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Statistics

Exercises in C++

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Module 1 · Data and variables

Data is a set of observations of variables

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Data is a set of observations of variables
, 2 to practiseVariables are categories or numbers
, 1 to practiseCounts are discrete, measurements are continuous
, 1 to practiseOrdered categories and numbers that are only labels
, 1 to practiseYou study a sample to learn about a population
The C++ this course uses
, 1 to practiseA tidy table has one row per observation
, 1 to practiseEvery measurement has error
, 1 to practiseWhere data comes from shapes what it can tell you
, 1 to practiseA first look at a data set
, 2 to practiseTry it: count and check a column
, 4 to practisePractice problems
Recap

, 2 to practiseA frequency table counts each value
, 1 to practiseBar charts show categorical data
, 2 to practiseHistograms show numerical data in bins
, 1 to practiseThe bin width changes the story
, 1 to practiseDot plots and stem-and-leaf plots keep every value
, 1 to practiseShape: symmetric, skewed or bimodal
, 1 to practiseA box plot draws five numbers
, 1 to practiseCharts can mislead
, 3 to practiseTry it: count values into bins
, 4 to practisePractice problems
Recap

What is a typical value
, 3 to practiseThe mean shares the total out equally
The mean is the balance point
, 2 to practiseThe median is the middle value
, 2 to practiseComputing the median in code
, 1 to practiseThe mode is the most common value
, 1 to practiseOne extreme value drags the mean
, 1 to practiseSkew pulls the mean towards the tail
, 1 to practiseA weighted mean lets some values count more
, 1 to practiseA trimmed mean drops the extremes
, 1 to practiseChoosing a measure of centre
, 4 to practisePractice problems
Recap

, 1 to practiseThe centre is not the whole story
, 1 to practiseThe range is simple but fragile
, 2 to practiseQuartiles and the interquartile range
, 2 to practiseThe 1.5 × IQR rule flags outliers
Variance is the mean squared deviation
, 1 to practiseA sample divides by n − 1
Standard deviation is the width of the data
, 2 to practiseTry it: the sample standard deviation
, 1 to practiseA z-score measures distance in standard deviations
, 1 to practiseThe 68–95–99.7 rule for bell-shaped data
, 1 to practiseChebyshev’s inequality works for any shape
, 4 to practisePractice problems
Recap

, 1 to practiseA scatter plot shows two numerical variables
, 1 to practiseDescribe direction, form and strength
, 2 to practiseCovariance: do the deviations agree
, 2 to practiseCorrelation is covariance without units
, 1 to practiseCorrelation only sees straight lines
, 1 to practiseTry it: compute the correlation
, 1 to practiseRank correlation for any rising pattern
, 1 to practiseTwo-way tables relate two categorical variables
, 1 to practiseCorrelation is not causation
, 1 to practiseSimpson’s paradox: the trend can reverse
, 4 to practisePractice problems
Recap

, 1 to practiseProbability puts a number on chance
, 2 to practiseEqually likely outcomes: count and divide
Probability as long-run frequency
, 1 to practiseThree rules every probability obeys
, 2 to practiseThe complement: one minus the chance
, 1 to practiseThe addition rule: or
, 1 to practiseIndependent events multiply
, 1 to practiseCounting: arrangements and choices
, 2 to practiseTry it: choose and the birthday problem
, 1 to practiseWhen counting is hard, simulate
, 4 to practisePractice problems
Recap

, 2 to practiseConditional probability: once you know something
, 2 to practiseChances multiply along a chain
, 1 to practiseTree diagrams lay out the chain
, 1 to practiseThe law of total probability adds the paths
, 1 to practiseBayes’ theorem turns the condition around
A positive test for a rare condition
, 2 to practiseIgnoring the base rate is the classic mistake
, 1 to practiseThe prosecutor’s fallacy
, 1 to practiseThe Monty Hall problem
, 4 to practisePractice problems
Recap

A random variable is a number decided by chance
, 2 to practiseA probability mass function lists the chances
, 1 to practiseThe expected value is the long-run average
, 1 to practiseThe variance of a random variable
, 3 to practiseTry it: the mean and variance of a distribution
, 1 to practiseShifting and scaling a random variable
, 1 to practiseExpected values always add
, 1 to practiseVariances add for independent variables
, 1 to practiseGames and fair bets
, 1 to practiseExpected value is not the whole story
, 4 to practisePractice problems
Recap

, 1 to practiseA Bernoulli trial: success or failure
, 1 to practiseThe binomial counts successes in n trials
, 2 to practiseThe binomial formula
The shape of the binomial
, 1 to practiseMean np, variance np(1 − p)
, 2 to practiseTry it: the binomial pmf and cdf
, 1 to practiseThe geometric distribution: waiting for a success
, 2 to practiseThe Poisson distribution counts rare events
, 1 to practiseThe Poisson is the binomial for rare events
, 1 to practiseTry it: the Poisson pmf
, 1 to practiseWhich distribution fits
, 4 to practisePractice problems
Recap

, 1 to practiseFor continuous variables, probability is area
, 2 to practiseThe uniform distribution: flat
, 1 to practiseThe exponential distribution: waiting for the next event
The normal distribution: the bell curve
, 1 to practiseThe standard normal and its table
, 2 to practiseNormal calculations: standardise, then look up
, 2 to practiseTry it: normal probabilities in code
, 1 to practiseThe normal approximation to the binomial
, 1 to practiseChecking normality with a QQ plot
, 4 to practisePractice problems
Recap

, 1 to practiseYou study a sample to learn about a population
, 1 to practiseRandom sampling gives every member a fair chance
, 1 to practiseA biased method misses the same way every time
, 1 to practiseA statistic has a distribution of its own
, 2 to practiseThe standard error measures how much a statistic wobbles
, 2 to practiseThe law of large numbers: averages settle down
, 2 to practiseThe central limit theorem: means become normal
, 2 to practiseSimulating a sampling distribution
, 1 to practiseWhen the normal approximation can be trusted
, 4 to practisePractice problems
Recap

, 1 to practiseA point estimate is one best guess
, 1 to practiseAn interval is the estimate plus or minus a margin
, 1 to practiseMore confidence costs a wider interval
, 1 to practiseNinety-five per cent is about the method, not this interval
, 1 to practiseWhen sigma is unknown, use the t distribution
, 3 to practiseThe t interval for a mean
, 1 to practiseAn interval for a proportion
, 1 to practiseWhat makes the margin of error shrink
, 1 to practiseChoosing the sample size before you collect
, 1 to practiseHow confidence intervals are misread
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA test asks whether chance alone could explain the data
, 1 to practiseThe null hypothesis is the boring explanation
, 2 to practiseA test statistic measures distance in standard errors
, 1 to practiseThe p-value is how surprising the data would be under the null
, 1 to practiseThe significance level is the line you draw in advance
, 1 to practiseOne-sided and two-sided tests
, 2 to practiseWhen sigma is unknown, the t-test takes its place
, 1 to practiseTwo ways to be wrong
, 1 to practisePower is the chance of finding an effect that is there
, 1 to practiseTests and confidence intervals tell the same story
, 1 to practiseRun enough tests and something will be significant
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseA difference in means is only convincing next to the spread
, 2 to practiseThe two-sample t-test
, 1 to practisePaired data: test the differences
, 2 to practiseChi-squared compares counts with what you expected
, 1 to practiseThe chi-squared distribution
, 1 to practiseChi-squared for independence in a table
, 1 to practiseANOVA compares the spread between groups with the spread within
, 1 to practiseComputing the F statistic
, 1 to practiseSignificant is not the same as large
, 2 to practiseMany comparisons need a stricter bar
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

A line that predicts one variable from another
, 1 to practiseThe best line makes the squared misses smallest
, 2 to practiseThe least squares formulas
, 1 to practiseFitting the line in code
, 1 to practiseReading the slope and the intercept
, 2 to practiseR squared: the share of the variation the line explains
, 1 to practiseResidual plots show what the line missed
, 1 to practiseOutliers and leverage: some points pull harder
, 1 to practisePredicting inside the data, not beyond it
, 1 to practiseIs the slope real? Inference on the slope
, 4 to practisePractice problems
, 1 to practiseRecap

Real outcomes have more than one cause
, 2 to practiseEach coefficient holds the others fixed
, 1 to practiseCategories enter as 0 or 1
, 1 to practiseInteractions let one slope depend on another variable
, 1 to practisePredictors that move together blur each other
, 1 to practiseMore predictors always fit better, and that is the trap
Logistic regression predicts a probability
, 3 to practiseCoefficients act on the log odds
, 1 to practiseGradient descent walks downhill to the best fit
, 1 to practiseFitting a line by gradient descent
, 3 to practisePractice problems
, 2 to practiseMore practice problems
, 1 to practiseRecap

, 2 to practiseWhen the maths is hard, simulate
, 1 to practiseComputers fake randomness, on purpose
, 1 to practiseYour first Monte Carlo estimate
, 1 to practiseThe bootstrap: the sample stands in for the population
, 2 to practiseBootstrapping the mean wait at a coffee shop
, 1 to practiseWrite the bootstrap
, 1 to practiseWhere the bootstrap fails
, 1 to practisePermutation tests: shuffle the labels
, 2 to practiseWrite a permutation test
, 2 to practiseCross-validation: test on data the model has not seen
, 4 to practisePractice problems
Recap

, 1 to practiseProbability as a degree of belief
, 2 to practiseUpdating with two possible coins
, 1 to practiseA grid of possible values
, 1 to practiseThe beta distribution describes a proportion
, 2 to practiseBeta prior plus binomial data gives a beta posterior
, 1 to practiseCredible intervals say what people wish confidence intervals said
, 1 to practiseA normal prior for a mean: precisions add
, 1 to practiseChoosing a prior, and how much it matters
, 1 to practiseMCMC: sampling a posterior you cannot write down
, 1 to practiseBayesian and frequentist: two questions, often the same answer
, 3 to practisePractice problems
, 2 to practiseMore practice problems
Recap

, 1 to practiseWatching is not the same as intervening
, 1 to practiseA confounder drives both the cause and the effect
, 1 to practiseRandomisation balances what you cannot see
, 1 to practiseControl groups and blinding
, 2 to practiseAn A/B test is an experiment on a website
, 1 to practiseSelection and survivorship bias
, 3 to practiseExtreme results drift back toward the average
, 1 to practiseSimpson’s paradox: the whole can reverse the parts
, 1 to practiseCausal diagrams say what to adjust for
, 4 to practisePractice problems
Recap

, 1 to practiseLikelihood turns probability around
, 2 to practiseMaximum likelihood picks the best-explaining value
, 1 to practiseEvery model trades bias against variance
, 3 to practiseCompare models on what they have not seen
, 1 to practiseRegularisation shrinks coefficients toward zero
, 1 to practiseThe reproducibility crisis
, 1 to practisePre-register, then report the uncertainty
, 4 to practiseMastery questions
, 1 to practiseFinal challenge: an interval for a difference
, 3 to practisePractice problems: the whole course
, 3 to practiseMore practice: the whole course
Where to go next
Recap