Z- and normal distributions

The usual setup

#install.packages("tidyverse") # you don't need to run this line if you have already installed tidyverse before
library(tidyverse) # run this line to be able to use all the functions we need.

The pnorm(), qnorm(), and rnorm() functions

Which values do pnorm, qnorm, and rnorm return? How do I remember the difference between these?

I find it helpful to have visual representations of distributions as pictures. It is difficult for me to think of distributions, or differences between probability, density, and quantiles without visualizing the shape of the distribution. So I figured it would be helpful to have a visual guide to pnorm, qnorm, and rnorm. (We will not talk about the dnorm() function as not strictly necessary for the course, but you should aware that it exists!)

# Prep the distribution data
x <- seq(-3, 3, 0.005)
y <- dnorm(x)
normal <- as.data.frame(cbind(x,y))

rnorm()

Let’s start with the easiest of the functions – rnorm(). This function randomly samples from a normal distribution. It takes three arguments:

  1. n: the sample size (i.e., how many observations values you want to draw from the distribution)
  2. mean: the mean you want your sample to have.
  3. sd: the standard deviation you want your sample to have.

So, with this function we can sample n observations from a normal distribution having the given mean and sd, as defined in the function itself. The default values for the mean and sd arguments are: mean=0, sd=1, which means that R will sample n observations from the standard normal distribution (i.e., the z-distribution). So for example:

set.seed(42) # ignore this command for now!

# below, the mean and sd values may not be specified since they are the default values
sample <- rnorm(n=5, mean=0, sd=1) 
sample <- round(sample, 2)

If we want we can try to draw from another distribution, with a non-standard distribution (i.e., with mean different from 0 and sd different from 1).

set.seed(42)
sample2 <- rnorm(5, mean=5, sd=0.5)
sample2 <- round(sample2, 2)

In the following examples we will keep using the standard normal distribution (i.e., a normal distribution with a mean of 0 and sd of 1) just for simplicity, but keep in mind that any of these functions may work with any normal distribution with a given mean and standard deviation.

pnorm()

The pnorm() function gives the probability function for a normal distribution. Basically, for any given real value x of a sample from a distribution with a given mean and sd, it will give you the area under the curve until x. Given that the area under the curve is always equal to 1, the function will spit out the cumulative probability of x. The pnorm() function takes three arguments:

  1. q: the quantile, or the value you want to know the cumulative probability of
  2. mean: the mean you want your sample to have.
  3. sd: the standard deviation you want your sample to have.
  4. lower.tail: it specifies whether you want the leftward (TRUE) or rightward (FALSE) side of the tail

The default values for the arguments mean and sd are 0 and 1, respectively (thus drawing the probability distribution from a standard normal distribution).

For example:

# below, the mean and sd numerical values, and the lower.tail logical value may not be specified since they are the default values
pnorm(q=0, mean=0, sd=1, lower.tail=T)

The function above, gives you 0.5 because that the cumulative probability of the value 0 from the standard normal distribution (i.e., with mean=0 and sd=1).

What if I want the leftward side of the distribution? I’ll just need to change the argument lower.tail and set it to FALSE. Alternatively, I could just calculate the complement of the default pnorm() command (i.e, 1-pnorm() with keeping lower.tail to TRUE as per default):

# below, lower.tail=F will give you the area under the curve on the *left* side of the value of the argument q
pnorm(q=0, mean=0, sd=1, lower.tail=F) 

#alternatively, you can just use the formula 1-pnorm()
1-pnorm(q=0, mean=0, sd=1, lower.tail=T)

The two functions above give you the same value as pnorm(0, mean=0, sd=1, lower.tail=T) because we chose the mean of our distribution as our value q. So the cumulative probability for it will be the same (i.e., 0.5) regardless of the side of the tail you are selecting.

However, if the quantile value you want to know the cumulative probability of is different from the mean of the normal distribution at hand, the probability pit out by pnorm() will be different depending on which tail you want to look at. For example, let’s get the cumulative probability of q=1. If I want the probability of 1 of the lower/leftward tail, I say this:

pnorm(q=1, mean=0, sd=1, lower.tail=T)

If I want the upper/rightward tail, I will have to choose either of the following options:

pnorm(q=1, mean=0, sd=1, lower.tail=F)

1 - pnorm(q=1, mean=0, sd=1, lower.tail=T)

Which will spit out the exact same value, i.e., the area under the curve on the right from the value defined as q.

Choosing the correct tail is crucial to get the correct p-value for your one-tailed test statistic.

qnorm

The qnorm() function is the reverse function of the pnorm() function. It gives you the quantile value of a normal distribution with a given mean and sd, for a given cumulative probability p. The qnorm() function takes three arguments:

  1. p: the cumulative probability you want to know the quantile of
  2. mean: the mean you want your sample to have.
  3. sd: the standard deviation you want your sample to have.
  4. lower.tail: it specifies whether you want the leftward (TRUE) or rightward (FALSE) side of the tail

As per usual, the default values for the arguments mean and sd are 0 and 1, respectively (thus drawing the probability distribution from a standard normal distribution). So, if I want the quantile value corresponding to the cumulative probabilty of 0.5, I will have to say:

qnorm(p=0.5, mean=0, sd=1, lower.tail=T)

Here, the same thing we talked about above regarding the lower.tail argument applies here too. lower.tail=TRUE will give you the quantile corresponding to the lower/leftward tail; lower.tail=FALSE will give you the quantile corresponding to the upper/rightward tail.

Other distributions

While the plots above are specific to the standard normal distribution, the same concepts apply to distribution. In this class in particular, we will be dealing with two other distributions:

  1. the t-distribution: rt(), pt(), qt(), dt()
  2. the F-distribution: rf(), pf(), qf(), df()

But there are many more: the beta distribution, the binomial distribution, the gamma distribution, the logistic distribution, the Poisson distribution. While you are not asked to remember these names, it will be useful to remember that, no matter what kind of distribution you will end up dealing with in the case, in R they all have the same functions (called the pqrd family)!