Hi, my name is Dr. Ann Weaver of Argosy University. This WebEx is about something in statistics called z-

Transcription

1 Hi, my name is Dr. Ann Weaver of Argosy University. This WebEx is about something in statistics called z- Scores. I have two purposes for this WebEx, one, I just want to show you how to use z-scores in a way that you ll find very typically represented in a typical statistics book. The other purpose is to show you how to use Steinberg Z Tables in Statistics Alive, our required text, because her Z Tables are formatted in an unusual manner compared to the way you ll see z-scores in most textbooks. z-scores are a location on a map. The map is a bell curve such as shown here and the bell curve is formally known as the Standard Normal Distribution. Now the location of z-scores on a bell curve map are along the horizontal X axis. Now the bell curve is a theoretical frequency distribution of a population, in other words it is a frequency distribution and inventory of an infinite number of numbers. It is the way numbers behave in when they are found in infinite quantity. The symbols along the X axis here indicate that this is a theoretical dimension because our population, because these letters are in Greek. The letter in the middle right here is mu, the Greek letter mu. It looks like a fancy letter U and that s one way you can remember it, mu is the population mean. Sorry, hope that goes away. All right, the other letter down here is called sigma, it looks like an O in our alphabet with a little scarf blowing it off it in the wind. It stands for the standard deviation of the population. You build a bell curve by getting yourself a mu and sigma, a mean and standard deviation of a population and then adding standard deviation to the mean once, twice and three times and subtracting a standard deviation from the mean once, twice and three times. So let s say that we set our mu at 25 with a standard deviation of 5, so to get 25 here mu, it goes right in the middle. If we add one standard deviation, 525 plus 5 equals 30, 25 plus 5 plus 5 equals 35, 25 plus 5 plus 5 plus 5 equals 40. On the low end of the bell curve we subtract standard deviation from mean, so 25 minus 5 is 20, 25 minus 5 minus 5 is 15 and 25 minus 5 minus 5 minus 5 is 10. So that s how we get a line of data on a bell curve here for practical purposes for this demonstration. And that s what these symbols here mean. Page 1 of 12

2 All right, now here is a picture of what z-scores typically look like in a statistics book and this was taken from Weaver s Good Natured Statistics because it is the standard of how things go, and in my opinion, of course I wrote that book, an easy way to learn how to use z-scores. Now if you look at this table here there is two sets of 3 columns each, the first column let me make this bigger. All right, there is a z-score, there is an area between the mean and z and there is a proportion and a tail. Okay, what does this mean and how do we find out? Well first of all, z-scores range from zero to about 4, so I m just going to demonstrate a z-score of 1 and show you how to use z-scores. We said to get a z-score you add or subtract standard deviation from the mean. So here is an example of a bell curve. Now here is how z-scores work. I know this looks really clear but bear with me here. All right, here is your bell curve. Now a z-score is a location on the map and the map is the X axis of the bell curve. However, the way z-score tables are typically setup is they only use half of the bell curve at a time. The mean is in the middle. If a z-score has a positive sign with it, like down here, plus 1, we are talking about part of the bell curve above the mean. If a z-score has a negative sign, we are talking about the part of the bell curve below the mean. We have a z-score here of plus 1, which is what I showed you on the table here and I ll show you again. Here we go. Here is a z-score of plus 1, so here is the z-score of 1. Now the z- Score table isn t going to show you plus or minus, but you have to figure that out yourself. In other words, depending on which side is the z-score, which side of the bell curve you are working on. Every z-score comes with two numbers, these are proportions of data. In this case a z-score of 1 comes with the number.3413 and So let s move to the table the bell curve here. Now because we have a positive z-score all it means is you are working with the values above the means. A bell curve is a frequency distribution, z-scores tell you proportions of data. So you take half of the mountain of numbers, in this case the higher half, these are the data above the means of 25. And you are going to divide that half of the mountain, and it represents numbers of data points, you are going to divide it into two proportions, and that s what z-scores do. So here your mean in the middle of 25, we added to the mean of 25 one standard deviation, so we added 5 because we ve established that the mean was 25 and the standard Page 2 of 12

3 deviation was 5, so 25 plus 5 equals 30. All right so the 25 here, and at one standard deviation above the means we have a value point of 30. z-scores are standard deviation units, so when we add one standard deviation to a mean that is a z-score of 1. It is always, a z-score of 1 is always associated with these two proportions, the.3413 and the What you have here is the proportion of data between the mean itself and the z-score and then right at that z-score and the proportion of data higher. If I go back here, once again I m going to show you on the table where I got these two numbers.3413 and Right? Here is the z-score table. The z-score itself is 1, they are in between the mean and z. From the middle out to the z point is in other words, 34% of the data fall between the mean and z. And the remaining data that are more extreme in value fall at a z-score of 1 and there is about 16% of them are I will go through this example many, many times. So anyway you get a z-score, you look it up on the table, you will get two numbers associated with it, they are proportions of data. They stay the same, you don t have to figure them out, you just have to go find them. So here is we said that the mu, the mean in the middle was 25, if you add one standard deviation to it, 5, you get 30. What this means here is this, that 34% of the data lie between the values of 25 and corresponding to this number rounded off,.16% - sorry, 16% of the data fall at 30 or higher and that corresponds to what you see in this tail here, okay? So that s how you work z-scores. Let me show you another example. Here is another z-score table, now we ll look at a z-score of 2. Now remember we built the bell curve by taking the mean and adding 1, 2 and 3 standard deviations to it and then subtracting 1, 2 and 3 standard deviations from it. So when we said when we add two standard deviations to the mean we get a z-score of 2, because the z-score is a unit of standard deviation as they say, we added two standard deviations, it turns into a z-score of 2. See here where it says 1.8, that s a z-score that means we added 1.8 standard deviation. Well for our purposes, we added two standard deviations now, so we get two classic proportions associated with the z-score,.4772 and Let s look at those numbers and how they apply on the map called the bell curve. Page 3 of 12

4 Now the z-score is plus 2, corresponds to the two standard deviations we added, here is the mean, we got this line by adding 1 standard deviation of 5, this line by adding 2. All right, this part of the bell curve is supposed to be blocked out of your mind, because you usually work with one side of a bell curve or the other, either the positive side, in other words up values of data higher than the mean, or the negative side, values of data lower than the mean. So this side, the lower side of the mountain, just block it out of your mind right now, we are just working in the upper half. The two numbers that come with the z-score of 2 represent proportions of data. So this is a plus 2, it means we added two standard deviations to the mean, so the lineation mark is right here. If I can outline this proportion of the data it corresponds to the first decimal between the mean and the z, on a z-score table If I can show you this side, this is the tail, this is the proportion of data in the tail, the other decimal that came with a z-score of plus 2, and it s If we round those off and understanding that decimals are the same as percentages we can say.48 or 48% of the data lie between the value of 25 and just up to 35 but excluding it, and just a tiny 2% of the data are at the value of 35 or higher, i.e. in the tail. So I ll do another example with a z-score now of 3. Here we have another look at a z-score table, now we have a z-score of 3. It means we added standard deviation to the main three times. We get two standard proportions that come with the z-score.4987 and Let s see those three numbers, positive 3,.49 and.0013 on a bell curve. Here we go. With a z-score of 3 plus 3, so we are still working on the top side of the mountain of numbers here from the mean higher, so we are concerned with this half of the mountain and this an attempt to block this part of the mountain out of your consciousness for a moment, and we get two numbers with every z-score. Every time we get two numbers with the z-score. In this case the first number was.4987,.0013, I ll show you those on the table again. There they are.4987 and The area between the mean and z corresponds to the first decimal, the proportion in the tail corresponds to the second. So if you draw that on a bell curve, which is a really good idea when you are Page 4 of 12

5 working with z-scores to draw. What we did is we said the z-scores plus 3 represents adding standard deviation to the mean once, twice, three times. So the proportion of data on this side of the mountain is cut off right here, so you ve got this proportion, if I can highlight it here. The 4987%, virtually half of the data, fall between the value of 25 and just up to 40 and just a tiny.0013%, less than 1% of the data fall from 40 or higher, i.e. in the tail. So the point of the z-scores is this, you get two decimals, one of them is the amount of data from the mean out to that z-score as shown here, and the second decimal is the percentage of data in the tail. Now bell curves have frequency distributions so we are always talking about what proportion of data are we talking about. Okay, so let s look at some more z-scores. Now I m going to go through the same z-scores only I m going to do on the lower side of the bell curve, the negative side. The only difference is it s that it s a mirror image. The z-scores and the proportions that go with them are the same. The first z-score on a table like this is still between the mean itself, in this case, our case, 25 and the z, the location on that X axis and the second number here corresponds to the proportion in the tail. So here is a picture of negative, z equals negative 1. Now all that negative 1 means, all the negative sign means is that we are working below the mean on a bell curve. So this here is an attempt to have you ignore this side of the bell curve in your mind. Now it s 1, a z-score of 1. z-scores are standard deviation units, in other words they correspond to how many standard deviations you have added or subtracted in this case from the mean. Now our mean was 25, our standard deviation was 5. So the z-score of negative 1 means the location where I take the mean and subtract one standard deviation, here it says mu minus 1 sigma or 25 minus 5 equals 20. The first number that we saw on that table corresponds to the proportion of data between the mean and this actual z-score, and here it s.3413 or 34% of the data. The other number on that table corresponds to the proportion of data in the tail. And that proportion up there was.1587 or about 16%. So one of the things to notice is that the numbers that come with the z-score add to 2.5. If you take.3413 and.1587 and add them right now, which I suggest you do, take.3413, grab a piece of paper and a pencil, and.1587, let me make that bigger so you can see it, okay, right here, I ll get it over to you. There you are. Okay, z- Page 5 of 12

6 Score of 1, if you take.3413 and.1587 and add them they will add to.5, and that.5 corresponds to half of a mountain. Smaller second negotiated. So when you add them up they correspond to half of the bell curve. So with z-scores you pay attention to their sign, they are either positive or negative. And usually we leave off the positive sign, so if you just see a z-score work above the mean, and if you see a z-score with a negative sign work below the mean. Go get the proportions of your data and figure out the proportion of data that you are concerned with. So right now let s just learn about z-scores and later on figure out what we are concerned with. Here is another example, we are going to do a z-score again, the same proportions prevail, this is the exact same table that I showed you before but now we are going to have a z-score of negative 2. It means we are working on the lower half of the bell curve, so we are concerned with this half of the bell curve now, ignore this, that s supposed to be just ignored, z is negative 2. We get two proportions with all z-scores, the negative part means we subtracted standard deviations from the mean, the 2 means we subtracted two of them, so here is the mean, here is one standard deviation and here is the next one. So we put a line right there of a z-score of 2, it s a location on an X axis. They you go ahead and yu get your two proportions associated with it. The first proportion in the list between the mean and z is In other words 48% of the data in this case lie between 25 and 15. How did I get that number? Well our mean was 25, the minus 2 means minus two standard deviations, so 25 minus 5, which is what this means, mu minus sigma, or mean minus standard deviation 25 minus 5 is 20. This means mu minus two standard deviations, okay a standard deviation, so 25 minus 5 minus 5 is 15. So you have two things going on when you are working with z-scores, one is the actual z-score themselves, the location here in standard deviation unit, and then the other is the actual data that you are working with. And I arbitrarily chose the mean of 25 and a standard deviation of 5 just to make things a little bit easier. What this means then is with a z-score of 2 you get two proportions. If you take a decimal you can turn it into a percentage, so if we just see this decimal as 48%, 48% of the data in this little world we ve concocted have a Page 6 of 12

7 value of 25 down to but excluding 15. And then just the remaining 2%, if you round this off you get 2%, have a value of 15 or less. So 48% of the data are 15 to 20 or 25 down to 15 but not 15, and just 2% of the data are 15 or less. What this translates into for F the mountain is that.4, you know.48 and.2 added together equal.5, that s half the data. So things can get more complicated when you can start adding data from the other half of the mountain. Let s do another example and we ll do a z-score of 3. Just like I showed you before, we get two customary proportions that come with that, in this case.4987, so virtually 50%, and a fractional proportion of data in the tail, if you round that off it comes to 0. So now we are real far out, we are real far away from the mean. See the z-score is a location on a map, the map is the bell curve. The bigger the z-score is the further out you are from the mean in the middle. And z-scores pretty much only get to about 4, because they represent the proportion of data. Well the proportion of data in the tail is getting pretty rare, so we use z-scores to figure out just exactly how rare those data points are. So if we take a z-score of negative 3 we automatically know that we re working on the negative side of the bell curve. A z-score of 3 whether it s positive of negative, whether you are working on the low side or the high side of the bell curve comes with two standard proportions, there were.4987 and So the first decimal with the z-score was the amount of data between the mean and that z-score. Now z-score means standard deviation unit, so a minus 3 z-score means you subtracted standard deviation from the mean 3 times. Our mean was 25, our standard deviation is 5, so this means subtract standard deviation from the mean once, this means subtract standard deviation from the mean twice, and this means subtract standard deviation from the mean three times. So if we did that, and we took our mean and subtracted standard deviation, which is 25, you get 25 minus 5 minus 5 minus 5 or 25 minus 15, you get 10. So those are the data we arbitrarily chose. Those will change depending on what data you are working with. These proportions of z-scores, or the mean in the middle and the bell curve being built on the addition of 1, 2 or 3 standard deviations up above the mean and the subtraction of 1, 2, 3 standard deviations below the mean will not change. That is standard and that s one Page 7 of 12

8 of the great things about statistics is that they ve gone ahead and figured all this out for you, all you have to do is go look up two numbers and figure out which of the two you need. So I mean you get a really good deal. So what we ve done here, the z-score negative 3 means we ve subtracted standard deviation from the mean three times, so here is the first subtraction line, the second subtraction line, so we went ahead and drew a line at the third subtraction. We applied those decimals to it, so 4987, 49.87% of the data have a value of between 25 up to but excluding 10. A vanishingly small amount of data, not even 1%, I mean.13% of the data have a value of 10 or less. That makes them very rare. That s how you use z-scores. Now z-scores range from 0 to 4 about, and you can have gradations. So for example the z-score in the middle of a bell curve is 0. If you look at the formula in the module you ll see why, right now take my word for it. But z-scores start at 0 in the middle and they go out to about 4. So they have all kinds of demarcations in between them, in fact they are continuous variables like we talked about in this module. So you can take you can say between a z-score of 0 and a z-score of 1 you can take half that distance, you get a z-score of half of that, which is.5. So let s take a look at where that would be a on a bell curve. Here is a z-score of.5, like all z-scores it comes with two proportions, the proportion of data between the mean in the middle and at the z, which is the location on the X axis, and then the remaining proportion in the tail. In this case a z-score of.5, in other words half the standard deviation, because z and standard deviation are the same, so half the standard deviation away from the mean the proportions are.19 and.31. Let s look at what this looks like on a picture. Now, this is a positive z-score so we are working on the upper half of the bell curve, in other words we are adding standard deviation. A z-score of.5 means we added half the standard deviation, and it s halfway between 0, which is the z-score right in the middle and 1, right? So we are halfway between these two points. This bottom half of the mountain right now is completely ignored. It as simple as can be, blacked out of your conscience. What we ve got here, we ll work on the upper half now because we have a z-score that s positive. Page 8 of 12

9 Normally you won t see the positive sign. It s halfway between 0 which is right in the middle, the mean, and 1 standard deviation above the mean is a z-score of 1, so I drew the line between these first two lines. The first proportion on the table is.1915, so 19% of the data have a value of 25 plus half the standard deviation from it. I didn t write that out. Oh, See, we took 5, which is our standard deviation and divided it half, added it to 25, so 25 plus 2.50 which is half of the standard deviation of 5, 27.5, what this means is this, z-score, all z- Scores come with two numbers. The first number is the proportion of data between the mean and that z- Score. The second number is the proportion of data in the tail. So I highlighted this little part that I shaded, here is the z-score right here, so that that little dot right there, it s halfway up to the z-score of 1. 19% of the data in this database have a value from 25 to 27.5 but excluding 27.5, now why we do all this will become obvious later but one thing at a time. All right, the other proportion is.3085, or 31%, so if we make a line from the z-score up above here it is,.3085,.3085, 31% of the data have a value of 27.5 or more. So 31% of the data are in the tail when the z-score is.5. What you need to do in this lesson is just figure out how to use z-scores, why you use them and what they are good for will become obvious in the future. Let s take another example, z-scores range from 0 to 4 and there is a million demarcations of z-scores in there as you can see. Here is another z-score of.75, like all z-scores it comes with two proportions. In this case 27% and 23%, 27 and 23 add up to 50%, the upper half or lower half of the mountain depending on which side you are working at. So let s take a z-score of positive.75, so again we are working above the mean here in the upper half. A z-score of.75 means we added 75%,.75 of the standard deviation. In this case we had a standard deviation of 5, so we re looking at a I don t know why that spontaneously did that, but my phone is ringing and I apologize. All right, so anyway working the upper half of the bell curve here, and working the upper half of the bell curve but we haven t quite made it to on standard deviation, we ve only got 75% of the way. And so we start from the mean in the middle up to one standard deviation, there we are about 75% of the way. So this is what I mean by z-scores being a location on a map, the map is the bell curve, the location of z-scores is on the horizontal X axis. Here we are about 75% of the way up to one standard deviation, so not Page 9 of 12

10 quite 5 points above the mean. Standard deviations ah z-scores come with two proportions, in this case 27% and 23%. Here is the 27%. So what this means is this, if you figure out 75% of 5, and add it to 25 you get So this here means 27% of the data in this pretend world we ve come up with have a value of 25 all the way up to but not quite up there. And then right at the z of.75 in the tail another 23% of the data, rounding this off, have a value of 28% or higher. So the proportions that come with z-scores represent the proportion of data that you have close to the mean and then further from it. So let me give you another example here. Now what I would like to do is go through the same z-scores that we ve reviewed and show you how they differ in Steinberg because she did something a little different. What Steinberg did and when you look in her book Statistics Alive, in her z table you will see that all she did, all she does is give you the proportion of the bell curve starting from the lower left hand side. So she ll go ahead and take it as far up as she wishes depending on the z-score, but it makes it a little tough to interpret. So let me attempt to bridge that gap. Let me make this a little bit smaller so that you can see both at the same time. Now on the top of these, now I m going to show you pairs of z-scores and on top of these is the typical way that you see it, and the bottom way is Steinberg s. So here we go. If we have a z-score of 1 it means we added 5 to 25, we added one standard deviation to our mean. But right now let s just work with proportions of the data. When you look on a z-score Table at a 1 you ll see two proportions,.34 and.16 rounded off, because you either work with one side or the other, you either work with the positive upper half or the negative lower half but the proportions associated with the z-score are the same. So it s a positive side, so we are working in the upper half, so I added one standard deviation from the mean, which here it says z equals plus 1, so 34% of the data are between mean in the middle and just shy of the first standard deviation, and the remaining 16% of the data in that half of the mountain have a value of in this case 25 plus 5 equals 30, 30 or more. Now again notice that the two proportions that come with a z-score, they always add up to 5. Page 10 of 12

11 So here is what Steinberg has done. She starts from this side and moves this way, so if you have a positive z- Score of 1 the only number she gives you is How did she get it? Well she took the proportion of the data between the mean and the z-score, which is the first number on the table in Weaver s Good Natured Statistics and then a typical statistics book. And then she added the other half. So she took half of the data here which is.5 here, she added that to this proportion here and she gets.8413, and that s what you ll see associated in Steinberg with a z-score of positive 1. All right, I just have a couple more to go through. All right, now we add a z-score of positive 2, typically you ll see a z-score of just 2 with two proportions 48% and 2%. In this case it s positive, we work on the upper half of the mountain, take that upper half and divide it into two proportions, the first proportion is from the mean to z and that s represented by the first decimal, the second proportion is the amount of data in the tail. Now in Steinberg if you look up a positive z-score of 2, she gives you the number What she did is took this proportion of data here which was here and here, and added the other half of the mountain of data. So.5 plus.4772 equals Just a couple more here, data, a z of plus 3, two proportions are associated with it,.49 and.0013 plus a z of 3 in Steinberg is What she did is she took the bottom half of the mountain and the first proportion of data going out to that z, the.49 and added them together and got her 99.8 or Now with her negative z-scores you will get the proportion in the tail. So in a typical statistics book like Good Natured Statistics where you get a negative z-score of 1 and the two same proportions you ll only get the second proportion in Steinberg when the z-scores are negative. So she s starting from the left side. So here s another example that negative 2, the proportions are the same but this is the proportion in the tail, this is a representation of the proportion in the tail and Steinberg, all she does is give you the proportion in the tail when the z-scores are negative. Here s another example of that, z-score of negative 3, the typical proportions you d get with it, Steinberg only gives you this fraction of a percentage right down here, so that s the only number you ll see with a z of 3. Page 11 of 12

12 If you stuck with this you learned an awful lot and more power to you. I hope it helps. Thanks, over and out, Dr. Weaver. Page 12 of 12