Chapter 6 moves on from discrete distributions to the normal distribution,

#a continuous distribution that will be the basis of most statistics in the book.

### Section 6.1 Frequency Distributions of Continuous Variables

A brief discussion of the section header's topic, nothing to code here.### Section 6.2 Properties of the Normal Distribution

#Let's check where it is.

?Distributions

?dnorm

#We can draw the normal distribution in a custom function as a plot using curve()

#and equation 6.1 on page 95.

normal.manual<-function(mean, sd){

curve((1/(sd*sqrt(2*pi)))*exp((-((x-mean)/sd)^2)/2),

-4, 4, #go from -4 to +4 standard deviations.

add = FALSE,

ylab="freq",

xlab="Y",

type = "l")

}

normal.manual(0,1)

#You can get this automatically in R with dnorm.

curve(dnorm(x, mean=0, sd=1),

-4, 4,

col="white",

lty="dashed",

add=TRUE)

#I added a white dashed line over the original black line so you can see that they follow the same path.

#You can draw any of the distributions in Figure 6.2 using these functions

#(either the R one or the one we just made).

#Figure 6.3 shows the cumulative normal distribution function.

#Let's do that and then plot the normal probability density function on it.

#Cumulative first since that plot will go from 0 to 1.

curve(pnorm(x, mean=0, sd=1),

-4, 4,

lty="solid")

#Then add the normal probability density function (like the one we made earlier, just with the default line color).

curve(dnorm(x, mean=0, sd=1),

-4, 4,

lty="dashed",

add=TRUE)

#To get out some of the values shown in page 96, we use pnorm().

#http://stackoverflow.com/questions/34236648/r-function-to-calculate-area-under-the-normal-curve-between-adjacent-standard-de

pnorm(0, mean=0, sd=1)

#Just adding one value gives you a point estimate.

#50% will be found up to zero (the mean).

#The other values shown in Figure 6.3

pnorm(-1)

pnorm(-2)

#0 and 1 are the default values for mean and standard deviation, so you can leave them out if you want in this case.

#If you add more than one, it will tell you the values.

(pnorm(0:1))

#Use diff as suggested here to get the area under the curve.

diff(pnorm(0:1))

#This works because it uses the cumulative function to calculate it.

#If you go from -1 to 1 you can add them up to get the 68.27% shown under Figure 6.2 and on page 96.

sum(diff(pnorm(-1:1)))

#To go the other way, i.e. to see where 50% of the items fall, you have to use qnorm.

#Let's see what qnorm looks like.

curve(qnorm(x))

#It is essentially tipping Figure 6.3 on its side.

#The frequency is now x on the x-axis, and the standard deviations are the y axis.

#You can get the point values in Figure 6.3 this way:

qnorm(0.5)

qnorm(0.1587)

qnorm(0.0228)

#The last two are close with rounding since those values in Figure 6.3 were rounded.

#How to get the values on page 96, of where 50%, 95%, etc of items are found?

qnorm(0.5+0.5*0.5)

qnorm(0.95+0.5*0.05)

qnorm(0.99+0.5*0.01)

#The values that you input are not the percent values given on page 96.

#You put them in proportions (the quantile you want to get) PLUS

#half of the remaining value.

#If you put in 0.95, it will go from 0 to 0.95. That isn't centered at the mean of 0,

#where the 50% quantile is. There is 0.05 left at the end. So we divide it by two

#to put it on either end.

#You would get the negative number for the standard deviation if you put in the opposite value.

qnorm(0.5*0.01)

#This also explains it nicely down where it starts talking about qnorm.

#https://cran.r-project.org/web/packages/tigerstats/vignettes/qnorm.html

#On pg. 98, they show calculation of standard deviates.

#This is described in such a way that they seem to be z scores although they are never named as such in the book.

#You can easily calculate this manually.

z.score<-function(x){

(x-mean(x))/sd(x)

}

x<-c(1,2,3)

z.score(x)

#You can also use the scale() function.

#https://www.r-bloggers.com/r-tutorial-series-centering-variables-and-generating-z-scores-with-the-scale-function/

scale(x,

center=TRUE,

scale=TRUE)

#help notes that scale=TRUE divides by the standard deviation when

#center=TRUE, and center=TRUE substracts each number by the column mean.

#This is the same thing we did in the z-score function.

#Table 6.1 has expected frequencies for the normal distribution in column 2 for a sample of 1000 individuals.

#We can generate this with pnorm() and thinking about what the class marks mean.

#Because the class marks are separated by 0.5, we need to go 0.5 around each class mark and start at -5.25.

boundaries<-seq(from=-5.25,

to=5.25,

by=0.5)

pnorm.results<-pnorm(boundaries)

expected.freqs<-abs(diff(pnorm.results))

cbind(boundaries[-length(boundaries)]+0.25,

#this takes the last entry off because we only need the lower bound for each class,

#and adds 0.25 to get the class mark.

round(expected.freqs*1000, 1))

#Then you can have a look at the table and see that it gives the same results as column 2 in Table 6.1.