Z-score introduction

one of the most commonly used tools in all of statistics is the notion of a z-score and one way to think about a z-score is it's just the number of standard deviations away from the mean that a certain data point is so let me write that down number of standard deviations I'll write it like this number of standard deviations from our population mean for a particular particular data point now let's make that a little bit concrete let's say that you're some type of marine biologist and you've discovered a new species of winged turtles and there's a total of seven winged turtles the entire population of these winged two turtles is seven and so you go and you're actually able to measure all the winged turtles so and you care about their length and you also want to care about how are those lengths distributed lengths of winged winged turtles all right and let's say and this is all in centimeters these are very small turtles so you discover and these are all adults so there's a two centimeter one there's another two centimeter one there's a three centimeter one there's another two centimeter one there's a five centimeter one a 1 centimeter one and A six centimeter one so we have seven data points and from this and you I encourage you at any point if you want pause this video and see if you want to calculate what is the population mean here we're assuming that this is the population of all the winged turtles well the mean in this situation is going to be equal to you could add up all of these numbers and divide by seven and you would then get three and then using these data points and the mean you can calculate the population standard deviation and once again as review I always encourage you to pause this video and see if you can do it on your own but I've calculated that ahead of time the population standard deviation in this situation is approximately I'll round to the hundredths place one point 6:9 so with this information you should be able to calculate the z-score for each of these data points pause this video and see if you can do that so let me make a new column here so here I'm going to put our z-score and if you just look at the definition what you're going to do for each of these data points let's say each data point is X you're going to subtract from that the mean and then you're going to divide that by the standard deviation the numerator I D over here is going to tell you how far you are above or below the mean but you want to know how many standard deviations you are from the mean so then you'll divide by the population standard deviation so for example this first data point right over here if I want to calculate it's a z-score I will take two from that I will subtract three and then I will divide by one point six nine I will divide by one point six nine and if you got a calculator out this is going to be negative one divided by one point six nine and if you use a calculator you would get this is going to be approximately negative zero point five nine and the z-score for this data point is going to be the same that is also going to be negative zero point five nine one way to interpret this is this is a little bit more than half a standard deviation below the mean and we could do a similar calculation for data points that are above the mean let's say this data point right over here what is it's a z-score pause this video and see if you can figure that out well it's going to be six minus our mean so minus 3 all of that over the standard deviation all of that over one point six nine and this if you have a calculator and I calculated it ahead of time this is going to be approximately one point seven seven so more than one but less than two standard deviations above the mean I encourage you to pause this video and now try to figure out the z-scores for these other data points now an obvious question that some of you might be asking is why do we care how many standard deviations above or below the mean a data point is in your future statistical life z-scores are gonna be a really useful way to think about how usual or how unusual a certain data point is and that's going to be really valuable once we start making inferences based on our data so I will leave you there just keep in mind it's a very useful idea but at the heart of it a fairly simple one if you know the mean you know the standard deviation take your data point subtract the mean from the data point and then divide by your standard deviation that gives you your z-score