1 00:00:00,090 --> 00:00:06,060 So we've talked about different ways to measure the center of a data set and the spread of that data 2 00:00:06,060 --> 00:00:06,540 set. 3 00:00:06,660 --> 00:00:13,590 Now we want to talk about the quartiles and I.Q. R or interquartile range of a data set so that we can 4 00:00:13,590 --> 00:00:17,190 understand what they can tell us about our data. 5 00:00:17,340 --> 00:00:22,740 So the easiest way to explain quartiles and IQ is to look at a data set. 6 00:00:22,950 --> 00:00:25,530 Let's say that we have this data set here. 7 00:00:25,680 --> 00:00:31,650 There are 18 values in this set, ranging from 66 to 75. 8 00:00:31,800 --> 00:00:38,810 And we're going to say that this data represents the golf score of 18 golfers. 9 00:00:38,820 --> 00:00:42,900 So maybe the golf course is a par 72 course. 10 00:00:43,020 --> 00:00:48,870 The best golfer shot a 66 and the worst golfer shot a 75. 11 00:00:48,960 --> 00:00:50,580 So we have 18 golfers. 12 00:00:50,580 --> 00:00:52,110 This is our data set. 13 00:00:52,110 --> 00:00:58,500 And the first thing we want to say is that this idea of a quartile is related to what we already know 14 00:00:58,500 --> 00:01:01,200 about the median of the data set. 15 00:01:01,200 --> 00:01:05,850 So if you remember when we talked about median, we talked about it as the center of the set. 16 00:01:05,850 --> 00:01:12,330 When we ordered the set from smallest to largest and found the center value. 17 00:01:12,450 --> 00:01:13,620 So let's start there. 18 00:01:13,620 --> 00:01:15,030 Let's think about the median. 19 00:01:15,120 --> 00:01:19,500 If we look at this data set, there are 18 values in the set, as we said. 20 00:01:19,500 --> 00:01:24,390 And so we have here the lower nine values and the upper nine values. 21 00:01:24,390 --> 00:01:34,740 Therefore, we know that the median is found by taking the mean of these two center values here. 22 00:01:35,010 --> 00:01:41,970 So if we take the mean of these two values, we would add 69 and 69 and then divide by two. 23 00:01:41,970 --> 00:01:44,970 And of course, the result there will be 69. 24 00:01:44,970 --> 00:01:49,770 So the median here of this data set is 69. 25 00:01:49,770 --> 00:01:56,070 Now, when we talk about the quartiles of a data set, we usually talk about the first, second and 26 00:01:56,070 --> 00:01:57,390 third quartiles. 27 00:01:57,390 --> 00:02:04,290 And in general, just at a broad level, you want to think about the first quartile as the median of 28 00:02:04,290 --> 00:02:06,060 this lower half of the data. 29 00:02:06,360 --> 00:02:11,790 You want to think about the third quartile as the median of the upper half of the data. 30 00:02:12,030 --> 00:02:16,230 And you want to think about the second quartile as being equal to the median. 31 00:02:16,230 --> 00:02:24,330 So we can refer to the first quartile sometimes as Q one, sometimes we'll refer to it as the first 32 00:02:24,330 --> 00:02:26,490 quartile or the lower quartile. 33 00:02:26,490 --> 00:02:32,430 So we'll call it the first quartile or we'll call it the lower quartile. 34 00:02:32,460 --> 00:02:35,040 We can also think about it as the. 35 00:02:35,800 --> 00:02:38,860 25th percentile of the data. 36 00:02:38,980 --> 00:02:45,610 On the other hand, the third quartile will often call it Q three or the third quartile. 37 00:02:45,610 --> 00:02:53,050 So instead of the first, we'll call it the third or we'll call it the upper quartile and we can think 38 00:02:53,050 --> 00:02:57,400 about it also as the 75th percentile of the data. 39 00:02:57,400 --> 00:03:02,290 And then the median here we call that the second quartile or Q two. 40 00:03:02,320 --> 00:03:09,190 So Q Two or the second quartile instead of the lower or the upper quartile, we would say that this 41 00:03:09,190 --> 00:03:13,900 is the median because the second quartile is equal to the median. 42 00:03:13,900 --> 00:03:22,390 And we would also refer to this as the 50th percentile because in the same way that the median divides, 43 00:03:22,390 --> 00:03:28,750 the lower half of the data from the upper half of the data, meaning that 50% of the data points will 44 00:03:28,750 --> 00:03:33,370 fall below the median and 50% of the data points will fall above the median. 45 00:03:33,370 --> 00:03:35,830 And therefore it's the 50th percentile. 46 00:03:35,830 --> 00:03:45,010 In that same way, 25% of the data points will fall below the first quartile, whereas 75% of the data 47 00:03:45,010 --> 00:03:48,970 points will fall below the third quartile or below the upper quartile. 48 00:03:49,000 --> 00:03:56,770 So the upper quartile divides the lower 75% of the data points from the upper 25% of the data points. 49 00:03:56,770 --> 00:04:04,210 The first quartile divides the lower 25% of the data points from the upper 75% of the data points. 50 00:04:04,210 --> 00:04:07,150 So these are our three quartiles. 51 00:04:07,150 --> 00:04:13,240 And one of the reason that these values are interesting to us is because they lead us to what we call 52 00:04:13,240 --> 00:04:14,860 a five number summary. 53 00:04:15,010 --> 00:04:21,880 And the five number summary gives us a really good picture of the center and the spread of the data 54 00:04:21,880 --> 00:04:23,500 set at the same time. 55 00:04:23,500 --> 00:04:29,320 So in the past we looked at the center of the data set as mean median or mode, and then we looked separately 56 00:04:29,320 --> 00:04:34,060 at the spread of the data set as the variance or standard deviation. 57 00:04:34,060 --> 00:04:39,190 But the five number summary kind of gives us a picture of both the center and the spread at the same 58 00:04:39,190 --> 00:04:39,760 time. 59 00:04:39,760 --> 00:04:45,610 So that five number summary is always going to include the same five values. 60 00:04:45,610 --> 00:04:49,510 It will include the minimum and maximum values from the data set. 61 00:04:49,510 --> 00:04:52,270 So the lowest value and the largest value. 62 00:04:52,270 --> 00:05:01,090 In the case of this data set, of course, that's 66 for the minimum value here and 75 for the maximum 63 00:05:01,090 --> 00:05:01,690 value. 64 00:05:01,720 --> 00:05:05,140 It will include the median, which of course we're already familiar with. 65 00:05:05,140 --> 00:05:11,830 And for this data set, we already calculated that that median was 69 and then it'll include the first 66 00:05:11,830 --> 00:05:18,190 and third quartile or the lower quartile and the upper quartile and without even calculating the lower 67 00:05:18,190 --> 00:05:23,620 and upper quartiles for this particular data set, you can already start to get a sense that this five 68 00:05:23,620 --> 00:05:29,500 number summary gives us a pretty good picture of the center and spread of the data set, because not 69 00:05:29,500 --> 00:05:36,460 only do we have the median as the center and we get the full range from the minimum to the maximum value, 70 00:05:36,490 --> 00:05:43,990 let's go ahead and say while we're at it, that we define the range of a data set as being equal to 71 00:05:43,990 --> 00:05:48,040 the maximum value minus the minimum value. 72 00:05:48,040 --> 00:05:51,640 So in this case, the range of the data set is 75. 73 00:05:52,360 --> 00:05:56,860 -66 or in our case, a range of nine. 74 00:05:56,980 --> 00:05:59,800 So we get the minimum, the maximum, the total range. 75 00:05:59,800 --> 00:06:01,930 We can see the center as the median. 76 00:06:02,140 --> 00:06:04,540 And these Q one and Q three values. 77 00:06:04,540 --> 00:06:11,650 Give us a picture of the middle 50% of the data set because we know everything below. 78 00:06:11,680 --> 00:06:19,060 Q One below the first quartile is the first 25% of the values in the set, and we know everything above. 79 00:06:19,090 --> 00:06:27,970 Q Three is the last 25% or the greatest 25% of the values in the set, but everything between Q one 80 00:06:27,970 --> 00:06:35,170 and Q three between the first and third quartiles represents that middle 50% of the data set of the 81 00:06:35,170 --> 00:06:36,540 values in the data set. 82 00:06:36,550 --> 00:06:39,160 And so those two values taken together. 83 00:06:39,190 --> 00:06:40,600 Q one and Q three. 84 00:06:40,630 --> 00:06:46,410 Give us a little bit of a picture of how tightly clustered the data is around the median. 85 00:06:46,420 --> 00:06:52,660 If the difference between Q one and Q three is fairly small, then we know that most of the data is 86 00:06:52,660 --> 00:06:54,340 tightly clustered around the median. 87 00:06:54,340 --> 00:07:00,190 But if the difference between Q one and Q three is large, then we know that the data is more spread 88 00:07:00,190 --> 00:07:00,700 out. 89 00:07:00,790 --> 00:07:07,720 So this five number summary given by these quartiles here is just another way to get a sense of what 90 00:07:07,720 --> 00:07:11,030 our data set is doing now while we're here. 91 00:07:11,050 --> 00:07:14,860 Let's go ahead and talk about how to calculate these quartile values. 92 00:07:14,860 --> 00:07:18,880 We already know how to calculate the second quartile because it's equal to the median and we know how 93 00:07:18,880 --> 00:07:19,900 to find the median. 94 00:07:19,900 --> 00:07:20,830 But how do we find. 95 00:07:20,860 --> 00:07:22,560 Q one and Q three. 96 00:07:22,570 --> 00:07:29,500 Well, the answer to that is a little nuanced, because technically there are different ways of calculating 97 00:07:29,500 --> 00:07:35,820 the first and the third quartile, and they're not always going to give exactly the same answer. 98 00:07:35,830 --> 00:07:42,780 There's no universally agreed upon method for finding the exact value of Q one and Q three. 99 00:07:42,790 --> 00:07:49,270 Just know that any accepted method we use is still going to give us a good approximation of what Q one 100 00:07:49,270 --> 00:07:55,030 and Q three actually are one of the methods you can use for calculating these two quartiles? 101 00:07:55,030 --> 00:08:02,320 And maybe the most straightforward one is to start by considering whether the original data set has 102 00:08:02,320 --> 00:08:05,260 an even or an odd number of data points. 103 00:08:05,260 --> 00:08:08,980 So in this case, this data set has 18 data points. 104 00:08:08,980 --> 00:08:12,940 So we have a data set with an even number of data points. 105 00:08:12,940 --> 00:08:19,600 If there are an even number of data points in the original set, then of course we already know, as 106 00:08:19,600 --> 00:08:26,650 we saw here, that the median is going to be found by taking the mean of these two numbers in the center, 107 00:08:26,650 --> 00:08:31,150 because with an even number of data points, the data is going to split into two halves perfectly. 108 00:08:31,150 --> 00:08:35,890 In this case, we have nine values in the lower half and nine values in the upper set. 109 00:08:35,890 --> 00:08:40,450 So of course, the median is going to be found by taking the mean of these two middle numbers. 110 00:08:40,450 --> 00:08:46,990 And then in order to find the first and third quartile, we can just look at each half of the data set 111 00:08:46,990 --> 00:08:50,590 and find the median of the lower and upper halves. 112 00:08:50,590 --> 00:08:57,490 So if we just look at the lower half here, starting with this lowest value, 66 and going up to this 113 00:08:57,490 --> 00:09:00,760 69 value right here, there are nine values. 114 00:09:00,760 --> 00:09:05,350 Now in this lower half, we can consider the median of these nine values. 115 00:09:05,350 --> 00:09:10,270 Well, of course, if we have nine values and we're looking for the median, we know we're going to 116 00:09:10,270 --> 00:09:15,760 have four values on the low side of this half for values on the high side of this half. 117 00:09:15,760 --> 00:09:20,860 And then the median is this fifth value in the center right here, 68. 118 00:09:20,860 --> 00:09:24,100 And then we would consider the same thing with the upper half. 119 00:09:24,100 --> 00:09:31,960 The upper half contains nine data points ranging from this 69 value right here up to 75. 120 00:09:31,960 --> 00:09:39,280 So if we ignore the first four data points, 69, 69, 70 and 70, and we ignore the last four data 121 00:09:39,280 --> 00:09:48,280 points, 71, 72, 73 and 75, the median of this upper half is this value here, 71 and we can fill 122 00:09:48,280 --> 00:09:53,940 in our five number summary and say 68 and 71. 123 00:09:53,950 --> 00:10:00,550 Now once we have Q one and Q three, going back to what we said earlier about the five number summary, 124 00:10:00,550 --> 00:10:06,100 the difference between the first and third quartiles gives us a picture of how spread out the data is 125 00:10:06,100 --> 00:10:12,310 around the median, because what this five number summary tells us is that 50% of all of the values 126 00:10:12,310 --> 00:10:18,250 in the data set fall between 68 and 71 around a median of 69. 127 00:10:18,250 --> 00:10:22,900 That gives us a sense of how tightly clustered the data is around the median. 128 00:10:22,900 --> 00:10:25,450 And that value is important. 129 00:10:25,450 --> 00:10:29,080 That's the interquartile range that we mentioned at the beginning. 130 00:10:29,200 --> 00:10:37,990 The IQR of a data set is equal to the difference between the third quartile and the first quartile. 131 00:10:37,990 --> 00:10:48,280 So in our case, 71 for the third quartile -68 for the first quartile or three. 132 00:10:48,610 --> 00:10:51,160 So the interquartile range of the IQR of this. 133 00:10:51,210 --> 00:10:53,250 Particular data set is three. 134 00:10:53,250 --> 00:10:54,760 It's Q three minus. 135 00:10:54,780 --> 00:10:57,900 Q one the difference between the first and the third quartiles. 136 00:10:57,900 --> 00:11:00,720 So that's what our five number summary looks like. 137 00:11:00,720 --> 00:11:01,700 That's how we calculate. 138 00:11:01,710 --> 00:11:08,130 Q one, Q three and the median, when we have an odd number of data points in our set. 139 00:11:08,160 --> 00:11:10,950 But what about if we have an even number of data points in the set? 140 00:11:10,950 --> 00:11:19,290 So this is the same data set except for the fact that we have removed one value of 68 from the set. 141 00:11:19,290 --> 00:11:24,150 So now we have 17 golf scores instead of 18 golf scores. 142 00:11:24,150 --> 00:11:30,990 We have an odd number of values in the data set and one common method for computing the first and the 143 00:11:30,990 --> 00:11:37,020 third quartiles, when we have an odd number of data points, is to again first find the median. 144 00:11:37,020 --> 00:11:44,280 So if we look at the lower and upper halves of the data because we have 17 values here in the set, 145 00:11:44,280 --> 00:11:50,700 we've underlined the first eight and the last eight, leaving us with just this center value here of 146 00:11:50,700 --> 00:11:54,900 69 as the median and one method for computing. 147 00:11:54,900 --> 00:12:02,100 Q one and Q three when we have an odd number of data points, is just to exclude this median value when 148 00:12:02,100 --> 00:12:04,050 we calculate Q one and Q three. 149 00:12:04,050 --> 00:12:10,740 So excluding the median, we just look at this lower half of the data, these eight values here, and 150 00:12:10,740 --> 00:12:14,370 we find the median of these eight values. 151 00:12:14,370 --> 00:12:23,940 So looking at the median here, we have four values 66, 67, 67, 68, and then four values 68 through 152 00:12:23,940 --> 00:12:24,780 69 here. 153 00:12:24,780 --> 00:12:27,690 So the median should fall right here. 154 00:12:27,720 --> 00:12:34,920 And we know, of course, that that means that we are taking the mean of these two values here. 155 00:12:34,920 --> 00:12:44,070 And so the mean of these two is 68, and the first quartile will be 68 to find Q three, the upper quartile, 156 00:12:44,070 --> 00:12:46,560 we consider just these eight values. 157 00:12:46,650 --> 00:12:53,610 We know that the median should fall between the lower four values and the upper four values. 158 00:12:53,610 --> 00:13:00,900 So right here and therefore we know that that means that we are taking the mean of these two values 159 00:13:00,900 --> 00:13:04,440 and we get an upper quartile of 71. 160 00:13:04,440 --> 00:13:13,470 So in this case, removing that value of 68 didn't change the value of the median or Q one or Q three. 161 00:13:13,500 --> 00:13:15,900 All three values stayed the same. 162 00:13:15,900 --> 00:13:18,600 But of course that won't always be the case. 163 00:13:18,600 --> 00:13:24,450 So these methods that we showed here to calculate Q one and Q three when we had an even number of data 164 00:13:24,450 --> 00:13:26,490 points, are an odd number of data points. 165 00:13:26,490 --> 00:13:28,560 You can always use these methods. 166 00:13:28,560 --> 00:13:35,700 They'll work, but other methods sometimes have us include this median value here when we calculate 167 00:13:35,700 --> 00:13:39,930 Q one and Q three instead of excluding it as we did with our method. 168 00:13:39,930 --> 00:13:46,500 Sometimes the method for an even number of data points up here, we'll have us calculate this median 169 00:13:46,500 --> 00:13:53,280 and then include this median as if we had an extra value of 69 right here and then go ahead and find 170 00:13:53,280 --> 00:13:54,810 the lower and upper quartile. 171 00:13:54,810 --> 00:14:01,680 So there are slight differences, but as you can imagine, they're always going to produce fairly similar 172 00:14:01,680 --> 00:14:02,580 results. 173 00:14:02,580 --> 00:14:08,250 So the most important thing here is just understanding what the first and third quartiles are, how 174 00:14:08,250 --> 00:14:13,800 they fit into this idea of a five number summary to give us an idea of the center and spread of the 175 00:14:13,800 --> 00:14:18,300 data set and at least one method for calculating the three quartiles. 176 00:14:18,300 --> 00:14:24,690 And of course we calculated these quartile values by hand, but we can also use computers to help us 177 00:14:24,690 --> 00:14:27,900 find these values, especially as the data set gets larger. 178 00:14:27,900 --> 00:14:34,650 For instance, you can use the quartile function in Excel to calculate any of the five values in the 179 00:14:34,650 --> 00:14:40,950 five number summary The quartile function will just ask you for your array, which will be the range 180 00:14:40,950 --> 00:14:46,470 of cells where you have your data set in the spreadsheet and then for the quartile that you want the 181 00:14:46,470 --> 00:14:48,210 function to return. 182 00:14:48,210 --> 00:14:54,570 And so you would enter the array and then either the number zero, if you want the minimum value, the 183 00:14:54,570 --> 00:15:01,110 number one, if you want the first quartile or the numbers two, three or four, if you want the median 184 00:15:01,110 --> 00:15:06,120 third quartile or maximum value of that array of that data set. 185 00:15:06,120 --> 00:15:12,450 So instead of trying to do this by hand, if we had inputted this entire data set into Excel, we could 186 00:15:12,450 --> 00:15:18,690 have used the quartile function, highlighted this data set, and then input the number one into the 187 00:15:18,690 --> 00:15:21,480 function for Excel to return to us. 188 00:15:21,480 --> 00:15:25,200 The first quartile 68 of this data set. 189 00:15:25,200 --> 00:15:32,430 So the last thing we want to say is that it's very common for us to use this concept of quartiles to 190 00:15:32,430 --> 00:15:34,890 identify outliers in the data set. 191 00:15:34,890 --> 00:15:40,380 We talked a little bit about outliers before when we talked about measuring the center of a data set, 192 00:15:40,380 --> 00:15:47,520 but determining which values in the set technically qualify as outliers can be more of an art than a 193 00:15:47,520 --> 00:15:48,000 science. 194 00:15:48,000 --> 00:15:50,580 But one method that people use is to. 195 00:15:50,900 --> 00:15:56,840 Boy, these quartiles to give sort of benchmarks for outliers in the data set. 196 00:15:56,840 --> 00:16:08,570 And so sometimes we will say that any value in the set that is lower than the first quartile -1.5 times 197 00:16:08,690 --> 00:16:17,450 the IQ r of the data set will be an outlier in the data or similarly, any value that is larger than 198 00:16:17,450 --> 00:16:25,790 the third quartile plus 1.5 times the IQ r that will consider any value larger than this value to be 199 00:16:25,790 --> 00:16:28,070 an outlier on the upper end of the data. 200 00:16:28,070 --> 00:16:34,550 So you can imagine with this data set right here, we said that the first quartile was equal to 68, 201 00:16:34,550 --> 00:16:43,580 so let's say 68 and then we'll say -1.5 times the IQ r, which we calculated to be three. 202 00:16:43,940 --> 00:16:51,560 So we have 68 -4.5, which gives us a value of 63.5. 203 00:16:51,560 --> 00:16:58,580 And so by this method, if we choose to use this method for determining outliers, we would say that 204 00:16:58,580 --> 00:17:05,270 any value less than 63.5 would be considered an outlier of this data set. 205 00:17:05,270 --> 00:17:13,339 So if we have 66, 67, 67 all the way to 75, if this is our data set, anything less than 63 and a 206 00:17:13,339 --> 00:17:15,319 half, we would consider an outlier. 207 00:17:15,319 --> 00:17:24,740 And similarly on the high end, if we take Q three, which we know is 71 for this set, so 71 plus 1.5 208 00:17:24,740 --> 00:17:34,100 times an IQ R of three is 71 plus 4.5 or 75.5. 209 00:17:34,190 --> 00:17:41,600 So any value greater than 75.5 we could consider to be an outlier for this data set. 210 00:17:41,600 --> 00:17:47,180 Again, this isn't some objective, universally accepted standard for outliers. 211 00:17:47,180 --> 00:17:54,170 It's just one way that we could maybe set some parameters to start getting an idea for what would we 212 00:17:54,170 --> 00:17:58,460 even consider an outlier on the low end or an outlier on the high end? 213 00:17:58,460 --> 00:18:05,600 For a data set like this one, it sort of gives us some boundaries, Some gates around the data set, 214 00:18:05,600 --> 00:18:13,250 a sense of which values are unusually low or unusually high given the rest of the data that we have. 215 00:18:13,250 --> 00:18:19,820 So again, a lot that we can do with this concept of quartiles and interquartile range, but all of 216 00:18:19,820 --> 00:18:27,590 this information is really just more context around the center and the spread of a data set.