Transcription of Chapter 1
1 Chapter 1 Basic statisticsStatistics are used forecasts estimate the probability that it will rain tomorrow based on a variety ofatmospheric measurements. Our email clients estimate the probability that incoming emailis spam using features found in the email message. By querying a relatively small group ofpeople, pollsters can gauge the pulse of a large population on a variety of issues, includingwho will win an election. In fact, during the 2012 US presidential election, Nate Silversuccessfully aggregated such polling data to correctly predict the election outcome of all 50states!1On top of this, the past decade or so has seen an explosion in the amount of data we collectacross many fields.
2 For example, The Large Hadron Collider, the world s largest particle accelerator, produces 15 petabytesof data about particle collisions every year2: that s 1015bytes, or a million gigabytes. Biologists are generating 15 petabytes of data a year in genomic information3. The internet is generating 1826 petabytes of data every day. The NSA s analysts claimto look at of that traffic, which comes out to about 25 petabytes per year!And those are just a few examples! statistics plays a key role in summarizing and distillingdata (large or small) so that we can make sense of statistics is an essential tool for justifying a variety of results in research projects,many researchers lack a clear grasp of statistics , misusing its tools and producing all sortsof bad science!
3 4 The goal of these notes is to help you avoid falling into that trap: we ll armyou with the proper tools to produce sound statistical particular, we ll do this by presenting important statistical tools and techniques whileemphasizing their underlying principles and Daniel Terdiman, Obama s win a big vindication for Nate Silver, king of the quants, CNET,November 6, 20122 See CERN s Computing site3 See Emily Singer, Biology s Big Problem: Theres Too Much Data to Handle, October 11, 20134 See The Economist, Unreliable research: trouble at the lab , October 19, for Research ProjectsChapter 1We ll start with a motivating example of how powerful statistics can be when they re usedproperly, and then dive into definitions of basic statistical concepts, exploratory analysismethods, and an overview of some commonly used probability : Uncovering data fakersIn 2008, a polling company called Research 2000 was hired by Daily Kos to gather approval data on toppoliticians (shown belowa).
4 Do you see anything odd?FavorableUnfavorableUndecidedTopicMe nWomenMenWomenMenWomenObama4359543437 Pelosi225266381210 Reid283660541210 McConnell311750701913 Boehner261651673317 Cong.(D)2844645482 Cong.(R)311358741113 Party(D)3145644659 Party(R)3820577159 Several amateur statisticians noticed that within each question, the percentages from the men almostalways had the same parity (odd-/even-ness) as the percentages from the women. If they truly had beensampling people randomly, this should have only happened about half the time. This table only showsa small part of the data, but it happened in 776 out of the 778 pairs they collected. The probability ofthis happening by chance is less than 10 228!Another anomaly they found: in normal polling data, there are many weeks In Research 2000 s data,this almost never happened: they were probably afraid to make up the same number two weeks in arow since that might not look random.
5 These problems (and others) were caught thanks to statisticalanalysis!aData and a full description at Daily Kos: Research 2000: Problems in plain sight, June 29, IntroductionWe start with some informal definitions: Probabilityis used when we have some model or representation of the world andwant to answer questions like what kind of data will this truth produce? Statisticsis what we use when we have data and want to discover the truth ormodel underlying the data. In fact, some of what we call statistics today used to becalled inverse probability .We ll focus on situations where we observe some set of particular outcomes, and want tofigure out why did we get these points?
6 It could be because of some underlying model ortruth in the world (in this case, we re usually interested in understanding that model), or2 statistics for Research ProjectsChapter 1because of how we collected the data (this is calledbias, and we try to avoid it as much aspossible).There are two schools of statistical thought (see this relevant xkcd5): Loosely speaking, thefrequentistviewpoint holds that the parameters of probabilisticmodels are fixed, but we just don t know them. These notes will focus on classicalfrequentist statistics . TheBayesianviewpoint holds that model parameters are not only unknown, but alsorandom. In this case, we ll encode our prior belief about them using a comes in many types.
7 Here are some of the most common: Categorical: discrete, not ordered ( , red , blue , etc.). Binary questions such aspolls also fall into this category. Ordinal: discrete, ordered ( , survey responses like agree , neutral , disagree ) Continuous: real values ( , time taken ). Discrete: numeric data that can only take on discrete values can either be modeled asordinal ( , for integers), or sometimes treated as continuous for ease of variableis a quantity (usually related to our data) that takes on randomvalues6. For a discrete random variable, probability distributionpdescribes how likelyeach of those random values are, sop(a) refers to the probability of observing distributionof some data (sometimes informally referred to as just thedistribution of the data) is the relative frequency of each value in some observed ll usually use the notationx1,x2.
8 ,xnto refer to data points that we observe. We llusually assume our sampled data points areindependent and identically distributed, or ,meaning that they re independent and all have the same probability a random variable is the average value it takes on:E[x] = poss. valuesap(a) aWe ll often use the notation xto represent the expectation of random islinear: for any random variablesx,yand constantsc,d,E[cx+dy] =cE[x] +dE[y].5Of course, this comic oversimplifies things: here s (Bayesian) statistician Andrew Gelman s , a random variable is a function that maps random outcomes to numbers, but this loosedefinition will suit our purposes and carries the intuition you ll the random variable is continuous instead of discrete,p(a) instead represents aprobability densityfunction, but we ll gloss over the distinction in these notes.
9 For more details, see an introductory probabilitytextbook, such asIntroduction to Probabilityby Bertsekas and for Research ProjectsChapter 1 This is a useful property, and it s true even whenxandyaren t independent!Intuition for linearity of expectationSuppose that we collect 5 data points of the form (x,y): (1,3),(2,4),(5,3),(4,3),(3,4). Let s write eachof these pairs along with their sum in a table:xyx+y134246538437347To estimate the mean of variablex, we could just average the values in the first column above ( , theobserved values forx): (1 + 2 + 5 + 4 + 3)/5 = 3. Similarly, to estimate the mean of variabley, weaverage the values in the second column above: (3 + 4 + 3 + 3 + 4)/5 = Finally, to estimate themean of variablex+y, we could just average the values in the third column: (4 + 6 + 8 + 7 + 7)/5 = ,which turns out to be the same as the sum of the averages of the first two that to arrive at the average of the values in the third column, we could ve reordered valueswithin column 1 and column 2!
10 For example, we scramble column 1 and, separately, column 2, and thenwe recompute column 3:xyx+y134235336448549 The average of the third column is (4 + 5 + 6 + 8 + 9)/5 = , which is the same as what we had before!This is true even thoughxandyare clearly not independent. Notice that we ve reordered columns 1and 2 to make them both increasing in value, effectively making them more correlated (and thereforeless independent). But, thanks to linearity of expectation, the average of the sum is still the same summary, linearity of expectation says that the ordering of the values within column 1, and separatelywithin column 2 don t actually matter in computing the average of the sum of two variables, which neednot be a random variable is a measure of how spread out it is:var[x] = poss.
