0:00:00.520,0:00:02.820 Let's say I've got[br]a set of numbers. 0:00:02.820,0:00:08.710 2, say I've got three 3's, I've[br]got a couple of 4's, and 0:00:08.710,0:00:10.620 I've got a 10 there. 0:00:10.620,0:00:14.210 And what we want to do is find[br]the middle of these numbers. 0:00:14.210,0:00:17.820 We want to represent these[br]numbers with the center of the 0:00:17.820,0:00:19.900 numbers, or the middle of the[br]numbers, just so we have a 0:00:19.900,0:00:24.350 sense of where these numbers[br]roughly are. 0:00:24.350,0:00:27.360 And this central tendency that[br]we're going to try to get out 0:00:27.360,0:00:31.020 of these numbers, we're going[br]to call the average. 0:00:31.020,0:00:34.990 The average of this[br]set of numbers. 0:00:34.990,0:00:38.840 And you've, I'm sure, heard the[br]word average before, but 0:00:38.840,0:00:41.110 we're going to get a little[br]bit more detailed on the 0:00:41.110,0:00:44.270 different types of averages[br]in this video. 0:00:44.270,0:00:46.890 The one you're probably most[br]familiar with, although you 0:00:46.890,0:00:49.950 might have not seen it referred[br]to in this way, is 0:00:49.950,0:00:58.290 the arithmetic mean, which[br]literally says, look, I, the 0:00:58.290,0:01:01.270 arithmetic mean of this set of[br]numbers, is literally the sum 0:01:01.270,0:01:03.420 of all of these numbers divided[br]by the number of 0:01:03.420,0:01:04.209 numbers there are. 0:01:04.209,0:01:07.440 So the arithmetic mean for this[br]set right here is going 0:01:07.440,0:01:17.070 to be 2 plus 3 plus 3 plus 3[br]plus 4 plus 4 plus 10, all of 0:01:17.070,0:01:19.230 that over, how many[br]numbers do I have? 0:01:19.230,0:01:22.870 1, 2, 3, 4, 5, 6, 7. 0:01:22.870,0:01:24.720 All of that over 7. 0:01:24.720,0:01:25.660 And what is this equal to? 0:01:25.660,0:01:34.560 This is 2 plus 9, which is 11,[br]plus 8, which is 19, plus 10, 0:01:34.560,0:01:35.520 which is 29. 0:01:35.520,0:01:41.290 So this is going to be equal to[br]29/7, or you could say it's 0:01:41.290,0:01:43.540 equal to 4 and 1/7. 0:01:43.540,0:01:45.240 If I got my calculator out,[br]we could figure out 0:01:45.240,0:01:46.480 the decimal of this. 0:01:46.480,0:01:50.990 But this is a representation[br]of the central tendency, or 0:01:50.990,0:01:52.450 the middle of these numbers. 0:01:52.450,0:01:53.670 And it kind of makes sense. 0:01:53.670,0:01:57.670 4 and 1/7, it's a little[br]bit higher than 4. 0:01:57.670,0:02:01.070 We're kind of close to the[br]middle of our number range 0:02:01.070,0:02:01.810 right there. 0:02:01.810,0:02:03.590 And you might say, well, it's[br]a little skewed to the right 0:02:03.590,0:02:04.700 and what caused that? 0:02:04.700,0:02:06.950 And well, gee, 10 is a little[br]bit larger than all of the 0:02:06.950,0:02:07.840 other numbers. 0:02:07.840,0:02:09.320 It's kind of an outlier. 0:02:09.320,0:02:13.370 Maybe that skewed this average[br]up, the arithmetic mean. 0:02:13.370,0:02:16.680 So there are other types of[br]averages, although this is the 0:02:16.680,0:02:19.200 one that, if people just say,[br]hey, let's take the average of 0:02:19.200,0:02:21.660 these numbers, and they don't[br]really tell you more, they're 0:02:21.660,0:02:23.940 probably talking about[br]the arithmetic mean. 0:02:23.940,0:02:30.270 The other forms of average,[br]though, are the median, and 0:02:30.270,0:02:32.290 this literally is the[br]middle number. 0:02:35.970,0:02:38.360 If there are two middle numbers,[br]you actually take the 0:02:38.360,0:02:40.730 arithmetic mean of those[br]two middle numbers. 0:02:40.730,0:02:43.310 You actually find the number[br]halfway in between those two 0:02:43.310,0:02:44.330 middle numbers. 0:02:44.330,0:02:47.260 So the median of this set[br]right here-- let me just 0:02:47.260,0:02:48.350 rewrite them. 0:02:48.350,0:02:57.060 So I have a 2, a 3,[br]3, 3, 4, 4, 10. 0:02:57.060,0:02:59.690 So, let's see, we have seven[br]numbers right here. 0:02:59.690,0:03:02.800 The middle number, if[br]I go 1, 2, 3, to the 0:03:02.800,0:03:04.140 right, we're there. 0:03:04.140,0:03:06.470 If we go 1, 2, 3 to the[br]left, we're there. 0:03:06.470,0:03:09.850 The middle number is[br]that 3 right there. 0:03:09.850,0:03:13.360 I just listed them in order, and[br]I said, well, look, 3, you 0:03:13.360,0:03:16.640 could think of it as the fourth[br]number from the right, 0:03:16.640,0:03:19.550 and it's also the fourth[br]number from the left. 0:03:19.550,0:03:21.360 3 is the middle number. 0:03:21.360,0:03:23.940 And this case, it[br]is the median. 0:03:23.940,0:03:29.280 So in this case, 3, if you use[br]the median, is our average. 0:03:29.280,0:03:30.360 And that also makes sense. 0:03:30.360,0:03:32.030 I mean, it's literally the[br]middle number, and if you look 0:03:32.030,0:03:35.670 at this set of numbers, it kind[br]of does represent the 0:03:35.670,0:03:37.790 central tendency of this set. 0:03:37.790,0:03:40.880 Now just to be clear, it was[br]very clear what the middle 0:03:40.880,0:03:45.350 number was, because I had an[br]odd number of numbers. 0:03:45.350,0:03:48.560 I had three on each side of the[br]three, so it was very easy 0:03:48.560,0:03:50.790 to figure out the median,[br]the middle number. 0:03:50.790,0:03:53.800 But if I had a situation-- let's[br]say I have the situation 0:03:53.800,0:03:57.850 where I have 2, 3, 4, and 5. 0:03:57.850,0:04:01.510 Let's say that's my[br]set of numbers. 0:04:01.510,0:04:04.350 Well, here, there is no[br]one middle number. 0:04:04.350,0:04:07.460 The 3 is closer to the left[br]than it is to the right. 0:04:07.460,0:04:10.040 The 4 is closer to the right[br]than it is to the left. 0:04:10.040,0:04:12.220 There's actually two middle[br]numbers here. 0:04:12.220,0:04:17.380 The two middle numbers here[br]are the 3 and the 4. 0:04:17.380,0:04:20.640 And here, when you have two[br]middle numbers, which occurs 0:04:20.640,0:04:24.430 when you have an even number[br]in your data set, there the 0:04:24.430,0:04:27.630 median is halfway in between[br]these two numbers. 0:04:27.630,0:04:31.070 So in this situation, the median[br]is going to be 3 plus 4 0:04:31.070,0:04:35.670 over 2, which is equal to 3.5. 0:04:35.670,0:04:37.610 And if you look at this data[br]set, that's not what our 0:04:37.610,0:04:44.430 original problem was, but if[br]you look at this data set 0:04:44.430,0:04:47.770 right there, you're actually[br]going to find that the 0:04:47.770,0:04:52.070 arithmetic mean and the median[br]here is the exact same thing. 0:04:52.070,0:04:53.230 Let's calculate it. 0:04:53.230,0:04:55.470 What 's the arithmetic[br]mean over here? 0:04:55.470,0:05:00.240 It's going to be 2 plus 3 plus[br]4 plus 5, which is what? 0:05:00.240,0:05:09.490 5 plus 9, which is equal[br]to 14, over 4. 0:05:09.490,0:05:10.910 And what's this equal to? 0:05:10.910,0:05:16.740 14/4 is 3 and 2/4, or 3 and[br]1/2, the exact same thing. 0:05:16.740,0:05:19.310 So for this data set, they[br]were the same thing. 0:05:19.310,0:05:24.020 For this data set, our median[br]is a little bit lower. 0:05:24.020,0:05:28.090 It's 3, while our arithmetic[br]mean is 4 and 1/7. 0:05:28.090,0:05:30.080 And I really want you to think[br]about why that is. 0:05:30.080,0:05:32.840 And it has a lot to do with this[br]10 that sits out there. 0:05:32.840,0:05:37.190 All of these other numbers are[br]pretty close to whichever 0:05:37.190,0:05:39.590 average you want to pick,[br]whether it's the arithmetic 0:05:39.590,0:05:42.560 mean or it's the median. 0:05:42.560,0:05:47.860 But this 10 is kind of[br]an outlier, or it 0:05:47.860,0:05:50.650 skews the data set. 0:05:50.650,0:05:53.790 Maybe it's so much larger than[br]the other numbers, that it 0:05:53.790,0:05:57.680 makes the arithmetic mean seem[br]larger than maybe is 0:05:57.680,0:05:59.670 representative of[br]this data set. 0:05:59.670,0:06:01.500 And that's something important[br]to think about. 0:06:01.500,0:06:07.570 When you're finding the average[br]for something, most 0:06:07.570,0:06:09.740 people will immediately go[br]to the arithmetic mean. 0:06:09.740,0:06:13.520 But in a lot of cases, median[br]will make a lot more sense, if 0:06:13.520,0:06:16.970 you have these really large or[br]really small numbers that 0:06:16.970,0:06:18.460 could skew the data set. 0:06:18.460,0:06:21.530 I mean, you can imagine, if this[br]wasn't a 10-- or let's 0:06:21.530,0:06:22.790 imagine adding another[br]number here. 0:06:22.790,0:06:27.810 If I added the number 1 million,[br]if I added 1 million 0:06:27.810,0:06:30.390 to this data set, if that was[br]the eighth number, the 0:06:30.390,0:06:32.260 arithmetic mean is going[br]to be this huge number. 0:06:32.260,0:06:36.400 It's going to be much larger[br]than what is representative of 0:06:36.400,0:06:38.470 most of the numbers[br]in this data set. 0:06:38.470,0:06:40.240 But the median is still[br]going to work. 0:06:40.240,0:06:43.730 The median is still going to be[br]about 3 and a half, right? 0:06:43.730,0:06:47.750 If you had 1 million here,[br]it would be 1, 2, 3, 4. 0:06:47.750,0:06:49.200 The middle two numbers[br]would be that. 0:06:49.200,0:06:50.580 It would be 3 and 1/2. 0:06:50.580,0:06:53.900 So the median is less sensitive[br]to one or two 0:06:53.900,0:06:58.430 numbers at the extremes that[br]otherwise would skew the mean. 0:06:58.430,0:07:01.130 Now, the last form of average[br]I want to talk 0:07:01.130,0:07:02.840 about is the mode. 0:07:05.400,0:07:08.000 It has nothing to do[br]with ice cream. 0:07:08.000,0:07:10.910 The mode is literally the[br]most frequent number. 0:07:16.800,0:07:19.880 And in this data set, it's[br]pretty clear what the most 0:07:19.880,0:07:21.130 frequent number is. 0:07:21.130,0:07:25.820 I only have one 2, I have three[br]3's, I have two 4's, I 0:07:25.820,0:07:28.445 have one 10, and even if want to[br]include the million, I only 0:07:28.445,0:07:29.710 have one million there. 0:07:29.710,0:07:31.990 So here, the number[br]that occurs most 0:07:31.990,0:07:35.360 frequently is the 3. 0:07:35.360,0:07:38.330 So, once again, the mode seems[br]like a pretty good measure of 0:07:38.330,0:07:41.260 central tendency or a[br]pretty good average 0:07:41.260,0:07:43.300 for this data set. 0:07:43.300,0:07:45.950 Now the mode, it's a little[br]tricky to deal with, and you 0:07:45.950,0:07:49.070 won't see it used that often,[br]because it becomes a little 0:07:49.070,0:07:52.920 ambiguous when-- you know,[br]look at this data set: 0:07:52.920,0:07:55.270 2, 3, 4, and 5. 0:07:55.270,0:07:56.370 What is the mode there? 0:07:56.370,0:07:59.200 All of these numbers are[br]equally frequent. 0:07:59.200,0:08:01.360 So if you have a situation[br]like this, then you might 0:08:01.360,0:08:03.930 just-- the mode really[br]loses its meaning. 0:08:03.930,0:08:06.350 It might force you anyway to[br]take the median or the 0:08:06.350,0:08:07.530 mean in some form. 0:08:07.530,0:08:10.700 But if you really do have[br]numbers that one shows up a 0:08:10.700,0:08:14.640 lot more than the other, then[br]the mode starts to make sense. 0:08:14.640,0:08:18.220 So, hopefully, this has given[br]you a pretty good overview of 0:08:18.220,0:08:25.400 how to represent the central[br]tendency of a data set. 0:08:25.400,0:08:26.430 Very fancy word. 0:08:26.430,0:08:28.180 But it's just saying, look,[br]we're trying to represent with 0:08:28.180,0:08:30.200 one number all of this data. 0:08:30.200,0:08:31.810 And you might say, hey, why do[br]we even worry about that? 0:08:31.810,0:08:34.059 It only has seven numbers here[br]or eight numbers here. 0:08:34.059,0:08:36.770 But you can imagine if you had 7[br]million numbers or 7 billion 0:08:36.770,0:08:39.440 numbers, and you don't want to[br]show someone all of that data. 0:08:39.440,0:08:42.210 You just want to give someone a[br]sense of what those numbers 0:08:42.210,0:08:45.345 are on average. 0:08:45.345,0:08:49.420 And as we said, the arithmetic[br]mean is what I see being used 0:08:49.420,0:08:53.170 the most. But in situations[br]where you might have numbers 0:08:53.170,0:08:55.940 that would skew the arithmetic[br]mean, because they're so large 0:08:55.940,0:08:58.670 or they're so small,[br]the median might 0:08:58.670,0:09:00.480 make a lot of sense.