1 00:00:00,120 --> 00:00:05,670 Now we want to start talking about distribution plots and specifically the histogram, which is a type 2 00:00:05,670 --> 00:00:07,010 of distribution plot. 3 00:00:07,020 --> 00:00:14,820 So this is an example of a histogram we have here age across the horizontal axis and population in thousands 4 00:00:14,820 --> 00:00:16,560 along the vertical axis. 5 00:00:16,560 --> 00:00:25,020 So this value here 50 means 50,000, 100 is 100,000 etc. and the ages along the horizontal axis are 6 00:00:25,020 --> 00:00:26,610 ages zero and up. 7 00:00:26,610 --> 00:00:32,009 And then for the second part of the histogram here, ages ten and up, which means this first bar or 8 00:00:32,009 --> 00:00:35,790 this first bin is ages 0 to 9. 9 00:00:35,790 --> 00:00:44,160 This bin right here is ages 10 to 19, this bin is 20 to 29, etc. all the way up to this last bin, 10 00:00:44,160 --> 00:00:47,820 which is 100 to 109. 11 00:00:47,820 --> 00:00:52,620 This is a histogram that, for example, might represent the population of a city. 12 00:00:52,620 --> 00:00:58,350 So maybe in this city, if we're trying to read the histogram, what it tells us is that there are about 13 00:00:58,350 --> 00:01:06,360 25,000, zero, ten nine year olds living in the city, maybe about, let's say 40,000, 10 to 19 year 14 00:01:06,360 --> 00:01:12,570 olds living in the city all the way up to the largest group, which is the 40 to 49 year olds, of which 15 00:01:12,570 --> 00:01:15,720 it looks like there might be about 170,000. 16 00:01:15,750 --> 00:01:19,500 Here are the things that we want to understand about a histogram. 17 00:01:19,500 --> 00:01:25,230 First of all, this horizontal axis feature, whatever characteristic were plotting along the horizontal 18 00:01:25,230 --> 00:01:29,250 axis, in this case, its age needs to be continuous. 19 00:01:29,250 --> 00:01:36,930 So age is a continuous characteristic, time is continuous, height and weight are both continuous because 20 00:01:36,930 --> 00:01:43,110 all of these things can be broken down into smaller and smaller and smaller measurements, whereas a 21 00:01:43,110 --> 00:01:45,810 different kind of characteristic might be discrete. 22 00:01:45,810 --> 00:01:52,410 So in the last lesson, we looked at line plots and we plotted months along the horizontal axis. 23 00:01:52,410 --> 00:01:58,980 So January, February, March, April, etc. While months are similar to time, if we're specifically 24 00:01:58,980 --> 00:02:04,410 talking about months, there's nothing in between January and February, there's nothing in between 25 00:02:04,410 --> 00:02:05,700 February and March. 26 00:02:05,700 --> 00:02:08,699 Those are distinct discrete buckets. 27 00:02:08,699 --> 00:02:11,820 We're not talking about a continuous characteristic. 28 00:02:11,820 --> 00:02:18,360 Compare that to something like this here with age where in this 0 to 9 years old bucket, we could have 29 00:02:18,360 --> 00:02:24,330 someone who is exactly four years old, but we could also have someone who is four years, three months, 30 00:02:24,330 --> 00:02:31,530 27 days, 4 hours, 6 minutes and 32.3 seconds years old, and they would fall into this bucket and 31 00:02:31,530 --> 00:02:32,790 there's a clean cut off. 32 00:02:32,790 --> 00:02:37,740 The moment that person turns ten years old, they would graduate from this first bucket into this second 33 00:02:37,740 --> 00:02:38,190 bucket. 34 00:02:38,190 --> 00:02:44,340 So this feature along the horizontal axis needs to be continuous if we're going to use a histogram to 35 00:02:44,340 --> 00:02:45,570 represent the data. 36 00:02:45,570 --> 00:02:53,760 Now, because of the continuity of the data set, we sketch a bar for each bin or class that we list 37 00:02:53,760 --> 00:03:02,010 out here along the horizontal axis, and we include no gaps in between each bar in our plot and including 38 00:03:02,010 --> 00:03:08,580 no gaps, suggests the continuous nature of this characteristic along the horizontal axis. 39 00:03:08,580 --> 00:03:14,610 When we talk about bar plots or bar charts, which we'll look at later, they'll look a lot like histograms, 40 00:03:14,610 --> 00:03:20,640 except that we will leave a little space or a little gap in between each bar here with the histogram, 41 00:03:20,640 --> 00:03:23,130 we leave no space or no gap between the bars. 42 00:03:23,130 --> 00:03:29,520 So right here or right here, there's no gap between these bars. 43 00:03:29,520 --> 00:03:34,970 And that's because of the continuous nature of this characteristic along the horizontal axis. 44 00:03:34,980 --> 00:03:41,520 Now, that being said, of course, like we've talked about, each bar is a count of the number of occurrences 45 00:03:41,520 --> 00:03:45,090 that fall into this bin or class or interval. 46 00:03:45,090 --> 00:03:51,990 So the height of each bar represents the number of occurrences that we find of people ages 0 to 9 in 47 00:03:51,990 --> 00:03:53,010 this city. 48 00:03:53,010 --> 00:03:58,410 This bar here, the height of it, represents the number of occurrences of people that we find between 49 00:03:58,410 --> 00:04:00,840 ages ten and 19 in this city. 50 00:04:00,840 --> 00:04:06,810 Now, one of the reasons that a histogram is really helpful is because it simplifies what could otherwise 51 00:04:06,840 --> 00:04:11,010 be a very large and messy data set here. 52 00:04:11,010 --> 00:04:15,120 We can clearly tell the population of this city is very large. 53 00:04:15,120 --> 00:04:16,980 It's multiple hundreds of thousands. 54 00:04:16,980 --> 00:04:23,430 If we were to try to plot a value along our chart here for the age of every single person in the city, 55 00:04:23,430 --> 00:04:29,430 the chart would quickly get very out of control because again, we would need a bar for a person of 56 00:04:29,430 --> 00:04:30,300 each age. 57 00:04:30,300 --> 00:04:36,840 And so if we had two people whose ages were different by even one day or even a couple of hours or a 58 00:04:36,840 --> 00:04:40,710 second, that would be a different individual bar on our chart. 59 00:04:40,710 --> 00:04:47,730 It would be a different individual age or timestamp along our horizontal axis that would get extremely 60 00:04:47,730 --> 00:04:48,690 overwhelming. 61 00:04:48,690 --> 00:04:53,040 It would make our chart so messy that it would be unreasonable to interpret it. 62 00:04:53,040 --> 00:04:58,680 What makes a lot more sense is that when we have several hundreds of thousands of people, we group 63 00:04:58,680 --> 00:04:59,820 them into bins. 64 00:05:00,330 --> 00:05:06,990 And then create this histogram that gives us a much clearer, simpler picture of the population when 65 00:05:06,990 --> 00:05:08,820 we're creating a histogram. 66 00:05:09,000 --> 00:05:14,520 We want to make sure that our buckets or bins or classes, whatever we're using here along the horizontal 67 00:05:14,520 --> 00:05:19,620 axis that that bin size, that that class width is always the same. 68 00:05:19,620 --> 00:05:27,240 So we have here for this example, ten year increments, we have ages 0 to 9, 10 to 19, 22, 29, 69 00:05:27,240 --> 00:05:31,800 32, 39, 42, 49, all the way up to 100 to 109. 70 00:05:31,800 --> 00:05:35,070 Keeping that bin width or that class width. 71 00:05:35,070 --> 00:05:38,940 The same is the only way that the histogram makes sense. 72 00:05:38,940 --> 00:05:47,910 If we instead have one bin for ages 0 to 9, another one for 10 to 19 and that have been for 20 to 49 73 00:05:47,910 --> 00:05:52,320 and then maybe 50 to 59 and then 60 all the way up to 109. 74 00:05:52,320 --> 00:05:57,360 It's going to make the data almost meaningless because those bin widths aren't the same. 75 00:05:57,360 --> 00:06:04,710 So we can't interpret or compare different bins to get an accurate picture of what our age distribution 76 00:06:04,710 --> 00:06:05,670 is looking like. 77 00:06:05,670 --> 00:06:11,340 So we need to make sure that our bin width, that our class width is always equivalent so that this 78 00:06:11,340 --> 00:06:16,050 distribution is consistent across each bucket along the horizontal axis. 79 00:06:16,050 --> 00:06:18,870 So we know that we have to keep bin with the same. 80 00:06:18,870 --> 00:06:22,920 But how do we know how many bins to use based on our data? 81 00:06:23,070 --> 00:06:27,240 Well, that's a little bit more of an art than it is a perfect science. 82 00:06:27,240 --> 00:06:32,880 But again, as we've said before, with all of these kinds of charts and plots, we're just trying to 83 00:06:32,880 --> 00:06:35,670 communicate a clear picture of the data. 84 00:06:35,670 --> 00:06:41,550 So we need to use whatever number of bins will communicate the idea we're trying to convey. 85 00:06:41,550 --> 00:06:48,270 So for instance, with this age data, for these age intervals across our city's population, we can 86 00:06:48,270 --> 00:06:52,950 see that we have people from ages zero all the way to 109. 87 00:06:52,950 --> 00:07:00,480 So we could certainly break that data into two buckets, classifying people into either the 0 to 54 88 00:07:00,480 --> 00:07:07,110 group or the 55 to 109 group and just have literally two bins or two classes. 89 00:07:07,110 --> 00:07:13,260 And while that's not wrong, it certainly wouldn't give us as clear of a picture as this histogram. 90 00:07:13,260 --> 00:07:17,820 This gives us much more information about the age distribution of our population. 91 00:07:17,820 --> 00:07:23,040 So with an age range like this, we probably wouldn't want to use two buckets, but we also wouldn't 92 00:07:23,040 --> 00:07:30,300 want to use, let's say 55 buckets because that would break this up into only two year bins or two year 93 00:07:30,300 --> 00:07:33,960 increments and that would maybe be more detailed than we need. 94 00:07:33,960 --> 00:07:37,380 So it's about finding something reasonable in the middle. 95 00:07:37,410 --> 00:07:43,170 It gives us a good picture of the distribution without giving us more detail than we really need. 96 00:07:43,170 --> 00:07:50,550 And hopefully you have a good gut feel for if our age range is essentially 0 to 110, then breaking 97 00:07:50,550 --> 00:07:56,370 up that interval into ten year buckets gives us 11 bars in our distribution. 98 00:07:56,370 --> 00:07:58,950 That's probably pretty reasonable. 99 00:07:58,950 --> 00:08:04,410 An 11 bar distribution should give us a pretty good picture without having buckets that are too big, 100 00:08:04,410 --> 00:08:08,070 that we don't get enough data or too small, that we get too much data. 101 00:08:08,070 --> 00:08:13,470 So we're just looking for a good middle ground and anything within that middle ground should give us 102 00:08:13,470 --> 00:08:15,150 a decent histogram. 103 00:08:15,150 --> 00:08:20,550 If you build a histogram and you feel like the bin width was too wide or too narrow, you can always 104 00:08:20,550 --> 00:08:25,500 make an adjustment to get a less detailed distribution or a more detailed distribution. 105 00:08:25,500 --> 00:08:31,800 Now some of the simple math that goes along with this, let's say that we wanted to build a histogram 106 00:08:31,800 --> 00:08:36,240 from the raw data of the ages of the people who live in this city. 107 00:08:36,240 --> 00:08:41,580 We want to think about the smallest value in the dataset and the largest value in the dataset. 108 00:08:41,580 --> 00:08:44,370 So in this case, the smallest value is age zero. 109 00:08:44,400 --> 00:08:46,890 The largest value is age 109. 110 00:08:46,890 --> 00:08:54,330 So to get the entire interval, because we're including both zero and 109, we have to to find the interval, 111 00:08:54,330 --> 00:08:59,400 say 109 -0, the largest value minus the smallest value. 112 00:08:59,400 --> 00:09:05,640 But if we're including both of those values in the dataset, both of those endpoints, then the subtraction 113 00:09:05,640 --> 00:09:10,530 is what's called inclusive and we have to add one to get the full range of the data. 114 00:09:10,530 --> 00:09:15,570 In other words, if you were to use your fingers to count starting at zero, so counting on your fingers, 115 00:09:15,570 --> 00:09:19,440 you say zero one, two, three, four, all the way up to 109. 116 00:09:19,440 --> 00:09:21,810 You would actually count on your fingers to. 117 00:09:22,570 --> 00:09:27,030 110 because you're including that zero value and that 109 value. 118 00:09:27,040 --> 00:09:29,080 That's why we add that plus one. 119 00:09:29,080 --> 00:09:33,190 So we have this range of 110 values. 120 00:09:33,190 --> 00:09:40,240 So then we just need to pick whatever seems to be a reasonable class interval or class width or bin 121 00:09:40,240 --> 00:09:40,690 width. 122 00:09:40,690 --> 00:09:45,520 And let's say that we want to divide this up into every ten years like we did. 123 00:09:45,520 --> 00:09:52,870 So if we divide this by ten, then that means that we will have 11 different buckets or 11 different 124 00:09:52,870 --> 00:09:56,080 classes, each with a width of ten. 125 00:09:56,080 --> 00:09:59,380 So then we take our smallest value, which is zero. 126 00:09:59,410 --> 00:10:01,330 We know the interval is ten. 127 00:10:01,330 --> 00:10:07,840 So if we again count on our fingers from zero, so 012 all the way up until we get to our 10th finger, 128 00:10:07,840 --> 00:10:11,230 then the upper bound of that first interval is nine. 129 00:10:11,230 --> 00:10:17,590 So we know that our first bin, our first class is ages 0 to 9, and then from there we can find the 130 00:10:17,590 --> 00:10:23,380 rest of our classes ages 10 to 19, ages 20 to 29. 131 00:10:23,380 --> 00:10:30,220 And notice that the difference between all of the class lower bounds will always be the bin width. 132 00:10:30,220 --> 00:10:33,880 So the difference between zero and ten, ten and 20 is always ten. 133 00:10:33,880 --> 00:10:39,010 And the difference between the upper bounds of all of our intervals will also be that same bin width. 134 00:10:39,010 --> 00:10:44,140 So the difference between nine and 19, between 19 and 29 is always ten. 135 00:10:44,140 --> 00:10:50,470 So whatever width we find for each class or each bin, we should see reflected here between the lower 136 00:10:50,470 --> 00:10:53,140 bounds and here between the upper bounds. 137 00:10:53,140 --> 00:11:02,320 And if we keep going here 30 to 39, all the way down to our highest value, we get to 102, 109. 138 00:11:02,320 --> 00:11:09,100 If we then count the number of bins that we have, we should have 11 bins since we calculated that bin 139 00:11:09,100 --> 00:11:13,060 count earlier, so 11 bins each with a width of ten. 140 00:11:13,090 --> 00:11:20,320 Then once we have the range of each bin, we can in our raw data, count the number of occurrences that 141 00:11:20,320 --> 00:11:21,400 fall within that bin. 142 00:11:21,400 --> 00:11:28,420 So if we have our raw data, we would count up the number of people in our data ages 0 to 9, and we 143 00:11:28,420 --> 00:11:30,190 would record that value. 144 00:11:30,190 --> 00:11:33,700 So let's say that's 25,000 people. 145 00:11:33,700 --> 00:11:37,630 We would count up the number of people who have an age between ten and 19. 146 00:11:37,630 --> 00:11:40,600 And let's say based on our graph here, that that's about. 147 00:11:41,320 --> 00:11:42,370 40,000 people. 148 00:11:42,370 --> 00:11:46,660 So we would count the number of occurrences in each bin or in each class. 149 00:11:46,660 --> 00:11:52,480 And then to sketch the histogram, we just place each class, each bin along the horizontal axis in 150 00:11:52,480 --> 00:11:58,810 order, and then we place counts here along our vertical axis and we plot the height of each bar in 151 00:11:58,810 --> 00:12:03,910 the histogram according to the number of occurrences that we see in each bin or each class. 152 00:12:03,910 --> 00:12:08,320 And that is how we build a histogram from a raw data set. 153 00:12:08,320 --> 00:12:14,440 That's the basic idea behind a histogram, which is our first type here of distribution plot. 154 00:12:14,470 --> 00:12:20,380 These are going to become more and more important as we go forward because what we realize is that this 155 00:12:20,380 --> 00:12:29,650 distribution can eventually create for us a probability distribution if we connect the top of each bar 156 00:12:29,920 --> 00:12:37,900 in the histogram, if we find the midpoint of each class and then we sketch a point at the top of each 157 00:12:37,900 --> 00:12:38,680 bar here. 158 00:12:38,680 --> 00:12:46,030 And if we eventually connect these points with a smooth curve, what we get is a distribution that models 159 00:12:46,030 --> 00:12:52,660 our raw data and eventually we can use that distribution to answer probability questions about how likely 160 00:12:52,660 --> 00:12:58,810 it is, for instance, that choosing a person at random in our city will give us a person between the 161 00:12:58,810 --> 00:13:02,560 ages of 40 and 49 or between the ages of 70 and 79. 162 00:13:02,560 --> 00:13:08,530 So not only is the histogram an important plot or chart like the scatter plots and line plots that we've 163 00:13:08,530 --> 00:13:12,580 looked at so far and like the charts and graphs that we'll look at next. 164 00:13:12,580 --> 00:13:17,620 But this kind of distribution plot will also be important for us to remember as we move forward to learn 165 00:13:17,620 --> 00:13:20,890 more about probability and statistics.