A curve has been found representing the frequency distribution of standard deviations of samples drawn from a normal population.

-Gosset. 1908, Biometrika 6:25.

• Use of the histogram
• Gaussian ain’t normal
• Poisson
• Binomial
• Diagnosing the distribution
• Practice exercises

# Let's simulate some fake weight data for 10,000 cats
set.seed(42)
cats <- rnorm(n = 10000, mean = 4, sd = 0.5)

• Bars are counts of observations
• ‘Bins’ non-overlapping
• Shape is diagnostic

Things you can measure with continuous precision

• The Gaussian is sometimes referred to as the ‘normal’ distribution
• Implies it is typical
• The Gaussian ain’t necessarily typical!
• Described by mean and std. dev. (the Gaussian parameters)

Mean

# Data
myvar <- c(1,4,8,3,5,3,8,4,5,6)

# Mean the "hard" way
(myvar.mean <- sum(myvar)/length(myvar))

## [1] 4.7

# Mean the easy way
mean(myvar)

## [1] 4.7

Standard Deviation

# (NB this is the sample variance with [n-1])
(sum((myvar-myvar.mean)^2 / (length(myvar)-1)))

## [1] 4.9

# Variance the easy way 
var(myvar)

## [1] 4.9

# Std dev the easy way
sqrt(var(myvar))

## [1] 2.213594

Counts of rare events (like deaths from being kicked by a horse in the Prussian army…)

• Usually low mean value
• Described by a single parameter \(\lambda\)
• \(\lambda\) is both the mean and std. dev

set.seed(42)
mypois <- rpois(n = 100, lambda = 3)
hist(mypois,
     main = "Ewes with triplets",
     xlab = "Count of Triplets")

Counts of events with exactly two outcomes, one of which might be a “success” (like ‘deaths from being kicked by a horse in the Prussian army…’heads’ or ‘tails’, live or die, disease or healthy, etc.)

• Sometimes we are interested in the probability of success
• Described by 2 parameters, p{success}, and the number of trials

A very common task faced when handling data is “diagnosing the distribution”. Just like a human doctor diagnosing an ailment, you examine the evidence, consider the alternatives, judge the context, and take a guess.

• Expectation based on the type of data
• Graph the data and look
• compare expected theoretical dist. with several known dist’s
• try transformation (e.g. to ‘coerce’ to Gaussian)