I want to explain the difference between the mean and the median. I think it's one of those things that it's easy to kind of get confused, or I find that a lot of people want to talk about averages, and what's the average, and really, the median is a more representative way of talking about a dataset. So, I'm just going to use this visual example to try and show you the difference between the two, and how an average can end up being kind of skewed or distorted in some way based on some outlying information and outlying data. So, if you have a number line like this, let's say we're talking about the age of a group of people. If we put those ages on this number line, and so we have five people. They all happen to be kids, and their ages are 2, 3, 5, 7, and 8 years old. I said, "How can we describe the ages of that group?" Well, you could do it based on the mean or same thing as saying the average. So, if you took the average of those ages and literally add them up, divide by the total number, you get a mean of 5. So, if I said, "How old are those kids?" and you said, "Well, the average they're five years old." Then, I would know what you're talking about and I would say, "Okay, I know how old they are." It turns out that the median is also five. The median just takes the the number of values and divides them up into two categories. So, in this case, the values have been sorted from lowest to highest and it says which value is in the middle. So we have half of the values are above the median, half the values are below the median. It doesn't matter what those values actually are as long as we know that they've been sorted and we've divided them into half above, half below. So, here we have a mean and a median that are equal. So, I could use either one to describe this dataset, and it would make sense if you would understand what we're talking about. What happens though if one of the kids is not as close in age to the other ones, and maybe is 13 years old instead of the other ones being between 2, 3, 5, and 7, so now, the mean has been changed, because remember we're adding up the total and then dividing by the number of values. Now, our mean has gone from 5 to 6. The median though has not changed, because that last value, all it means is that it's still above the halfway mark in terms of how many values we had when we sort them, put them into two groups. So, our mean is now not as representative necessarily, because four of the people in this group are 2, 3, 5, and 7. So, if you said that the mean is 6, yeah, I guess it's still not that bad. But what happens if we have a kid that's not 13 years old, but it's actually really just a kid at heart and it's actually 53 years old? So, now we have five people in this group and four of them are between the ages of 2 and 7 and one of them is 53 years old. So look what happens to the mean here. Now, we have a mean of 14 years old. So, if I said, how old is to the people in that group, and you said, 14 is the average. Then, I would say, "Okay, so they're all teenagers. Or you'd kind of have them in your mind when really four of them are on the age of seven or less and one of them is as an adult. So, notice though that the median is still five. This is really, I hope a good way of visualizing the difference between the mean and the median. That's why often when you're talking to anyone who's sort of more versed in statistics is, if you say an average, yes, it's something that people can relate to. If it really is generally representative, there's no harm done, but if there's any kind of outlier happening if you have somebody that's way outside of the group, like for example if you had income for a group and you know everyone makes around the same amount of money, but then you have some billionaire added to that group. Then you've got one person who's completely skewing the average, but the median would still be representative. So, I just wanted to make sure that those two concepts are clear, it really does make a difference when you're looking at data, when you're classifying it for mapping. If we look at how the mean and the median relate to our mapping data here, so this is the classification dialog box and ArcMap and here's our distribution, and so here's the median for this dataset, and here's the mean, and so the average is a little bit higher than the median. That makes sense because you can see that there's a couple of outliers here that are dragging the mean higher than the median. In other words, they're skewing the data a little bit because of the fact that we have these outliers. With the data values I'm using here, which are income values for census tracts, I'm showing both the median income and the average income, and to show you the difference. So, these are actually being calculated for each of these areas, and you'll notice that towards the low end of the data range, the values are not that far apart. The average is still a little bit higher than the median, but maybe $7,000 difference, something like that. But if you look at the high end of the range, you actually have a difference between the average and the median of about $160,000. So, what is more representative of that group? This is probably, you've got some very wealthy people living in that neighborhood, but not that many of them necessarily, because the median is much lower. So, that tells me that the median is probably going to give us some of representative number in terms of describing the income levels of the people that live in that particular census tract.
