1 00:00:00,000 --> 00:00:20,260 Hello everyone, welcome to the magnificence of mathematics. I'm your host, Eddie Kingston. 2 00:00:20,260 --> 00:00:24,760 Last time we talked about probability and some counterintuitive ways it can behave. 3 00:00:24,760 --> 00:00:29,580 Now we're going to talk about an adjacent field, statistics. Stats is a field that hits 4 00:00:29,580 --> 00:00:33,640 close to home for me, because that's what I got a master's degree in. I only have 5 00:00:33,640 --> 00:00:38,360 a bachelor's in mathematics. I have placed both the math and stats master's programs 6 00:00:38,360 --> 00:00:42,880 in my university, and quite frankly the only reason I enrolled in the stats program is 7 00:00:42,880 --> 00:00:47,560 that they got back to me first, and I figured that it would be a more lucrative option. 8 00:00:47,560 --> 00:00:51,760 I'll talk more about my grad school experience later, but first I want to talk about what 9 00:00:51,760 --> 00:00:54,160 stats even is. 10 00:00:54,160 --> 00:00:58,880 Statistics has to do with the collection, analysis, and interpretation of data. Statisticians 11 00:00:58,880 --> 00:01:02,800 use models to describe patterns in data and interpret them in ways that make sense to 12 00:01:02,800 --> 00:01:08,720 us. It's important to note that models are just that, models. In the real world, no data 13 00:01:08,720 --> 00:01:14,800 set will follow any particular distribution perfectly. There are two main flavors of stats, 14 00:01:14,800 --> 00:01:19,360 descriptive and inferential. You might already be familiar with descriptive stats. Basics 15 00:01:19,360 --> 00:01:24,040 of descriptive stats are what you learned in school, mean, median, standard deviation, 16 00:01:24,040 --> 00:01:29,640 variance, etc. These are used to describe features of an entire population, like a census. 17 00:01:29,640 --> 00:01:34,400 However, a lot of the time, getting an entire population's worth of data isn't feasible, 18 00:01:34,400 --> 00:01:39,080 so researchers take a representative sample, or a subset, of the population. This just 19 00:01:39,080 --> 00:01:43,040 means taking a relatively small amount of people that are bunched into one particular 20 00:01:43,040 --> 00:01:47,040 group. Getting a representative sample is usually done by randomly selecting members 21 00:01:47,040 --> 00:01:52,000 of the population to be in the sample. Then, researchers use the second flavor, inferential 22 00:01:52,000 --> 00:01:57,000 stats, to draw conclusions and make general statements about the entire population. This 23 00:01:57,000 --> 00:02:01,720 tends to take the form of hypothesis testing, say whether or not the proportion of listeners 24 00:02:01,720 --> 00:02:07,280 of this podcast who live in the US is equal to 50%, or estimation, say a more specific 25 00:02:07,280 --> 00:02:12,080 estimate of the proportion of listeners who live in the US. Inferential stats can extend 26 00:02:12,080 --> 00:02:15,960 as far as forecasting stock prices and mining for bitcoin. 27 00:02:15,960 --> 00:02:19,960 Let's talk a bit about inferential stats. There are two main camps that statisticians 28 00:02:19,960 --> 00:02:24,640 live in, the Frequentist camp and the Bayesian camp. Frequentists believe that population 29 00:02:24,640 --> 00:02:29,080 parameters and probabilities are fixed. For example, the probability of landing heads 30 00:02:29,080 --> 00:02:33,260 on a coin can be found by flipping a coin over and over and over again and analyzing 31 00:02:33,260 --> 00:02:38,480 its long term behavior. You should see that that number converges to 50%. Frequentists 32 00:02:38,480 --> 00:02:42,760 use what's known as hypothesis testing to either see whether there's any relationship 33 00:02:42,760 --> 00:02:46,640 among two variables, like whether the amount of roses the floor of cells goes up when they 34 00:02:46,640 --> 00:02:51,240 lower the price of the roses, or to make a guess at how much the data changes over time, 35 00:02:51,240 --> 00:02:55,360 for example whether the floors can sell more than 30 bouquets if they lower the price by 36 00:02:55,360 --> 00:02:58,440 $5. When running a hypothesis test, researchers 37 00:02:58,440 --> 00:03:02,400 come up with a null hypothesis, which is a starting statement that the researcher might 38 00:03:02,400 --> 00:03:07,200 try to disprove, and an alternative hypothesis, which provides a different idea of how the 39 00:03:07,200 --> 00:03:11,720 data behave. For example, a professor might want to see if a change in their syllabus 40 00:03:11,720 --> 00:03:16,800 in a course they teach improves students' grades. The null hypothesis could be that 41 00:03:16,800 --> 00:03:21,520 students' grades are unchanged, and the mean grade in the course is the same. The alternative 42 00:03:21,520 --> 00:03:26,880 hypothesis could be that students' mean grades are higher, or at least unequal. This, in 43 00:03:26,880 --> 00:03:31,120 particular, would be a great example for which one could use Welch's t-test, named after 44 00:03:31,120 --> 00:03:36,240 20th century British statistician Bernard Lewis Welch. In general, the basic idea of 45 00:03:36,240 --> 00:03:40,260 Welch's t-test is that you want to compare the mean of two different populations and 46 00:03:40,260 --> 00:03:44,960 test whether those means are equal. The defining features and assumptions of Welch's t-test 47 00:03:44,960 --> 00:03:49,480 are that the variances of the populations, that is, how far the data in each population 48 00:03:49,480 --> 00:03:53,920 tend to stray from the mean, are unequal, and that the sample means in both populations 49 00:03:53,920 --> 00:03:58,540 are normally distributed. This means that, for example, if a professor takes a bunch 50 00:03:58,540 --> 00:04:03,840 of random samples of grades from their class, the means of those samples follow a neural 51 00:04:03,840 --> 00:04:08,600 distribution. You may have seen what's called a bell curve when learning about stats. It's 52 00:04:08,600 --> 00:04:12,760 a kind of figure where a lot of the data are centered around a certain value, and there's 53 00:04:12,760 --> 00:04:18,560 fewer data points the further away from that value you go in either direction. From here, 54 00:04:18,560 --> 00:04:23,040 statisticians collect some sample data, compute the means and standard deviations of the samples, 55 00:04:23,040 --> 00:04:28,280 then use those values to calculate a test statistic. The idea is that if the null hypothesis 56 00:04:28,280 --> 00:04:34,080 is true, this test statistic would come from a certain distribution, in this case a t-distribution, 57 00:04:34,080 --> 00:04:38,720 which is in some respects similar but in other ways different from a normal distribution. 58 00:04:38,720 --> 00:04:44,440 Then researchers use this value to calculate a p-value. A p-value is the probability of 59 00:04:44,440 --> 00:04:49,540 observing a certain value assuming the null hypothesis is true. If we were comparing the 60 00:04:49,540 --> 00:04:55,080 test statistic to, say, a normal distribution with mean zero, and we observed a test statistic 61 00:04:55,080 --> 00:04:59,920 of, say, three, that's extremely unlikely that that test statistic came from a normal 62 00:04:59,920 --> 00:05:06,380 distribution. So a researcher would reject the null hypothesis in favor of the alternative. 63 00:05:06,380 --> 00:05:11,880 If the test statistic was closer to zero, say 0.3, they would say they fail to reject 64 00:05:11,880 --> 00:05:17,400 the null hypothesis. Threshold p-values vary from field to field, but the most common significance 65 00:05:17,400 --> 00:05:23,520 level, that is, the highest p-value which one would reject the null hypothesis, is 0.05, 66 00:05:23,520 --> 00:05:29,560 or 5%. This would mean that if the p-value a researcher calculates is less than 5%, meaning 67 00:05:29,560 --> 00:05:34,440 there is a less than 5% chance of observing a certain test statistic if the null hypothesis 68 00:05:34,440 --> 00:05:39,560 was true, then the researcher rejects the null hypothesis in favor of the alternative. 69 00:05:39,560 --> 00:05:45,720 Conversely, if the p-value is anything more than 5%, they fail to reject the null hypothesis, 70 00:05:45,720 --> 00:05:50,360 which just means they lack sufficient evidence to rule out the possibility of the null hypothesis 71 00:05:50,360 --> 00:05:55,560 being true. It's important to note that it is improper to say you accept the null hypothesis 72 00:05:55,560 --> 00:06:00,080 is true. One of the most common methods to introduce students to hypothesis testing is 73 00:06:00,080 --> 00:06:06,400 in a court setting. The null hypothesis is that the defendant is innocent. During a trial, 74 00:06:06,400 --> 00:06:10,640 evidence is collected that may or may not point towards the defendant's guilt. The jury 75 00:06:10,640 --> 00:06:15,440 then returns a verdict, either guilty or not guilty. Guilty, of course, means the jury 76 00:06:15,440 --> 00:06:19,480 thinks that there is sufficient evidence to convict the defendant, so they reject the 77 00:06:19,480 --> 00:06:24,520 null hypothesis of them being innocent. Not guilty doesn't necessarily mean the jury 78 00:06:24,520 --> 00:06:28,520 thinks the defendant is innocent, it just means that the jury doesn't have enough 79 00:06:28,520 --> 00:06:32,000 evidence to convict the defendant, so they acquit them. 80 00:06:32,000 --> 00:06:36,640 Another important idea in statistics is the idea of regression. The most common starting 81 00:06:36,640 --> 00:06:41,560 point is talking about simple linear regression, meaning there is only one dependent variable, 82 00:06:41,560 --> 00:06:48,040 also known as a response variable, y, and one independent variable, aka an explanatory 83 00:06:48,040 --> 00:06:53,840 variable x. For example, x could be the price of roses at a florist shop and y could be 84 00:06:53,840 --> 00:06:58,200 the number of roses sold. Researchers can take a look at a graph of the number of roses 85 00:06:58,200 --> 00:07:02,960 sold on a certain day versus when the roses were a certain price, and perhaps make a linear 86 00:07:02,960 --> 00:07:07,060 approximation of how many roses would be sold when the price is a different amount. When 87 00:07:07,060 --> 00:07:11,520 conducting a linear regression, researchers find an estimated intercept, which is how 88 00:07:11,520 --> 00:07:15,720 many roses would be given away if they were free, and an estimated slope, which is how 89 00:07:15,720 --> 00:07:20,640 many roses would be sold every time the price decreased by $1, or whatever interval they're 90 00:07:20,640 --> 00:07:25,760 looking at. Remember y equals mx plus b from your high school algebra class? It's the same 91 00:07:25,760 --> 00:07:29,280 idea here, b is your intercept and m is your slope. 92 00:07:29,280 --> 00:07:32,920 What researchers do with this information depends on their goals. It could be either 93 00:07:32,920 --> 00:07:36,720 to predict how many roses would be sold if the price was another amount that wasn't 94 00:07:36,720 --> 00:07:41,640 looked at yet, or to estimate how the explanatory variable affects the response variable, ie 95 00:07:41,640 --> 00:07:46,300 just finding and reporting about the slope. There's also multiple regression, which involves 96 00:07:46,300 --> 00:07:52,280 multiple explanatory variables and one response variable. Multilinear regression, which involves 97 00:07:52,280 --> 00:07:58,160 one explanatory variable and multiple response variables. And multiple multilinear regression, 98 00:07:58,160 --> 00:08:02,360 which involves multiple explanatory variables and multiple response variables. And this 99 00:08:02,360 --> 00:08:06,600 is just for continuous data. There's also ways to do regression with non-continuous 100 00:08:06,600 --> 00:08:11,840 or discrete data. This is with what's called a generalized linear model. We've barely 101 00:08:11,840 --> 00:08:17,040 started to draft plans to scratch the surface of statistics. There's so much more to it, 102 00:08:17,040 --> 00:08:21,640 more tests to learn, more theory to talk about, than it couldn't possibly fit in one episode. 103 00:08:21,640 --> 00:08:26,240 Instead, what I'll do is talk more about my experience as a statistics student and offer 104 00:08:26,240 --> 00:08:30,280 my perspective of a former graduate student in general so that those of you listeners 105 00:08:30,280 --> 00:08:33,940 who might be thinking about graduate school can get an idea of what it's like. 106 00:08:33,940 --> 00:08:39,140 So my university went by a quarter trimester system, which we call terms. In any given 107 00:08:39,140 --> 00:08:44,400 school year we have three terms, fall, winter, and spring, each consisting of 10 weeks of 108 00:08:44,400 --> 00:08:47,960 classes, including a few extra days in the fall to make up for Thanksgiving towards the 109 00:08:47,960 --> 00:08:52,800 end of the term, and one finals week. I took my first stats class in my second term of 110 00:08:52,800 --> 00:08:57,960 freshman year, so January to March of 2019, where I learned about the absolute basics 111 00:08:57,960 --> 00:09:02,680 of stats, like what I talked about here. Then I took two terms of a class called methods 112 00:09:02,680 --> 00:09:07,480 of data analysis in each of my second and third terms of my sophomore year, where I 113 00:09:07,480 --> 00:09:11,400 learned more about all the different kinds of tests that exist and the basics of multiple 114 00:09:11,400 --> 00:09:16,040 linear regression. After that I took two terms of mathematical statistics, which start out 115 00:09:16,040 --> 00:09:19,880 with the basics of probability that I talked about in my last podcast, but then quickly 116 00:09:19,880 --> 00:09:24,680 go into the more mathematical side of hypothesis testing and do a lot of calculations involving 117 00:09:24,680 --> 00:09:26,000 calculus. 118 00:09:26,000 --> 00:09:29,480 By this point I had applied and gotten into grad school, so I was fortunate enough to 119 00:09:29,480 --> 00:09:33,760 have finished my bachelor's in math in three years and decided to jump right into my first 120 00:09:33,760 --> 00:09:37,920 year of grad school and what would have otherwise been my senior year of college. Grad school 121 00:09:37,920 --> 00:09:42,680 lasted five terms for me, so a whole academic year and then fall and winter terms of the 122 00:09:42,680 --> 00:09:47,800 year after that. The electives I took included multivariate analysis, talking about multilinear 123 00:09:47,800 --> 00:09:52,720 and multiple multilinear regression, time series, which involved the analysis of temporal 124 00:09:52,720 --> 00:09:58,840 data, one of my favorites, data visualization, where I got to make pretty graphs using R, 125 00:09:58,840 --> 00:10:02,680 real analysis, which got into the depths of measure theory, which I talked about last 126 00:10:02,680 --> 00:10:07,840 time, gradual level probability theory, which uses a lot of measure theory, and statistical 127 00:10:07,840 --> 00:10:13,480 methods for genomics research. Besides these, my cohort and I took two whole three term 128 00:10:13,480 --> 00:10:18,320 sequences on statistical methods and the theory of statistics, which got super mathematical 129 00:10:18,320 --> 00:10:23,000 and took up the vast majority of all of our time. Those sequences made us learn a lot 130 00:10:23,000 --> 00:10:28,560 because we were all tested on these classes via comprehensive exams or comps in each subject, 131 00:10:28,560 --> 00:10:32,720 theory and methods. We studied for those comps throughout the year and especially throughout 132 00:10:32,720 --> 00:10:36,840 the whole summer after our first year, with the immense help of one of our professors 133 00:10:36,840 --> 00:10:41,920 who went above and beyond to hold study sessions on past comp exams to help prepare us better. 134 00:10:41,920 --> 00:10:46,360 Huge, huge shout out to Sarah if you're listening to this, we all really appreciate your enormous 135 00:10:46,360 --> 00:10:51,320 help. I was really particularly weak in a lot of the second portion of methods and theory, 136 00:10:51,320 --> 00:10:55,400 mostly because my son was born two days after finals ended that term, and I was more stressed 137 00:10:55,400 --> 00:10:59,720 about when he would come than about learning the material. I had to relearn a lot of that 138 00:10:59,720 --> 00:11:04,240 stuff and practice it over and over again during the summer. Thanks to Sarah, my friend 139 00:11:04,240 --> 00:11:09,160 Toy studied all summer with, and the department allowing us to use cheat sheets on each exam, 140 00:11:09,160 --> 00:11:13,860 I was able to pass both exams at what was known as a PhD pass, which meant I scored 141 00:11:13,860 --> 00:11:18,580 higher than the minimum required on each test to qualify for my school's PhD program, 142 00:11:18,580 --> 00:11:22,320 as opposed to an MS pass, which was enough to graduate but not enough to qualify for 143 00:11:22,320 --> 00:11:27,280 the PhD. I originally thought about staying and going for a PhD, but with raising an infant 144 00:11:27,280 --> 00:11:31,520 son and just being burned out in general, I decided to just stop after five terms with 145 00:11:31,520 --> 00:11:35,440 a masters, and whenever I think about it, I'm so glad that that's what I ended up 146 00:11:35,440 --> 00:11:40,940 doing. I just finished this past March, so grad school for me went from September 2021 147 00:11:40,940 --> 00:11:46,400 to March 2023. Don't get me wrong, grad school was overall very rewarding to me. I made a 148 00:11:46,400 --> 00:11:50,740 few great friends who I still keep in contact with regularly over Discord, and got to work 149 00:11:50,740 --> 00:11:54,300 with some absolutely brilliant professors, one of whom I learned in the middle of my 150 00:11:54,300 --> 00:11:58,760 third term was the mom of a guy I went to elementary school with. But I would be lying 151 00:11:58,760 --> 00:12:02,820 if I said I didn't notice my stress levels even in my last two terms when comps were 152 00:12:02,820 --> 00:12:07,680 over versus when I was done and started a new job and moved into a new house in March. 153 00:12:07,680 --> 00:12:11,500 I miss the people, but I don't miss working around the clock on homework and devoting 154 00:12:11,500 --> 00:12:16,560 a significant amount of time each weekend to that plus grading for my teaching assistantship. 155 00:12:16,560 --> 00:12:19,700 Speaking of which, those of you who went to college probably remember taking at least 156 00:12:19,700 --> 00:12:23,500 one class that had a lab component where a graduate student walked you through how to 157 00:12:23,500 --> 00:12:28,300 code something or how to dissect an animal or something depending on the class. I almost 158 00:12:28,300 --> 00:12:32,700 never got the chance to lead a lab at all. All of my first three terms consisted of me 159 00:12:32,700 --> 00:12:37,400 grading holding office hours for an intro stats class for engineers, and my fourth term 160 00:12:37,400 --> 00:12:41,760 was split between grading for mathematical stats and for an intro class for a graduate 161 00:12:41,760 --> 00:12:46,500 data analytics program, neither of which had me leading a recitation or lab. It wasn't 162 00:12:46,500 --> 00:12:51,520 until my last term that I had my first class where I actually led a coding lab, which coincidentally 163 00:12:51,520 --> 00:12:56,020 and poetically was for the very class that I first took as a freshman. I was a bit nervous 164 00:12:56,020 --> 00:13:00,400 about the idea of publicly speaking and leading labs, but I actually started to enjoy it after 165 00:13:00,400 --> 00:13:05,420 a few weeks. It helped that a lot of it was just me doing some basic live coding and R, 166 00:13:05,420 --> 00:13:09,220 a programming language that I had been using for four years up to that point, not having 167 00:13:09,220 --> 00:13:13,460 to make much eye contact with the students. It was mostly just 20 minutes of me going 168 00:13:13,460 --> 00:13:17,220 through how to do some analysis that the students would have to replicate on their homework 169 00:13:17,220 --> 00:13:20,660 and an hour of me walking around the room answering questions as students worked on 170 00:13:20,660 --> 00:13:24,940 their lab assignments. The last term of the program involved working with a professor 171 00:13:24,940 --> 00:13:28,940 on a project. For the master's program, this doesn't have to be anything novel. It could 172 00:13:28,940 --> 00:13:32,880 just be you learning about something you didn't touch on in any of your previous classes and 173 00:13:32,880 --> 00:13:37,500 writing a 10 to 15 page report and making a presentation on it. For example, one person 174 00:13:37,500 --> 00:13:41,900 of my cohort who graduated the same term as me did their project on Deming regression, 175 00:13:41,900 --> 00:13:47,540 a special kind of regression. My project involved using social networks to model relationships 176 00:13:47,540 --> 00:13:51,640 and alliances and an online game of survivor, which was really fun and one of the highlights 177 00:13:51,640 --> 00:13:55,900 of the program for me. I'll talk more about that project if and when I do an episode on 178 00:13:55,900 --> 00:14:00,640 graph theory. If I had to give some advice for future grad students, here are some suggestions 179 00:14:00,640 --> 00:14:06,060 I have. One, go to office hours, especially if you're in a situation like me where you'll 180 00:14:06,060 --> 00:14:10,340 be assessed on the classes you take through not only a regular final exam, but also a 181 00:14:10,340 --> 00:14:15,220 huge comprehensive one at the end of the year. Your professors want to see you succeed and 182 00:14:15,220 --> 00:14:19,340 they want to do what they can to help you learn. Nobody will let you fail out of the 183 00:14:19,340 --> 00:14:24,180 program as long as you do what you need to do to learn. Two, lean on your cohort for 184 00:14:24,180 --> 00:14:27,980 support. I'm not saying y'all have to be best friends or even necessarily friends at 185 00:14:27,980 --> 00:14:32,060 all, but at least work together on assignments and pick their brains every now and then. 186 00:14:32,060 --> 00:14:35,720 More likely than not, y'all have different backgrounds and experiences and therefore 187 00:14:35,720 --> 00:14:39,940 unique perspectives when it comes to problem solving. You might have certain strengths 188 00:14:39,940 --> 00:14:43,940 and intuition to impart on your cohort and they might have different strengths and intuition 189 00:14:43,940 --> 00:14:50,500 to impart on you. Three, join clubs or otherwise hang out with grad students in different departments. 190 00:14:50,500 --> 00:14:53,900 Not only does this expand your worldview, but you have a greater chance of finding some 191 00:14:53,900 --> 00:14:58,660 people who share some of your hobbies. Four, take some time for yourself when you need 192 00:14:58,660 --> 00:15:02,700 it. Taking a couple hours to take your mind off of schoolwork will ultimately be more 193 00:15:02,700 --> 00:15:07,460 productive than working all day on a problem. It also helps you become less burned out in 194 00:15:07,460 --> 00:15:12,780 the long run. Five, don't stress about grades. As long as you get the grades you need to 195 00:15:12,780 --> 00:15:16,660 stay in the program, which can usually be achieved by just staying on top of your work, 196 00:15:16,660 --> 00:15:20,980 you'll be fine. Employers don't care about your grad school grades and if they do, they're 197 00:15:20,980 --> 00:15:25,580 not a good place to work for in the first place. If you want to apply to a top PhD program, 198 00:15:25,580 --> 00:15:29,220 however, then grades matter a bit more, but ultimately it's about the experiences you 199 00:15:29,220 --> 00:15:33,140 get in your program. In the grand scheme of things, a grad school is what you make of 200 00:15:33,140 --> 00:15:37,120 it. I'm personally glad I went through it and took the path I did, but I'm also glad 201 00:15:37,120 --> 00:15:42,020 I stopped when I did and kickstarted my career. The important thing to note is you don't 202 00:15:42,020 --> 00:15:46,140 have to be an absolute genius to survive grad school. A strong work ethic, a willingness 203 00:15:46,140 --> 00:15:50,900 to learn, and collaboration skills are all you need to get by. At the end of the day, 204 00:15:50,900 --> 00:15:54,340 don't work yourself to death if you don't want or have to. If you only want to get your 205 00:15:54,340 --> 00:15:59,020 masters in skitaddle, it's only a year or two of your life. If you're going for a PhD 206 00:15:59,020 --> 00:16:03,220 on the other hand, that's a much larger chunk of time spent in graduate school, but I can't 207 00:16:03,220 --> 00:16:07,280 speak to what that's like. Maybe one day I'll revisit this topic and include someone 208 00:16:07,280 --> 00:16:10,780 who went through a PhD program to get their perspective. 209 00:16:10,780 --> 00:16:15,260 Thank you for listening to this episode of the Magnificent Mathematics. Next time, I'll 210 00:16:15,260 --> 00:16:32,700 talk more about calculus, the foundation for modern math. See you then! 211 00:16:32,700 --> 00:17:56,460 Algid Productions LLC Outro. Thank you for listening!