For this video, we are going to be talking about measures of variability or dispersion. That the concept of dispersion, it refers to the variety, diversity, or amount of variation among scores. The greater the dispersion of a variable, the greater the range of scores and the greater the differences between scores. When we're looking at measuring variability, we have to consider the different types of data that we have. And also why it is that we are trying to measure the variability. And so the ones that we're going to be talking about in this lecture are the ICU UV range, a deviation score or average deviation, variance and standard deviation. So if we have nominal data, then the one that we have to use is Miller and scholars index a qualitative variation. Range can be used for a number different types of variables. So this is the distance between over which particular proportions of scores are spread. Deviation score. It's a distance of scores from the mean of their distribution. Variance is the sum of the squared deviation scores divided by n. And then our standard deviation is the square root of the variance. And the standard deviation is important because it's the one that we use for decision-making about if something is significant or not. So very important part of understanding statistics. Okay, so here we're going to talk about I QV first. And so this particular is the formula I QV equals observed heterogeneity divided by maximum heterogeneity the times a 100. And so essentially to be able to find out how many of these observed and maximums that you need. You would use this particular formula, k times k minus one divided by two. And so here we have an example of nominal data and which we have different types of grapes. And so we have date rape, rate bag, close friend, by a family acquaintance, by a stranger, and by a relative. And so in order to find out our level of variance between these, we need to find out the observed and the expected. And so if there is nothing going on, if there was no types influences on the outside as to what leads to these different types of grapes, then we would expect for all of them to be even all to be 200. And so that would be our maximum heterogeneity, right? And so in this particular case, we would look at the number of products. And so here we have 12345 cases. Then we multiply it by that same number minus 1, so 5 plus 4. And then we would divide it by two. So that gives us 10. So we would have 10 different sets of products. And so what that's talking about here is for our observed, our first product would be 200 times 100. And then we would add that score to 200 times 200. And then we would add that score to 200 times 350 and then 200 times 150. And then we would go down to the next one. And we would have a 100 times 200100 times 350 and a 100 times a 150. And so going ahead and knowing how many products you should have, right? So product, this means how many times you're going to be multiplying something by. And that kinda lets us go ahead and lay out the equation and know exactly what it's supposed to be expected. And so since the base is the same. Then you can actually just go ahead and we know that it's going to be 10 products. So you could just take 200 times 200 and raise it to the tenth, which would give us 400 thousand. So that's pretty easy to do on this side. But essentially to know that we do a 10, right? So this is one, this is two, this is three, this is four. This is 567. This is eight, this is nine, this is 10. And so it just helps keep everything straight. And then you add all those together. And said If we did that, then we would end up with 382,500 divided by 400 thousand. And so that gives us 0.9562. And then we multiply that by a 100 in order to get it into a percentage. And so that would be 95.62. And so that tells us that there is a, a decent amount of diversity, but it's really not that far along because like Syria we have here that are 200 even and the a 150 is close. And so that tells us that essentially there's not a whole lot of dispersion there between observed and expected. Okay, so the range, range indicates the distance between the highest and lowest scores of a distribution. Range is often indicated with an r equals high score minus lowest score. It is a quick and easy indication. Variability. It can be used with ordinal or interval ratio data. And the reason that, again, that we can't use the range from a nominal variable is that it just wouldn't make sense that you would have a measurable distance between pigs and cows are between apples and oranges. So it can only be used for the higher levels of measurement. So here we have an example of an array that we're going to talk about the range for. And so we would indicate and find our highest and our lowest. So again, by putting them in array, that helps us to find that information. And so our lowest is 20 and our highest is 49. And so the range, so distance over which 100% of the scores in a distribution is spread would be 49 minus 20. So that would give us 29. So some textbooks would say that you, that you should use the lower limits and upper limits. And so again, it depends on the textbook, so always pay attention to that. So like if back when I was where you guys were at, I would actually hacked they take 49.5 and subtract it from night team 0.5 and so my range would be 30. So it really depends. And again, you've had a look at what your textbooks wanting and what your teachers wanting. So this follow the textbooks guidance on that one. So here we can also find our interquartile range. And so we'll go back to this slide here just a second. So the interquartile range is a type of range measurement. It considers only the middle 50 percent of cases of a distribution. It avoid some of the problems of the range by focusing on just the middle 50 percent of the scores. It is limited because the inner quartile range is based only on to scores. Yeah, it felt to yield any information from all of the other scores. So you're cutting off 50% of the scores. And so if we went back here and we wanted to find where our information would be, then I would need to locate my first third quartile. So the first one. So it would be our fifth case. So 1234. Five, so 28 is our first quartile. And then the last 5, 1, 2, 3, 4, 5. Though, that would be the start of the fourth. So I want the end of the third. And so this would give me 44 minus 28, which would give me a intake or inter-quartile range of 16, which is very different from the 2009 that we got before. So this when there really wouldn't be any need to do that because we don't have extreme outliers on either side. So that would be kind of misleading in order to use that type of range for this particular distribution. So again, you've got to look at what your data looks like and to determine what you should report. So kind of the same whenever we're wanting to find the range for group data, then we would just take the lowest from whichever said it is that we have so the 115 and 179. And so essentially the range would be 179 minus 1. That teen or if you're doing old school like I was raised to do when some 9.5 minus 1.514. So again, follow the textbook and do whichever way the textbook tells you to do. All right? And so here, the 175 minus 14.5.75.5 minus 1414.5 equals 61. So this is a very unstable measure because it is very sensitive to deviant scores. So it's a poor choice if you have outliers. And so we can also find the interquartile range or group data as well, kind of like we talked about with the last video. So this is actually the same chart that we're going to be looking at from, from that. And so here, if we were doing the halfway, would take it by five. So this is exactly what we went through with our last equation. And so go through that and we would just change it up a little bit. Where if we wanted to find the first third quartiles, then we would have to take the 111 and multiply it by point T5. And so that would give us the 27.5. And then we would take this and multiply the a 111 by 0.75 to give us the third quartile. And then based on that information, we would find the median score for each of those. So weird. Yeah, Essentially you'd have to go figure out which one it is and you'd have to go through and follow the exact same steps. And so if I wanted to find where 27.5 would be, I would go as close as I can get without going over. So there is 26. So I would need to borrow from this particular interval right here. So my lower than that would be my 129.5. And I at 26. So 27.5 minus 26 would be what, 1.5. So that is the number that I need. But I ended up finding 15. I only needed 1.5. And so that's why I would have to take and put my frequency needed and divide it by my frequency found times 5. And so remember when you're doing these equations that we have to go by order of operations. So again, that's why this, you gotta make sure Follow-up order of operations. Okay, so for this formula, remember. At this right here, then this right here is to multiply. It's not the parentheses. And so it's essentially meaning that we need to multiply the frequency needed divided by the frequency spans times the interval. And so in order to do that, we would take the 1.5, which is what we found, and the 15, and we multiply it by five. And so anytime you multiply a fraction by the, by a whole number, you would put one as being the number below. That would be the divider. And so that is essentially why this bill ends up as nine because nine times one is nine. And then I'm sorry, nine times. Yeah. And then five times the top number. And so here we would end up with is point 1 times 5, which gives us 0.5 if we were trying to find the 25th percentile. And so our actual first quartile would be a 130. So we would actually just use this first one right here. And so then we would go up and then we'll find the third quartile, and then we would subtract the two. And that's pretty much how you would find. The range for group data. Though, limitations of the range, range is based on only two scores. So this is distorted by a typically high or low score. So very similar to the mean. No information about variables between high and low scores. And so you don't really know how many are things going man finds. And so it really doesn't give us a good look at the data within itself. So when we look at the average deviation, the average deviation variation of scores from the mean of the distribution is essentially on average, how far is a distribution off? So this little symbol right here just means that it is the deviation score. And of course the sum you guys have already seen. And then this, the sidebars here means the absolute value of those. And then n is the sample size that you take each score and subtract it from the mean to get the deviation score. So here our mean is 29. And so, but say that this was a basketball player and he, he or she normally scored 29 points per game. And so for this particular game, they shy that are scored 23 points, which is six less than their average. And here 30. So that would be one more than average, and 31 would be two more, and so on and so forth. And so we did this over the course of five games, right? 12345. And so we would take the absolute value. So we would want to turn all of these and the positives. That's all that means is we will get rid of the negatives and added altogether. So if we added everything, then we would end up with 40 divided by five and our average deviation would be eight. And so essentially, what that means is if we were going to maybe bet on this particular person, then we would do it with their average mine and it's essentially plus or minus 8. And then we would be pretty often as N2 where they're probably going to fall. So really as far as that's concerned, this kinda goes along with like bedding and trying to figure out like what's a safe bet and what's not. So, you know, you never know for sure, but this gives you a decent idea of what to expect from a particular player on any given day. So this goes into where you can check and make sure that you didn't make any mistakes by like we were talking about the last video. Is that the mean was calculated correctly. The sum of all the deviation scores will always equal 0. So here we have 17 minus 14, right? Which gives us 3 plus 2 is 5, plus 1 is 6, minus 6 is 0. So we didn't make any mistakes here. That's just the way of going back and double checking yourself. The standard deviation calculation. So to solve, we have to subtract the mean from each score. Then we have to square the deviations. And then we sum the squared deviations. And we divide the sum of the squared deviations by n, and then we find the square root of those results. And so the first steps up until the Find the square roots is actually finding the variance first, right? And then, so S squared is variance. And then just plain old S is your standard deviation. So here, so variance is the sum of the squared deviation scores divided by n. And standard deviation is just the square root at those. And so here we were able to, we have our numbers. So let's say that we were looking at days missed for 11 kids. And so this person or maybe pays attended. So this person intended 20 days and this person intended 30 days. Let's say it's 30. So this one had perfect attendance. And so on average, how did the group do? And so we would take our total number of days and we would divide it by the number of students that we had. And so that would give us 275 divided by 11. Or maybe you would make more sense to use the example course because it does the test scores. And so it could really be whatever it is that you want it to be. But anyway, if you add all of those scores together, you're going to get to 175. And we have 11 kids, so 275 divided by 11 gives us 25. And so from each of the scores, we're going to subtract 25. And so that gives us our deviation score. And so if you add all these up, we get 0. So that means that we did it correctly. And then the next thing we want to do is to square the deviation scores, though, right? So here we have the sum of the difference between, so the sum of the deviation scores squared. That means that I have to square all the deviation scores, right? Because remember again, that you have to follow order of operations and this particular case, then this is parentheses, so it is done first and then exponents, right? Please excuse. And then adding would be last as far as that. And then dividing. So we take all this and then we'll divide it by that bit. Gotta go through this first. And so essentially, we, if we were to take all of these squared and we would get to a 110. We added everything together. And so a 110 divided by 11 would give us 10. So our variance is 10. And then in order to get the square root, that variance, then we would take the square root of 10, which is 3.16. And so that is pretty much how you calculate for their deviation. And we could do it for grouped that your books not going to. So I'm just going to go ahead and stop the lecture here.