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Head First Statistics

Name: Head First Statistics
Author: Dawn Griffiths
ISBN: 9780596527587

by Dawn Griffiths

August 2008

Beginner

716 pages

16h 41m

English

O'Reilly Media, Inc.

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Dedication
A Note Regarding Supplemental Files
Advance Praise for Head First Statistics
Praise for other Head First books
Author of Head First Statistics
How to use this Book: Intro
Who is this book for?Who should probably back away from this book?We know what you’re thinkingWe know what your brain is thinkingMetacognition: thinking about thinkingHere’s what WE didHere’s what YOU can do to bend your brain into submissionRead MeThe technical review teamAcknowledgmentsSafari® Books Online
1. Visualizing Information: First Impressions
Statistics are everywhereBut why learn statistics?A tale of two chartsManic Mango needs some chartsThe humble pie chartSo when are pie charts useful?Chart failureBar charts can allow for more accuracyVertical bar chartsHorizontal bar chartsIt’s a matter of scaleUsing percentage scalesUsing frequency scalesDealing with multiple sets of dataThe split-category bar chartThe segmented bar chartYour bar charts rockCategories vs. numbersCategorical or qualitative dataNumerical or quantitative dataDealing with grouped dataTo make a histogram, start by finding bar widthsManic Mango needs another chartA histogram’s bar area must be proportional to frequencyMake the area of histogram bars proportional to frequencyStep 1: Find the bar widthsStep 2: Find the bar heightsStep 3: Draw your chart—a histogramHistograms can’t do everythingIntroducing cumulative frequencySo what are the cumulative frequencies?Drawing the cumulative frequency graphChoosing the right chartManic Mango conquered the games market!
2. Measuring Central Tendency: The Middle Way
Welcome to the Health ClubA common measure of average is the meanMean mathLetters and numbersDealing with unknownsBack to the meanThe mean has its own symbolHandling frequenciesBack to the Health ClubEverybody was Kung Fu fightingOur data has outliersThe butler outliers did itWatercooler conversationFinding the medianBusiness is boomingThe Little Ducklings swimming classFrequency MagnetsFrequency MagnetsWhat went wrong with the mean and median?Introducing the modeIt even works with categorical dataCongratulations!
3. Measuring Variability and Spread: Power Ranges
Wanted: one playerWe need to compare player scoresUse the range to differentiate between data setsMeasuring the rangeThe problem with outliersWe need to get away from outliersQuartiles come to the rescueThe interquartile range excludes outliersQuartile anatomyFinding the position of the lower quartileFinding the position of the upper quartileWe’re not just limited to quartilesSo what are percentiles?Percentile usesFinding percentilesBox and whisker plots let you visualize rangesVariability is more than just spreadCalculating average distancesWe can calculate variation with the variance......but standard deviation is a more intuitive measureStandard deviation know-howA quicker calculation for varianceWhat if we need a baseline for comparison?Use standard scores to compare values across data setsCalculating standard scoresInterpreting standard scoresSo what does this tell us about the players?Statsville All Stars win the league!
4. Calculating Probabilities: Taking Chances
Fat Dan’s Grand SlamRoll up for roulette!Your very own roulette boardPlace your bets now!What are the chances?Find roulette probabilitiesYou can visualize probabilities with a Venn diagramComplementary eventsIt’s time to play!And the winning number is...Let’s bet on an even more likely eventYou can also add probabilitiesYou win!Time for another betExclusive events and intersecting eventsProblems at the intersectionSome more notationAnother unlucky spin......but it’s time for another betConditions applyFind conditional probabilitiesYou can visualize conditional probabilities with a probability treeTrees also help you calculate conditional probabilitiesBad luck!We can find P(Black l Even) using the probabilities we already haveStep 1: Finding P(Black ∩ Even)So where does this get us?Step 2: Finding P(Even)Step 3: Finding P(Black l Even)These results can be generalized to other problemsUse the Law of Total Probability to find P(B)Introducing Bayes’ TheoremWe have a winner!It’s time for one last betIf events affect each other, they are dependentIf events do not affect each other, they are independentMore on calculating probability for independent eventsWinner! Winner!

5. Using Discrete Probability Distributions: Manage Your Expectations
Back at Fat Dan’s CasinoWe can compose a probability distribution for the slot machineExpectation gives you a prediction of the results...... and variance tells you about the spread of the resultsVariances and probability distributionsSo how do we calculate E(X – μ)2?Let’s calculate the slot machine’s varianceFat Dan changed his pricesThere’s a linear relationship between E(X) and E(Y)Slot machine transformationsGeneral formulas for linear transformsEvery pull of the lever is an independent observationObservation shortcutsExpectationVarianceNew slot machine on the blockAdd E(X) and E(Y) to get E(X + Y)...... and subtract E(X) and E(Y) to get E(X – Y)You can also add and subtract linear transformationsAdding aX and bYSubtracting aX and bYJackpot!
6. Permutations and Combinations: Making Arrangements
The Statsville DerbyIt’s a three-horse raceHow many ways can they cross the finish line?Calculate the number of arrangementsSo what if there are n horses?Going round in circlesIt’s time for the novelty raceArranging by individuals is different than arranging by typeWe need to arrange animals by typeGeneralize a formula for arranging duplicatesIt’s time for the twenty-horse raceHow many ways can we fill the top three positions?Examining permutationsWhat if horse order doesn’t matterExamining combinationsIt’s the end of the race
7. Geometric, Binomial, and Poisson Distributions: Keeping Things Discrete
Meet Chad, the hapless snowboarderWe need to find Chad’s probability distributionThere’s a pattern to this probability distributionThe probability distribution can be represented algebraicallyThe pattern of expectations for the geometric distributionExpectation is 1/pFinding the variance for our distributionYou’ve mastered the geometric distributionShould you play, or walk away?Generalizing the probability for three questionsWhat’s the missing number?Let’s generalize the probability furtherWhat’s the expectation and variance?Let’s look at one trialBinomial expectation and varianceThe Statsville Cinema has a problemIt’s a different sort of distributionSo how do we find probabilities?Expectation and variance for the Poisson distributionWhat does the Poisson distribution look like?So what’s the probability distribution?Combine Poisson variablesThe Poisson in disguiseAnyone for popcorn?
8. Using the Normal Distribution: Being Normal
Discrete data takes exact values...... but not all numeric data is discreteWhat’s the delay?We need a probability distribution for continuous dataProbability density functions can be used for continuous dataProbability = areaTo calculate probability, start by finding f(x)...... then find probability by finding the areaWe’ve found the probabilitySearching for a soul sole mateMale modellingThe normal distribution is an “ideal” model for continuous dataSo how do we find normal probabilities?Three steps to calculating normal probabilitiesStep 1: Determine your distributionStep 2: Standardize to N(0, 1)To standardize, first move the mean...... then squash the widthNow find Z for the specific value you want to find probability forStep 3: Look up the probability in your handy tableSo how do you use probability tables?Julie’s probability is in the tableAnd they all lived happily ever afterBut it doesn’t stop there.
9. Using the Normal Distribution ii: Beyond Normal
Love is a roller coasterAll aboard the Love TrainNormal bride + normal groomIt’s still just weightHow’s the combined weight distributed?Finding probabilitiesMore people want the Love TrainLinear transforms describe underlying changes in values...So what’s the distribution of a linear transform?...and independent observations describe how many values you haveExpectation and variance for independent observationsShould we play, or walk away?Normal distribution to the rescueWhen to approximate the binomial distribution with the normalFinding the mean and varianceRevisiting the normal approximationThe binomial is discrete, but the normal is continuousApply a continuity correction before calculating the approximationAll aboard the Love TrainWhen to approximate the binomial distribution with the normalWhen λ is small...When λ is large...So how large is large enough?A runaway success!
10. Using Statistical Sampling: Taking Samples
The Mighty Gumball taste testThey’re running out of gumballsTest a gumball sample, not the whole gumball populationGumball populationsGumball samplesHow sampling worksWhen sampling goes wrongHow to design a sampleDefine your target populationDefine your sampling unitsDefine your sampling frameSometimes samples can be biasedUnbiased SamplesBiased SamplesSources of biasHow to choose your sampleSimple random samplingSampling with replacementSampling without replacementHow to choose a simple random sampleDrawing lotsRandom number generatorsThere are other types of samplingWe can use stratified sampling......or we can use cluster sampling......or even systematic samplingMighty Gumball has a sampleSo what’s next?
11. Estimating Populations and Samples: Making Predictions
So how long does flavor really last for?Let’s start by estimating the population meanPoint estimators can approximate population parametersLet’s estimate the population varianceWe need a different point estimator than sample varianceSo what is the estimator?Which formula’s which?Mighty Gumball has done more samplingIt’s a question of proportionPredicting population proportionBuy your gumballs here!Introducing new jumbo boxesSo how does this relate to sampling?The sampling distribution of proportionsSo what’s the expectation of Ps?And what’s the variance of Ps?Find the distribution of PsPs follows a normal distributionPs—continuity correction requiredHow many gumballs?There’s just one more problem...We need probabilities for the sample meanThe sampling distribution of the meanFind the expectation for X̄What about the the variance of X̄?So how is X̄ distributed?If n is large, X̄ can still be approximated by the normal distributionIntroducing the Central Limit TheoremUsing the central limit theoremThe binomial distributionThe Poisson distributionFinding probabilitiesSampling saves the day!You’ve made a lot of progress
12. Constructing Confidence Intervals: Guessing with Confidence
Mighty Gumball is in troubleThey need you to save themThe problem with precisionIntroducing confidence intervalsFour steps for finding confidence intervalsStep 1: Choose your population statisticStep 2: Find its sampling distributionPoint estimators to the rescueWe’ve found the distribution for X̄Step 3: Decide on the level of confidenceHow to select an appropriate confidence levelStep 4: Find the confidence limitsStart by finding ZRewrite the inequality in terms of μFinally, find the value of X̄You’ve found the confidence intervalLet’s summarize the stepsHandy shortcuts for confidence intervalsWhat’s the interval in general?Just one more problem...Step 1: Choose your population statisticStep 2: Find its sampling distributionX̄ follows the t-distribution when the sample is smallFind the standard score for the t-distributionStep 3: Decide on the level of confidenceStep 4: Find the confidence limitsUsing t-distribution probability tablesThe t-distribution vs. the normal distributionYou’ve found the confidence intervals!
13. Using Hypothesis Tests: Look At The Evidence
Statsville’s new miracle drugSo what’s the problem?Resolving the conflict from 50,000 feetThe six steps for hypothesis testingStep 1: Decide on the hypothesisThe drug company’s claimSo what’s the null hypothesis for SnoreCull?So what’s the alternative?The doctor’s perspectiveThe alternate hypothesis for SnoreCullStep 2: Choose your test statisticWhat’s the test statistic for SnoreCull?Step 3: Determine the critical regionAt what point can we reject the drug company claims?To find the critical region, first decide on the significance levelSo what significance level should we use?Step 4: Find the p-valueHow do we find the p-value?We’ve found the p-valueStep 5: Is the sample result in the critical region?Step 6: Make your decisionSo what did we just do?What if the sample size is larger?Let’s conduct another hypothesis testStep 1: Decide on the hypothesesIt’s still the same problemStep 2: Choose the test statisticUse the normal to approximate the binomial in our test statisticStep 3: Find the critical regionSnoreCull failed the testMistakes can happenLet’s start with Type I errorsSo what’s the probability of getting a Type I error?What about Type II errors?So how do we find β?Finding errors for SnoreCullLet’s start with the Type I errorSo what about the Type II error?We need to find the range of valuesFind P(Type II error)Introducing powerSo what’s the power of SnoreCull?The doctor’s happyBut it doesn’t stop there
14. The χ2 Distribution: There’s Something Going On...
There may be trouble ahead at Fat Dan’s CasinoLet’s start with the slot machinesThe χ2 test assesses differenceSo what does the test statistic represent?Two main uses of the χ2 distributionWhen v is 1 or 2When v is greater than 2v represents degrees of freedomSo what’s v?What’s the significance?How to use χ2 probability tablesHypothesis testing with χ2You’ve solved the slot machine mysteryFat Dan has another problemthe χ2 distribution can test for independenceYou can find the expected frequencies using probabilitySo what are the frequencies?How do we find the frequencies in general?We still need to calculate degrees of freedomGeneralizing the degrees of freedomAnd the formula is...You’ve saved the casino
15. Correlation and Regression: What’s My Line?
Never trust the weatherLet’s analyze sunshine and attendanceExploring types of dataAll about bivariate dataVisualizing bivariate dataScatter diagrams show you patternsCorrelation vs. causationWe need to predict the concert attendancePredict values with a line of best fitYour best guess is still a guessWe need to find the equation of the lineWe need to minimize the errorsIntroducing the sum of squared errorsFind the equation for the line of best fitLet’s start with bFinding the slope for the line of best fitWe use x̄ and ȳ to help us find bFinding the slope for the line of best fit, part iiWe’ve found b, but what about a?You’ve made the connectionLet’s look at some correlationsAccurate linear correlationNo linear correlationThe correlation coefficient measures how well the line fits the dataThere’s a formula for calculating the correlation coefficient, rFind r for the concert dataFind r for the concert data, continuedYou’ve saved the day!Leaving town...It’s been great having you here in Statsville!
A. Leftovers: The Top Ten Things (we didn’t cover)
#1. Other ways of presenting dataDotplotsStemplots#2. Distribution anatomyThe empirical rule for normal distributionsChebyshev’s rule for any distribution#3. ExperimentsSo what makes for a good experiment?Designing your experimentCompletely randomized designRandomized block designMatched pairs design#4. Least square regression alternate notation#5. The coefficient of determinationCalculating r2#6. Non-linear relationships#7. The confidence interval for the slope of a regression lineThe margin of error for b#8. Sampling distributions – the difference between two means#9. Sampling distributions – the difference between two proportions#10. E(X) and Var(X) for continuous probability distributionsFinding E(X)Finding Var(X)
B. Statistics Tables: Looking Things Up
#1. Standard normal probabilities#2. t-distribution critical values#3. X2 critical values
Index
About the Author
Copyright

Content preview from Head First Statistics

Chapter 8. Using the Normal Distribution: Being Normal

Discrete probability distributions can’t handle every situation.

So far we’ve looked at probability distributions where we’ve been able to specify exact values, but this isn’t the case for every set of data. Some types of data just don’t fit the probability distributions we’ve encountered so far. In this chapter, we’ll take a look at how continuous probability distributions work, and introduce you to one of the most important probability distributions in town—the normal distribution.

Discrete data takes exact values...

So far we’ve looked at probability distributions where the data is discrete. By this we mean the data is composed of distinct numeric values, and we’re been able to calculate the probability of each of these values. As an example, when we looked at the probability distribution for the winnings on a slot machine, the possible amounts we could win on each game were very precise. We knew exactly what amounts of money we could win, and we knew we’d win one of them.

If data is discrete, it’s numeric and can take only exact values. It’s often data that can be counted in some way, such as the number of gumballs in a gumball machine, the number of questions answered correctly in a game show, or the number of breakdowns in a particular ...

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Head First Statistics

by Dawn Griffiths

Chapter 8. Using the Normal Distribution: Being Normal

Discrete data takes exact values...

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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