Okay, So today we're going to be talking about measures of central tendency. Central tendency, it describes the points around which the rest of this course focus. So the three measures that we use for central tendencies are the mean, the median, and the mode. So the mean is just the arithmetic average score of all the scores in a set of scores. The median is considered the central score, or that point which divides a distribution into two equal parts with 50% of the distribution on one side of the median and 50 percent on the other. The mode is considered the typical or most frequent occurring score in a distribution of scores. Each has its own assumptions. And it is important to know these assumptions in order to know when, when is appropriate to report daring data analysis. So the mode is the most common score, as I just talked about. It can be used with variables at all three levels of measurement. It makes it very versatile. Most often used with nominal level variables. So when we want to find the mode, we count the number of times that each core occurs, and the score that occurs the most often is the mode at the variable is presented in a frequency distribution. The mode is the largest category or the one with the most frequencies. And at the variable is presented in a line chart, then the mode is the highest peak. So that would represent the, the one with the most. Okay, so here we have a couple of examples trying to find the mode. And so if we go through and count, we would see that we have two 25s where 326 is 21 at the rest of them, right? So that means that 26 is our mode. So for the next array, we would want to go through and kind of do the same. So we have three 25s, 320, sixes, and 327. So that means that this one actually has three modes. And so for the next one, we have two and then two sets and then 22 and then two. And so once we hit more than three, we would consider it to not have a mode kind of the same if it wasn't multiple of one particular score. So we can have unimodal with just one node. We can have bimodal with two modes, or three modes would be multimodal. But then after that, it's considered not to have a mode. So when we look at the mode for group scores, the mode for group data is defined as the midpoint of the interval containing the most frequencies. So three is the maximum number for group modes as well. So if there's a fourth, then is considered not to have a mode. Use the mid points of the intervals as the mode. Example here, we would have to find out where our one is with the most frequencies. And so that looks like it is here with the team. And so 15 is found in the interval of 40 to 44. And so the next step would be to find the midpoint of this particular interval. And so to find the midpoint, you take the lower limit of the variable that, or if sorry, of the interval that you're in. So the lower limit here would be 39.5. So that's the lowest line because it could round up and 240. And then you divide the interval size by two. So five divided by two would be equal to 2.5. And so now I want to add 90 or sorry, 39.5 to 2.5 and that gives me 42. So when you have very small interval sizes, of course it's easy to count and just see that it is 42. But that's the way that you would go about looking at it and seeing. And so over here we have three sets of 15. And so this one would be considered to be tri modal. And you would just go through the same process and just find the midpoint for each one of these intervals. So, so when we're looking at this, Typically have to realize that the most popular is not always the most central score. It can actually be very far away from the central tendency. So deviant scores are outliers. Score is located in one extreme or the other, can affect the mode. And so an example of this could be on a, on a quiz. And so let's say that you have five people who made a 90 on it. And then you have by people who made a 80 on it. And then you have for people who made a 100, right? And so really you would have the 22 modes, right? But that really doesn't an 80 and 90. And then all those hundreds doesn't really give you exactly very much useful information at all. So you've really got to consider why exactly that you are reporting your data as into which one it is that you want. But also, let's say with that same score, and you had to buy five zeros. And the zeros, we're just because somebody had taken them yet. And so that would be extremely misleading if that was the mode. So some of the limitations of the mode. So some distribution tab, no mode. Some distribution have multiple modes. And mode of a ordinal or interval ratio level variable may not be central to the whole distribution. And so again, that's why we use it a lot more for nominal variables even though it can be used for all three. So the median, the exact center of the distribution of scores. So essentially just the one in the middle. It can be used with variables measured at the ordinal or interval. And ratio levels cannot be used for nominal level variables. So we can't really have the middle colors. It doesn't make any sense. At least not numerically to have 50 percent of the site green and 50 percent about site green just doesn't work that way of where as for the mode, right, we could say that Green had the highest frequency. So in order to find the mean, you need to put all the scores into an array. So array the cases from low to high or from high to low. It really doesn't matter as long as you put them in order, you locate the middle case. So if n is odd, the median score in the middle case. If n is even, and the median is the average of the scores of the two middle cases. And so here we have a couple of arrays that we can use for district nutrition. And so here when we're looking for the one in the middle. So we'll, we have three cases. That would mean that 15 would be our median. So here we have a even number. So that means that the two in the middle we have to add together and then we divide by two. And so that would give us immediate 16. And so here though, is where the median could also be helpful and maybe a little bit more resistant to outliers. Because still here, even though we have a 100 instead of a 19, the median is still 16. And so here it's 15 and 35. And then here we'd have to do 15 plus 15 divided by 2 is 13. So that is pretty much it. It's really pretty easy to be able to find out which one that is. It gets a little bit more difficult if we're needing to find the median for group data. So the formula for finding the median for group data is the lower limit plus frequencies needed in the interval divided by frequencies found in the interval times the interval size. Okay? And so this is where having a cumulative frequency comes in handy. And so the first thing that I want to do is find out where exactly the sinner score is. And so to do that, I would take my a 111 and divide it by two. And so that would give me 55.5. And so I want to go up through here and find out how close I can get to 55.5 without going over. And so it looks like I can get to 51. Alright, so how many more do I need in order to hit my pity by 0.5? So 55.5 minus one gives me 4.5. So that would be my frequencies needed out the next interval. And so if I go up to the next interval, it looks like I find nine frequencies, so that's my frequencies pound. And so I want to use the lower limit of the interval from which I am going to be borrowing some of these frequencies from. So that would be a 139.5. And so I would just go through and fill it in. So a 139.5 plus 4.5 divided by nine times five. And so my essentially my median would be essentially a 142, would be our closest guess. And so that is how we would figure out where the median is by just finding out what these numbers are and then doing the math. So pretty simple. So one of the things about median is that it assumes that data can be measured at the ordinal or scale or higher. So again, it can't be like we've already talked about, used for actual, for nominal data is still. So this is a pretty stable measure of central tendency in the sense that it divides the scores in half. So it is the center of the data. So we don't have to look at just the 50 percent. Sometimes it will tell us more to look at maybe descent tiles or desk styles or quartiles. So 50 percent is the desk style, but it's also the second quartile. And so this, this gives us a little bit more information. So if you have ever taken an exam and you scored in the 75th percentile, that means that you scored better than 75% of all the other people who took that particular exam. And so if we're wanting to look at near 30 percent, 33 percent, then that would be the 33rd Gentile. And so it's very, some work easy way to look at it as like with our coins. It's like $0.01. It's 1%. A dime, 10, $0.10 quarter, 25 cents. So that's kind of an easier way to help you remember it. And so like if I wanted to find the 75th percentile or the third quartile for the dataset that we just had, then instead divided by 2. Or if I wanted to multiply, I would multiply it by 0.5, which will be the same equivalent. So here I would just take the 111 and multiply it by 0.75. And so the 75th percentile for that data set that we were just looking at is 83.2. And then from there we would just use a formula to plug in and find where exactly that particular score would be. So the mean is very simply, the average score. Requires variables measured at the interval ratio level. But it's often used with the ordinal level variables. Really shouldn't do that. It's really bad practice to do this because it violates a couple of assumptions, but some people do. Well. And when you get to the other classes, we'll talk more about how to handle that. But for the most part, it really just needs to be used for interval ratio. Because it doesn't essentially you can't have a 4.5 agreement or a level of agreement. So it doesn't make sense to use at the ordinal level variable. So cannot be used for nominal level variables at all. Definitely can't be 0.5 green. And so kind of the same reason that you really should not use it at the ordinal level either because you can't be 0.5 agreeable. So the mean or arithmetic average is by far the most commonly used measure of central tendency. So the formula for the mean is just the x bar. That's the symbol here, equals the mean. And so you have your sum symbol here, and you have this symbol for scores. So when you see this, this just means the sum of the scores. And then n is the number of cases that you have. So some of the characteristics of the mean, the mean balances out all of the scores in a distribution. So all scores cancel out around the mean. So if we was to take each individual score and minus it from the mean, and then some that together it should always equals 0. And we'll talk more about that in the next chapter's lecture video. The mean is the point of minimised variation of the scores. So the least squares principle. So again, that has to do a lot more with it. Levels of variants. And so again, we'll talk more about that next lecture video. The mean is affected by all scores. So all scores are used in the calculation of the mean. And so this makes it very difficult because when you have extreme scores and it really throws off the demand and the main can actually be very misleading. And so a lot of times you would want to look at your outliers and maybe report both means or report the median. And I mean, we're just kinda talk about it in some way so that you're not misleading your audience when you do have a skewed mean. So a strength of the mean is that the mean is used at the available information from the variable. So it uses all of the information. Some of the weaknesses though, is that it is affected by every score. So our high scores and low scores or outliers, do you affect it? If there are some very high or very low scores? So what we're talking about here would be a skewed distribution, then the mean can be very misleading, like I just talked about. Here, is kind of an example. And so all you do is, you know, this fancy formula let me into really is that you add up all the numbers that you have here. And so if we totaled them together, we would have 695. And then I have 21 scores. And so I take the 695 and I divide it by 21. And that would give me a mean of 33.1. And so when I'm talking about it being sensitive to outliers, Let's say that instead of the 60, we had 600. And so that would be a very far extreme from where our data actually is. That if we want to have a 600 instead of 60, that would change our mean, 258.8. And so 58.8 would be nowhere near our central of this particular data set. And so a lot of times the median is the most appropriate when there are extreme outliers like because in that case than the median will be centered, more central to where the rest of the cases are. So the mean for group data is simply the sum of the frequencies times and then points divided by n. And so we find the midpoints for each of our intervals. And then we multiply that by the number of frequencies. So here we have 8 times 8, which gives us 464. So you keep going down through. And once you find that Then you divide by n. And so in this particular case, N equals 100, 15. And so you would, if we added amount together, then we would have books like my numbers are off a little bit. Canada up to 404,615 divided by 15. And then that would give us 40, 0.1. And so, and so this, ignore this right here. This right here should, or you want to go to. And so that's really pretty simple. I mean, it's a lot of work. But again, don't let like all these fancy equations that you off. But I think that's where a lot of people get intimidated about stats. But if you break it down into, you just take this and you multiply it by that. And then you add all this together and divide by that, then truly not that hard. You just have to pay attention to detail. All right? And so means medians and skew. When a distribution has API very high or low scores, the mean will be pulled in the direction of the extreme scores. For a positive skew, the mean will be greater than the median. And for a negative score and the mean will be less than the median. And so that is essentially how you can tell the difference between the two. So some people say if the tails to the left or to the right, that then that gets complicated as unto, you know, which lacked what right. And so positive and negative is just a direction. But the best way to be able to tell which way it's skewed. It, It's positive than the mean will be greater than the median. And if it's negative, the me, an OB less than when an interval ratio level variable has a pronounced skew. The media and maybe the most trustworthy measure of central tendencies. And so that's pretty much sums up what we're talking about with making sure that we know why we're reporting them. And some of the assumptions behind in order to make sure that we don't report something that doesn't make sense. And so we would end up with what are those studies that says that the average household has 2.3 kids, um, because you can't really have 0.3 of my kids. So that's make sure that you know, all these assumptions so that you don't end up saying something like that, right? And that pretty much covers central tendency.