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One of the most common things we want to do with the data set is to get a measure of its center.

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And there are many ways to think about this idea of center, the first of which is the idea of mean.

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And this is the most common measure of center that we use.

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We usually think about mean as average.

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But when we're being a little more technical, we call it the mean or the arithmetic mean.

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And this here is the formula that we use to calculate mean.

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But really all this formula says is that we are to add up all of the data points in our data set, all

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the values in our data set, and then divide by the number of data points or the number of values that

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we have in the data set.

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So the way that we read this formula here is to say that this sigma notation here, this big E is called

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sigma.

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It means to sum up.

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And so we are adding up here all of the values of X and we use this little indicator I here to mean

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the first value of x, second value of x, third value of x, etc. So this I equals one notation below.

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The sigma notation means start with the first value and keep adding until you get to the nth value.

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So if there are ten values in the data set, this says start with the first value in the data set,

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the I equals one value and add that to the second value, the third value, the fourth value, etc.

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All the way up to the end equals 10th value.

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We're adding all the values together in the data set and then dividing by N equals ten the number of

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data points in the set.

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So this is just a fancy formula.

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To summarize what we're doing here, sum all the data points, divide by the number of data points in

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the set.

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So for instance, if we have a data set like this one, a very simple data set, we can see we have

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five points in the set one, two, four, six and seven.

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There are five values in this data set to use our formula here to find the mean.

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We would simply add all of these values together and then divide by five because there are five values

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in the data set.

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The result then, of course, the sum of all of the values in the numerator is 20, 20 divided by five

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is four.

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And so we can say here that the mean is 20 divided by five, which is equal to four.

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Now, the way that we indicate mean depends on what we're talking about.

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We mentioned earlier the idea of population versus sample.

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Well, if we're taking the mean of the population, let's say that this data set here accounts for the

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entire population, then we usually use the Greek letter MU, which looks like this U here.

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And we say that the mean is given by MU and so we would say mu is equal to for the mean of the population

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is equal to four.

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If on the other hand we're taking the mean of a sample, we usually indicate that with this notation,

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which we read as x bar, it's just x with a bar over the top of it.

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And so if you see x bar, it usually means the mean of a sample of the population.

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Whereas this notation here mu usually means the mean of the entire population.

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The formula is the same in both cases, right?

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This could be x bar or it could be mu.

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We would still use this same formula to sum all the data points and divide by the number of data points.

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But we would use x bar to indicate that value.

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If we are finding the mean of the sample we use.

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Mu.

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If we are finding the mean of the population, we really want to think about the mean either way, whether

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it's the mean of the sample or the mean of a population we really want to think about.

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The mean is the balancing point of the data.

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And we also want to realize that there are some different things we can do with mean that are interesting,

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including for instance, take a weighted mean.

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So let's say we work at a company and we've asked every department manager to survey all of the employees

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in their department and get from them an employee satisfaction score.

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On a scale of 1 to 10, ten being perfectly satisfied, one being extremely unsatisfied.

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So with this first data point here, we have a department that includes 20 employees and the department

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manager.

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The department head has reported to us that the mean employee satisfaction score in their department

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is 8.4.

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Another department manager has told us that of their seven employees, the mean employee satisfaction

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score is 6.1 in that department.

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So what you see here is that we have all different department sizes across our company and we're getting

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a different employee satisfaction score from each department head, each department head.

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Each department manager has already taken a mean for their own department.

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When we have a situation like this, we have to take a weighted mean if we want to get a mean employee

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satisfaction score for the entire company.

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Because obviously here we can get kind of an intuitive sense that this 8.4 value for satisfaction counts

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a little heavier or should count a little heavier than this employee satisfaction score here of 6.1,

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because there are 20 employees in this department, only seven employees in this department.

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So intuitively, we should wait this 8.4 a little heavier than we weight this 6.1.

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So how do we do that?

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Well, the formula is virtually the same as what we did up here, except that we're accounting for this

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different weighting.

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So what we do is we sum up all of the products of the actual score and the weighting, and then we divide

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that by the sum of all of the weights.

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So if we do that with this data set here, here's what that looks like.

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We take the product of 20 and 8.4, we add that to the product of seven and 6.1.

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Then we add to that 13 times 9.1 plus 25 times 7.8.

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So we're taking all these products in the numerator, adding them together, and then we're dividing

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by the total weight.

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This denominator should make intuitive sense to us because if we go back to our original formula for

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the mean, we're dividing by the number of data points down here, we know that if we add up all of

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these department sizes, we get 20 plus seven, plus 13 is 40, plus 25 is 65.

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We have 65 total employees across four departments.

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So this denominator here is really just like the denominator up here.

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We have 65 total employees, so we kind of have this same idea of total number of data points in the

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denominator of each formula.

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The only thing that's different is this numerator.

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And in the numerator here, that's where we account for the different weights based on these different

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department sizes.

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And if we were to actually do this math, what we'd see here is that we get an approximate employee

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satisfaction score of 8.1, and that is a weighted mean based on the data that was turned in by our

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department managers.

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So we can calculate a mean, we can calculate a weighted mean.

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It's also important to realize that we can calculate what's called a truncated mean.

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This comes into play when we have data where maybe we've introduced some outliers.

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For instance, if we have this data set here, notice that 16, 18, 21, etc. all the way to 33 this

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part of the data set seems to be fairly consistent and then all of a sudden we have this enormous 91

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value in the data set, which looks to be an outlier.

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It looks to be outside of the normal range of the rest of the data set.

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So it's possible, based on the context of how we collected this data, that this 91 value is an extreme

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value, it's an outlier.

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It doesn't really fit with the rest of the data.

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Depending on why we think this value occurred, we may choose to calculate a truncated mean.

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And what we would do in that case is say this 91 value looks like an outlier.

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So what we're going to do is we're going to ignore it, so we're going to ignore this 91.

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But because we're taking a value off of the high end of the data set, we're also going to take a value

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off of the low end of the data set to kind of balance it out, leaving us with just these six values

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in the data set between 18 and 33.

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From there, then we would just calculate the mean of this six point data set.

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We would add 18, 21, 27, 32, 32 and 33.

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We'd add those together and divide by six because we have six values here and we would calculate a mean,

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but it would be very important if we did that to indicate that we eliminated 25% of our data because

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there were eight values in the original dataset, we took away two of them.

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That means we took away 25% of our data set in order to calculate a truncated mean.

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If we're ever going to do something like this, we need to be really, really careful when we do it,

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because if we're taking out an outlier, doing so may give us a more accurate picture of what's going

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on.

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But it may also give us a less accurate picture of what's going on.

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Maybe this 91 value that we got in our survey or that's sitting out here in our sample or our population.

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Maybe this value is totally accurate and extremely important to include in the data set.

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Or maybe something just went wrong with this 91 value and it is more appropriate to eliminate it to

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get a more accurate mean.

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We may feel fairly confident about eliminating this 91 value or we may not.

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Either way, if we take away a value from our dataset, it's very important to indicate and to communicate

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to anybody looking at this data, looking at the mean that we calculated that we did remove some outlier

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data, that we removed 25% of the data set and that we're actually calculating here a truncated mean

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as opposed to a.

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FRUMIN With the original data set.

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We always want to be really, really, really careful whenever we are manipulating the original data

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set that we collected.

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But this is something that we can do if we have really good reason to believe that we'll get a more

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accurate picture of what's happening with the data.

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And then we just want to make sure to communicate that that's in fact what we have done.

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So this is kind of the general idea of mean.

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Let's look at a second measure of central tendency, which is the mode.

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Now, the mode of a data set is an even simpler concept than the idea of the mean of the set.

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It is simply the value that occurs most often in the set.

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So for instance, if we're looking at this data set, we can see that every value in the set occurs

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once except this value here of 32, which occurs twice.

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So because the value 32 occurs more than any other value in the set.

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We say that 32 is the mode of the data set.

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That would be the mode.

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We don't say that.

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Two is the mode of the data set.

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Sometimes people think that the count of that highest value will be the mode.

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The mode here is not.

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Two the number of times the 32 occurs, the mode is actually 32.

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So we could say that the mode of the data set is 32.

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Now, a couple of things to think about with mode, we can also have a bimodal data set.

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So this is the same set except that we have inserted an extra data point, an extra value of 18 here.

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So 18 occurs twice and 32 occurs twice.

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So we might say that this data set is bimodal, that it has two modes a mode of 18 and a mode of 32.

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So 18 and 32 and indicate that this is a bimodal data set.

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Or we can have the scenario where the data set has no mode.

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In this case, every data point occurs only one time.

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We have 16, 18, 21, 27, 32, 33, 91.

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There's no value in the set that occurs more than any other value in this set.

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There's no standout value.

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And so we would say that this data set has no mode.

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Now we'll talk a little bit more about mode when we talk about when to use different types of central

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tendency.

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But this is the idea for now and then the last measure of central tendency that we want to talk about

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is the idea of the median.

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Now, the median of a data set is the center of the data set that we find simply by ordering the entire

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data set and looking for the value in the center.

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So given this data set here, we've ordered the set from smallest to largest data point.

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And so all we want to do is work our way in toward the center, will eliminate here or ignore the smallest

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and the largest value.

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We keep working our way in until we get to the center and we see here then that the value left in the

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center is 27.

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So for this set, the median is 27.

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If we do the same thing for this set, working our way from the outside toward the inside, the data

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set is already ordered from smallest to largest.

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That's important when it comes to the median.

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We have to have the data in order.

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So if we start working our way in this way, we're left with now two values in the center, 27 and 32.

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Now, for the median in particular, we can't just pick the value of 27 or pick the value of 32 because

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neither one of them is exactly the center.

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So what we say is when we have an even number of data points in our set, we'll always be left with

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two data points that are at this center point position and then in order to calculate median will actually

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go ahead and take the arithmetic mean take the mean of these last two centralized data points.

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So with 27 and 32, we say 27 plus 32 and we have two data points here.

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So we divide by two.

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We end up with 59 over two.

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And the result then is 29.5.

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And so the median of this data set is 29.5, which is really just the mean of 27 and 32, the two most

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centralized data points when we order the data set from smallest to largest.

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So at the most basic level, that's how we calculate mean, median and mode.

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But what's even more interesting is thinking about when to use each of these measures of central tendency

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based on the kind of data we have and the context around our data.

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So earlier we talked about all different kinds of data, including continuous versus discrete data,

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nominal versus ordinal data, numeric versus non numeric data.

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And now we've just introduced these ideas of mean median and mode, these three different ways of measuring

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central tendency of measuring how the data collects itself around some center point, identifying the

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center point of a data set.

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What's far more interesting is thinking about when it's appropriate to use mean, when it's appropriate

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to use median, when it's appropriate to use mode.

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It might even be helpful to list more than one of these measures of center, depending on the data we're

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looking at.

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And not all of these measures are always going to make sense.

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For instance, remember we said earlier that nominal data was data that couldn't be ordered at.

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As an example, we said species of animals.

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So if we think about household pets like dogs, cats, birds, etc., that's nominal data.

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It's data that doesn't really have an order to it.

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If that's the case, if we can't put data in a specific logical order, then there's really no way to

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calculate a median for that data.

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It really just doesn't make sense.

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If you remember when we learned to calculate median, we said that we had to line up the data in order

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from some sort of smallest to largest or one end to the other spectrum and then find the center point.

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And that's just really not applicable at all if the data is not nominal.

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Similarly, when we calculated a mean, we add it up all the data points and then divide it by the number

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of data points, that's not going to make sense.

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If we have non numeric data, we obviously can't find a sum for a data set that is non numeric that

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isn't numbers.

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We can't add letters, we can't add animal species, we can't add colors in a mathematical numeric sense.

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So calculating a mean for non numeric data just doesn't really make sense.

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But in all these other cases we should be able to calculate some kind of mean median mode for the data.

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For instance, we could think about continuous data like a measurement of height for a group of people.

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And we could imagine, of course, that we could calculate a mean height for that group of people.

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We could also think about a set of discrete data, maybe the number of children per family in a data

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set of families.

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Of course, that's going to be a discrete number.

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Every family is going to have zero children, one child, two children, three children, etc. No single

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family is going to have 2.43 children.

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So that would be an example of discrete data.

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But of course, we could find a mean for that data simply by adding up the total number of children

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and then dividing by the total number of families.

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We would get a mean.

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But keep in mind that in that example we might come up with a mean of, let's say 3.17 children per

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one family, and we have to think about and interpret the context of that data.

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We understand that no individual family is going to have 3.17 children while at the same time understanding

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that the mean of the data set could be 3.17 children skipping around here.

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We can think about, for instance, the mode of numeric versus non numeric data.

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We've already looked at the mode of numeric data.

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We had that data set earlier where the data point 32 occurred twice in the data set, and we said that

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the mode of that data set was 32, but we could have non numeric data and indicate a mode.

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For instance, maybe our family has two dogs.

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And one cat.

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And one bird.

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As pets.

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We would say that the mode of this data set is dogs.

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This is non numeric data in terms of dogs, cat and bird, in the sense that our data set would be dog,

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dog, cat bird.

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That's non numeric data, but we could clearly see there that in the set dog, dog, cat bird, the

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mode is dog.

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And then maybe one last example.

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Let's look at the mean of nominal versus ordinal data here.

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Remember that nominal data is data that does not have some kind of specific order.

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It can't be ordered versus ordinal data can be ordered.

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We can put an ordinal data set in a logical specific order.

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And the question then of whether we can calculate a mean for a data set that can't be ordered or calculate

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the mean for a data set that can be ordered is dependent.

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It depends on whether or not the data is numeric or non numeric.

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For instance, you could imagine an ordinal data set of grades, letter grades in a class.

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So if a professor is handing out grades A through F, that's data that can clearly be ordered, but

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we wouldn't be able to calculate a mean of that data set because this is non numeric data.

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And so we can't add up these grades and then divide by the total number of grades if instead of letter

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grades were being given percentage grades, for instance, students have grades like 92% in the class

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or 73% in the class.

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Now we have numeric data and we could calculate a mean, but if all we have is letter grades A, through

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F, B plus and C minus, etc., we can order that, but it's not numeric, so we can't calculate a mean.

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So maybe we'll be able to calculate a mean for ordinal data if it's numeric data.

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Same thing with nominal data.

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Nominal data cannot be ordered, but we may still be able to calculate a mean if it's numeric data.

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Now, if it's numeric, presumably we should be able to order it.

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But the idea here is really just that we have to have numeric data if we're going to be able to calculate

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a mean.

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So this gives you a general idea of when we may be able to use mean median and mode depending on the

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kind of data we have.

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And of course these different types of data are not all exclusive.

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We can have continuous data that is ordinal and numeric, or we can have discrete data that is nominal

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and non numeric.

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So these are overlapping categories.

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We just have to think about which measures of central tendency are appropriate based on the kind of

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data we're looking at.

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And then the last thing is really where the art form of all of this gets introduced, which is this

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idea of when it's most appropriate to use each of these particular measures of central tendency.

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For example, in economics, in politics, we very often talk about annual household income for the

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population of a country.

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And when we talk about that statistic, we usually refer to median annual household income, not mean

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annual household income.

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And the reason is because there are very often a very small number of extremely, extremely, extremely

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high income earners.

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You think about the very wealthiest people in a country, the very wealthiest people in the world whose

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incomes are so, so, so large that including them in the dataset would drastically skew the mean of

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the dataset to the point we're talking about mean annual household income would really not give us a

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good representation of typical annual household income, whereas median is a much better representation

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of annual household income because when we look at the median, we eliminate all of those high income

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earners, we eliminate all of those extreme outlier high income earners.

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We just look at the center point, the median point for household income.

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That tends to give us a much, much more typical look at annual household income for the population

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of a country.

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So when we talk about this statistic, we almost always refer to the median instead of the mean.

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You can even think about a different example.

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Imagine we have a race, maybe it's an Ironman triathlon and the race has a time limit.

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Let's say there's a segment of the race that must be completed within 90 minutes.

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So we say that the maximum time allowed is 90 minutes for this segment of a race.

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And for any participants who don't complete that segment in 90 minutes, they're disqualified from the

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rest of the race.

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Their times don't count.

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If we have a scenario like that, we have a bunch of people who are going to finish under this 90 minute

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time cap, and then we're going to have a bunch of people who are disqualified who hit this 90 minute

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time cap and then they are disqualified from the race.

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If we want to take data of those participants and calculate some sort of a center for how long it.

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Took the participants to finish this particular leg.

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This particular segment of the race.

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In this scenario, median would be a much better tool to use to calculate that measure of center than

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mean, because, I mean, it doesn't really give us a way to handle all of the people who were disqualified

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when they hit this 90 minute time cap.

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But for median, you could imagine that we would line up all of the participants data points in order.

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So you can think about lining up our data set like this.

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And then we have a bunch of people at the top who were disqualified indicating those people here in

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red.

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Well, assuming we have enough participants, these people who were disqualified won't skew our measure

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of center if we use median because we simply cross them out as we move to the center here and we end

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up then with just these people in the center to calculate the median for the data set and get an actual

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idea of center.

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So in a scenario like that one, we may be forced to use median instead of mean.

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The same would be true here with our letter grades A, B, C, D, and F.

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We couldn't calculate a mean there, but we could find a median of letter grades.

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And the whole idea here, the whole takeaway is just that when we're thinking about how to look at the

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central tendency of the data, we need to first think about whether or not it's even possible to calculate

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a mean and or a median and or a mode.

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It may not even be possible to find all three of these measures for the data set.

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So our first step is to think about what we actually can calculate.

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And then once we know what we can calculate, to think about which measurement makes more sense.

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So going back to the annual household income example, in that case, it would be possible to calculate

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a mean, a median and a mode.

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We would be able to find all three measures of central tendency for that data.

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So that's our first step.

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We have all three of these possibilities available to us.

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But then our second step is to think about which of these three is most appropriate to represent the

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data set.

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And that's where we realize that median is going to give us a truer picture of what's going on with

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this data set.

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Then the mean will the mode may give us somewhat of an accurate picture, but the race here is probably

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going to be between median and mode in this example with the race with the time limit.

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Going back to our first step, which values can we calculate?

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We may not even be able to calculate a mean if some of our participants didn't finish in this 90 minute

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time cap.

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So then the only options available to us are median and mode.

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And based on the data set, we could decide which measure is more appropriate.

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Which measure gives a better picture of the typical finishing time of our participants.

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So as you're thinking through these measures of central tendency and getting comfortable working with

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them, always come back to your two step process of number one.

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Which of these measures can I calculate?

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Number two of the measures I can calculate which is the most appropriate measure to use to best represent

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the center of the data set.