 In this video we will discuss Z scores. So the motivation behind Z scores is imagine you're in a class with an instructor and you take a test and you get like a 90% but you have a friend who's in a different or the same class with a different instructor and then they get a 95% well who technically did better. Can you instantly say the person that made the 95% did better? Not necessarily. You both took different exams that were made by different instructors even if they were on the same topic. Maybe the one professor gave a more difficult exam. It had a tougher, tougher set of questions on it. So that's where Z scores come in handy. It allows us to kind of standardize our test scores to make accurate comparisons or fair comparisons between pieces of data or subjects across different groups. So Z score or standardized value is the number of standard deviations a given value is above or below the mean. So if you have a Z score of two it means your data value that you were working with is two standard deviations above the mean. So what the formula is to calculate the sample Z score is you take your data value and you subtract the mean and you divide by the standard deviation. For a sample the means X bar are the standard deviations S. For a population the mean is mu and the standard deviation sigma. So that's Greek letters mu and sigma. So those are the formulas. Data value minus mean divided by standard deviation will typically want to round your Z scores to two decimal places or whatever they tell you to do on your practice assignment or practice exercises. So first off the cool thing about Z scores is the vast majority of sample values lie within two standard deviations of the mean. So that means Z scores that are between negative two and two are considered to be called usual. So usual values are two standard deviations of the mean or within two standard deviations of the mean. So if you ever want to find the minimum usual value for a data set you take the mean and you subtract two times the standard deviation. You take the mean and you subtract the standard deviation twice. That will tell you your minimum usual value. To find your maximum usual value you will take your mean and then you will add two times the standard deviation. You will take the mean and you will add the standard deviation twice. That's how you find what is known as the minimum and maximum usual values. So in terms of Z scores, remember anything but within two standard deviations of the mean is considered usual. In terms of Z scores which tell you how many standard deviations of data value is above or below the mean, any Z score between negative two and positive two, so any data value between two standard deviations below the mean or above two standard deviations above the mean is called usual or ordinary. Anything outside of that is called unusual. It's called unusual. Whenever a value is less than the mean then it will have a negative Z score. So if I give you a data value and it's less than the mean you better get a negative calculation when you find it's Z score. We'll practice that in just a minute. So ordinary values have Z scores that are between negative two and two. So again this is just a mathematical representation of our picture or diagram at the top. And unusual values are less than negative two or have Z scores that are more than two. So you're looking at more than two standard deviations above or below the mean. So there's this class. And this class took a history exam. And the class average was 78 with the standard deviation of 4. One student made a Z score of negative two on the exam. What was the student's actual exam score? So for this question here we need to think about our Z score formula. Data value minus mean divided by standard deviation. So Z equals, so we have our data value which is typically X minus your mean. So for example it's X bar for a population it's mu, we're about to replace it for a number so we really don't care which degree notation to use. And we're going to divide by the standard deviation. So what this means is we go through the question. We're trying to figure out what was the student's actual exam score. We're trying to figure out what was that data value. So my Z score in this case was negative two. My data value was, we don't know, call it X. The mean was 78 and the standard deviation was 4. So it's up to us to do a little bit of algebra here to solve this. We're too much because literally it says basic as saying multiply both sides by 4. Both sides of the equation get multiplied by 4. So you're left with X minus 78 and to find the data value we'll just add 78 to both sides. What was the student's actual exam score? Well negative 8 plus 78 is just 70 and that is our data value. The student score was actually a 70 in this case. So occasionally if you have a Z score you will be able to convert it back to a data value if they ask you to do so. Let's talk a little about Jack, shall we? Well Jack scored an 83 on the exam. The class average was 78 with a standard deviation of 4. Find the Z score for Jack's exam grade. Alright so I'm looking for my Z score for Jack. So Z score equals, remember, data value minus mean over standard deviation. I will write it out one more time. Data value, subtract the mean, divided by the standard deviation, we'll round to about two decimal places I suppose. So we have data value of 83 minus a mean of 78, that's the class average, and the standard deviation of 4, divided by 4. So what you end up getting is 5 divided by 4, you get a Z score of 1.25. Remember what the Z score tells you, it tells you how many standard deviations above or below the mean a data value is. So I will say Jack scored 1.25 standard deviations above the mean. That's what that means, that's how you interpret a Z score. So that's good, he was above the class average clearly by 1.25 standard deviations. Usual exam scores, or ordinary depends on what word you like better, are Z scores between negative 2 and 2. Is Jack's score usual or unusual? Well, is 1.25 within negative 2 and 2, is it between those two numbers? I will say since 1.25 is between negative 2 and 2, Jack's score is usual. Now the poor student that scored like 4 standard deviations below the mean, who had a Z score of negative 4, now that's unusual there. That's where you don't want to be when you take a test. So Jack's score is usual because 1.25 is between negative 2 and 2, it's within 2 standard deviations of the mean. So the scenario I portrayed to you at the beginning of the video, let's talk about it. You have 2 students who take different math classes, had exams on the same day, on the same stuff. Jack's score was 83 while Jill's score was 79, which student did relatively better given the class data shown. So if you look, you see Jack's score to 83, Jill's score to 79, you're like well Jack did better. Not necessarily, maybe Jill's teacher gave a really easy test and pretty much everybody got hundreds, and Jack's teacher gave a really tough test. So we look at the Z score, when we talk about who did relatively better, let's look at the Z score. So I know Jack's Z score from the previous example, Jack actually had a Z score of 1.25, and then we need to do some investigation here with Jill. Jill's Z score, we don't know it yet, so we need to calculate it. So Z equals data value, which was 79, minus the mean, which is 71, 79 minus 71 divided by the standard deviation of 5, this is for Jill. So I pretty much have 8 divided by 5, Jill's Z score is 1.6, so I got 1.6 for Jill. So relatively speaking, Jack was 1.25 standard deviations above his class average, whereas Jill was actually more standard deviations above her class average, she was 1.6. Since Jill had the higher Z score, she did relatively better, Jill did relatively better since she had the higher Z score. So when you're talking about who did relatively better, you should look at Z scores, because that's how you standardize data values. Jill did relatively better since she had the higher Z score. So that's all I have for now. Thanks for watching.