2017-09-19

Statistics

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]

Why statistics?

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

Data

## 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 ...

Old Faithfull-pdPhoto by Jon Sullivan

Data summaries

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 quantile for a proportion \(p\) is a value \(x\) for which \(p\) of the data are less than or equal to \(x\). The first quartile, median, and third quartile are the quantiles for \(p=0.25\), \(p=0.5\), and \(p=0.75\), respectively.

Visual Summary 1: Box Plot

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

Visual Summary 1.5: Box Plot, Jitter Plot

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)

Visual Summary 2: Histogram

## 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')

Visual Summary 2.5: Histogram

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

Replicates

  • 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.

Randomness

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 Spaces and Events

  • 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: \([1,2] \times [50,70]\).

Event Probabilities

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

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

Interpreting probability:
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.)

Abstraction of data: Random Variable

  • 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.

Distributions of random variables

  • 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

Probabilities for random variables

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

  • Discrete random variables take values from a countable set
    • Coin flip \(X\)
      • \(\mathcal{X} = \{0,1\}\)
    • Number of snowflakes that fall in a day \(Y\)
      • \(\mathcal{Y} = \{0, 1, 2, ...\}\)

Probability Mass Function (PMF)

  • 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.

Probability Mass Function (PMF) Example

\(X\) is number of "heads" in 20 flips of a fair coin
\(\mathcal{X} = \{0,1,...,20\}\)

Cumulative Distribution Function (CDF)

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

Cumulative Distribution Function (CDF) Example

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

Continuous random variables

  • 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.

Probability Density Function (PDF)

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

Probability Density Function (PDF) Example