The systematic collection and arrangement of numerical facts or data of any kind;

(also) the branch of science or mathematics concerned with the analysis and interpretation of numerical data and appropriate ways of gathering such data. [OED]

2017-09-19

The systematic collection and arrangement of numerical facts or data of any kind;

(also) the branch of science or mathematics concerned with the analysis and interpretation of numerical data and appropriate ways of gathering such data. [OED]

- Can tell you if you should be surprised by your data
- Can help predict what future data will look like

## We'll use data on the duration and spacing of eruptions ## of the old faithful geyser ## Data are eruption duration and waiting time to next eruption data ("faithful") # load data str (faithful) # display the internal structure of an R object

## 'data.frame': 272 obs. of 2 variables: ## $ eruptions: num 3.6 1.8 3.33 2.28 4.53 ... ## $ waiting : num 79 54 74 62 85 55 88 85 51 85 ...

A "statistic" is a the result of applying a function (summary) to the data: `statistic <- function(data)`

E.g. ranks: Min, Quantiles, Median, Mean, Max

summary (faithful$eruptions)

## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 1.600 2.163 4.000 3.488 4.454 5.100

*Roughly*, a qua*n*tile for a proportion \(p\) is a value \(x\) for which \(p\) of the data are less than or equal to \(x\). The first qua*r*tile, median, and third qua*r*tile are the qua*n*tiles for \(p=0.25\), \(p=0.5\), and \(p=0.75\), respectively.

boxplot (faithful$eruptions, main="Eruption time", horizontal=T)

library(ggplot2);library(gridExtra); #boxplot relatives b1<-ggplot(faithful, aes(x="All",y=eruptions)) + labs(x=NULL) + geom_boxplot() #jitter plot b2<-ggplot(faithful, aes(x="All",y=eruptions)) + labs(x=NULL) + geom_jitter(position=position_jitter(height=0,width=0.25)) grid.arrange(b1, b2, nrow=1)

## Construct histogram of eruption times, plot data points on the x axis hist (faithful$eruptions, main="Eruption time", xlab="Time (minutes)", ylab="Count") points (x=faithful$eruptions,y=rep(0,length(faithful$eruptions)), lwd=4, col='blue')

## Construct different histogram of eruption times ggplot(faithful, aes(x=eruptions)) + labs(y="Proportion") + geom_histogram(aes(y = ..count../sum(..count..)))

- Common assumption is that data consists of replicates that are "the same."
- Come from "the same population"
- Come from "the same process"
- The goal of data analysis is to understand what the data tell us about the population.

We often assume that we can treat items as if they were distributed "*randomly*."

- "
*That's so random!*" - Result of a coin flip is random
- Passengers were screened at random

- "random" does not mean "uniform"

- Mathematical formalism:
*events*and*probability*

*Sample space*\({\mathcal{S}}\) is the set of all possible events we might observe. Depends on context.- Coin flips: \({\mathcal{S}}= \{ h, t \}\)
- Eruption times: \({\mathcal{S}}= {\mathbb{R}}^{\ge 0}\)
- (Eruption times, Eruption waits): \({\mathcal{S}}= {\mathbb{R}}^{\ge 0} \times {\mathbb{R}}^{\ge 0}\)

- An
*event*is a subset of the sample space.- Observe heads: \(\{ h \}\)
- Observe eruption for 2 minutes: \(\{ 2.0 \}\)
- Observe eruption with length between 1 and 2 minutes and wait between 50 and 70 minutes: \([1,2] \times [50,70]\).

Any event can be assigned a *probability* between \(0\) and \(1\) (inclusive).

- \(\Pr(\{h\}) = 0.5\)
- \(\Pr([1,2] \times [50,70]) = 0.10\)

Objectivist view

- Suppose we observe \(n\) replications of an experiment.
- Let \(n(A)\) be the number of times event \(A\) was observed
- \(\lim_{n \to \infty} \frac{n(A)}{n} = \Pr(A)\)
This is (loosely)

*Borel's Law of Large Numbers*Subjective interpretation is possible as well. ("Bayesian" statistics is related to this idea – more later.)

- We often reduce data to numbers.
- "\(1\) means heads, \(0\) means tails."

A

*random variable*is a mapping from the event space to a number (or vector.)Usually rendered in uppercase

*italics*\(X\) is every statistician's favourite, followed closely by \(Y\) and \(Z\).

"Realizations" of \(X\) are written in lower case, e.g. \(x_1\), \(x_2\), …

We will write the set of possible realizations as: \(\mathcal{X}\) for \(X\),

\(\mathcal{Y}\) for \(Y\), and so on.

Realizations are observed according to probabilities specified by the

*distribution*of \(X\)Can think of \(X\) as an "infinite supply of data"

Separate realizations of the same r.v. \(X\) are "independent and identically distributed" (i.i.d.)

Formal definition of a random variable requires measure theory, not covered here

Random variable \(X\), realization \(x\).

- What is the probability we see \(x\)?
- \(\Pr(X=x)\), (if lazy, \(\Pr(x)\), but don't do this)

- Subsets of the domain of a random variable correspond to events.
- \(\Pr(X > 0)\) probability that I see a realization that is positive.

- Discrete random variables take values from a countable set
- Coin flip \(X\)
- \(\mathcal{X} = \{0,1\}\)

- Number of snowflakes that fall in a day \(Y\)
- \(\mathcal{Y} = \{0, 1, 2, ...\}\)

- Coin flip \(X\)

- For a discrete \(X\), \(p_{X}(x)\) gives \(\Pr(X = x)\).
- Requirement: \(\sum_{x \in \mathcal{X}} p_{X}(x) = 1\).
- Note that the sum can have an infinite number of terms.

\(X\) is number of "heads" in 20 flips of a fair coin

\(\mathcal{X} = \{0,1,...,20\}\)

- For a discrete \(X\), \(P_{X}(x)\) gives \(\Pr(X \le x)\).
- Requirements:
- \(P\) is nondecreasing
- \(\sup_{x \in \mathcal{X}} P_{X}(x) = 1\)

- Note:
- \(P_X(b) = \sum_{x \le b} p_X(x)\)
- \(\Pr(a < X \le b) = P_X(b) - P_X(a)\)

\(X\) is number of "heads" in 20 flips of a fair coin

- Continuous random variables take values in intervals of \({\mathbb{R}}\)
- Mass \(M\) of a star
- \(\mathcal{M} = (0,\infty)\)

- Oxygen saturation \(S\) of blood
- \(\mathcal{S} = [0,1]\)

- For a continuous r.v. \(X\), \(\Pr(X = x) = 0\) for all \(x\).
*There is no probability mass function.* - However, \(\Pr(X \in (a,b)) \ne 0\) in general.

- For continuous \(X\), \(\Pr(X = x) = 0\) and PMF does not exist.
- However, we define the
*Probability Density Function*\(f_X\):- \(\Pr(a \le X \le b) = \int_{a}^{b} f_X(x) {\mathrm{\,d}}x\)

- Requirement:
- \(\forall x \;f_X(x) > 0\), \(\int_{-\infty}^\infty f_X(x) {\mathrm{\,d}}x = 1\)