to do is we're going to find the area to the left of our Z value of negative 1.5. So in order to do this we need to use our Z table. So let me show you what a Z table looks like.

Here we have a Z table, and you've got to be careful which type of Z table you use, because they're all a little bit different. In this particular one, all the areas given to you are to the left of the value of Z. Notice how in this picture, it shows you the area is to the left of our Z value. So make sure that you know which type of Z table that you're using. So we need to find the area to the left of our Z value of negative 1.5.

So if we go on the left side of our table, here we have a value of negative 1.5. This is the area of negative 1.50. If you go to the right, this is the area of negative 1.51, and so on and so forth. This is the area to the left of negative 1.52. So we need to find the area to the left of negative 1.50, which is this number right here.

0.0668. So we know that the area to the left of negative 1.5 is equal to 0.0668 or you could say that this area is 6.68% of the area under the curve. Alright, so now let's take a look at section 3. You can probably figure this out just by looking at it.

We know that this is half of the curve. because the average of 75 is directly in the middle. So we know that to the right of this 75 is half the curve, so we know that this area of section 3 is equal to 50% of the curve. Alright, so now let me erase some stuff on the right just to give myself a little bit more space.

Alright, so what do we know so far? We know that the area of section 1 is 6. point six 8 percent of the total area under the curve. We know that the area of section 3 is equal to 50 percent of the total area under the curve and we don't know what what the area of section 2 is at this point.

But we do know that the total area under the curve must equal 100 percent. So if we add the area of section 1 plus the area of section 2 and the area of section 3. If we add all those together it has to equal 100%. So we can solve this equation for the area of section 2 just by subtracting area 1 and area 3. So the area of section 2 is equal to 100% minus the area of 1 which is 6.68% minus the area of section 3 which is 50%.

So we know that the area of section 2 is equal to 100 minus 6.68 minus 50 which is equal to 43.32%. So now let's label this on our curve. We know that this area of section 2 is 43.32%.

of the area under the curve. So once again I'm running out of space so I'm going to erase some stuff here on the right. Okay so now we have everything that we need to know in order to solve this problem.

The problem is asking us what is the probability that a student scored above a 60. So we need to find the area above our value of 60. So the probability that a random student X scored greater than a 60 is equal to the area of section 2 plus the area of section 3. Those are the areas above 60. Or another way we could say this is the probability that our Z value is greater than negative 1.5 is equal to the area of section 2 plus the area of section 3. These are two ways of saying the probability that a student scored above a 60. Alright, so let's plug in our numbers. Our area of section 2 is 43.32% plus our area of section 3, which is 50%. So the probability that a student scored above a 60 is 43.32 plus 50, which is equal to 93. 0.32 percent. So I hope this gave you a better idea of z-scores and how to use them to find probability. In my next video I'm going to talk about t-scores.

It's not always possible to use this method of z-scores which I just taught you. So sometimes we need to use another method which we use t-scores instead. So stay tuned for my next video.

I really hope you're enjoying these and I will see you later.

Transcript for:Understanding Z-Scores and Their Applications

Transcript for:
Understanding Z-Scores and Their Applications