18.4.1 A new plotting tool

The function curve() draws a line on a plot, starting on the x-asis at the from = value and going to the to = value. It calculates y values from the function we give it. The function creates a large number of x values along the range we give it, then calculates a y value for each and connects them as a line. For example, we can plot the line y = x * 2:

# Plot simple function
curve(x*2,
      from = 0,
      to = 10)

We can build more complex expressions as well. Here, we can draw a parabola

# Plot parabola
curve(x^2,
      from = -10,
      to = 10)

or even a complicated multinomial function:

# Plot complicated line
curve(2*x^3 + 3*x^2 - 2,
      from = -10,
      to = 10)

This is great for plotting lines, but it doesn’t help us plot a normal curve.

18.4.2 Calculating densities

What we need is a function to produce the density of a normal distribution. In R, this is dnorm() which returns the density of a normal distribution, at the points given. For example.

# Show the denisty of a normal curve
dnorm(-2:2)

## [1] 0.05399097 0.24197072 0.39894228 0.24197072 0.05399097

But, we have said a few times that the normal curve is specified with a mean and a standard deviation. dnorm() (and other functions like it) uses default values of mean = 0 and sd = 1, which are often referred to as the “standard normal” distribution (just like the z-scores we calculated earlier). However, we can set a different mean to get the density of any normal curve.

# show density of a different curve
dnorm(8:12,mean = 10)

## [1] 0.05399097 0.24197072 0.39894228 0.24197072 0.05399097

# show density of a different curve
dnorm(8:12,mean = 10, sd = 2)

## [1] 0.1209854 0.1760327 0.1994711 0.1760327 0.1209854

We can, in this way, generate the density of any normal distribution that we want. Putting it together curve(), we can then plot that distribution.

# Plot a standard normal distribution
curve(dnorm(x),
      from = -3,
      to = 3)

By setting a different mean = and sd =, we can draw a different normal model.

# Plot a different normal distribution
curve(dnorm(x, mean = 10, sd = 2),
      from = 4,
      to   = 16)

What changed between these two plot? Notice the from = and to = arguments above. Think carefully about what that difference means.

The function curve also accepts a number of other arguments, including the ylab =, col = and lwd = options we’ve used extensively before.

# Plot a different normal distribution
curve(dnorm(x, mean = 10, sd = 2),
      from = 4,
      to   = 16,
      col = "purple", lwd=3,
      ylab = "Density")

18.4.3 Drawing with real data

First, let’s generate something with a nice, unimodal, symmetric, shape. Copy in the following to generate a bootstrap distribution of the speed of roller coasters (we also did this in an earlier lesson).

# Coaster bootstrapping

# Initialize vector
coasterBoot <- numeric()

# Loop through many times
for(i in 1:12489){
  # calculate the correlation
  # note, we could save these each as a vector first
  # but we don't need to 
  coasterBoot[i] <- mean(sample(rollerCoaster$Speed,
                                replace = TRUE)
                         )
}

# Plot the result
hist(coasterBoot,
     freq = FALSE,
     main = "Bootstrap of Coaster Speed",
     xlab = "Mean Speed (mph) in sample")

Now, let’s plot the normal curve of our coaster bootstrap. All we need to do is pass in our mean and sd, and the function does the rest. Because we are copying the bootstrap distribution, we just use the mean and standard deviation from our samples. Note that I set the from = and to = to match the range of the data. This range ensures that we can see the relavant part of the curve.

# Plot the coaster bootstrap distribution
curve(dnorm(x,
            mean = mean(coasterBoot),
            sd   =   sd(coasterBoot)),
      from = min(coasterBoot),
      to   = max(coasterBoot),
      col = "purple",
      lwd = 3,
      ylab = "Density",
      xlab = "Sample mean speed (mph)")

This gets us a good part of the way there, and we could use this as is. However, we can do one better, and why shouldn’t we? Let’s lay the curve on top of our histogram.

Almost all of the plotting functions in R include an option to add the plot to the existing plot, instead of overwriting it. You can use this by setting add = TRUE in most functions, including in curve.

A few notes are necessary here. First, we need to set freq = FALSE in the histogram to make sure the density is plotted. If you don’t, curve() will draw a line that looks very much like 0 (because the densities are so much smaller than counts). In addition, when using add = TRUE, it isn’t necessary to set the from = and to = options. Instead, curve() automatically matches the range of the plot.

# Plot the result
hist(coasterBoot,
     freq = FALSE,
     main = "Bootstrap of Coaster Speed",
     xlab = "Mean Speed (mph) in sample")

# Plot the coaster bootstrap distribution
# Add a curve to the plot
curve(dnorm(x,
            mean = mean(coasterBoot),
            sd   =   sd(coasterBoot)),
      col = "purple", lwd=3,
      add = TRUE)

As before, we see that the curve is a very close fit to our observed data. Sometimes if the curve looks like a bad fit, it can be caused by having a poorly chosen number of bins in your histogram. Recall that we can change the number of bins by setting the breaks = argument to histogram. Thus, this plot:

# Plot the result
hist(coasterBoot,
     breaks = 60,
     freq = FALSE,
     main = "Bootstrap of Coaster Speed",
     xlab = "Mean Speed (mph) in sample")

# Plot the coaster bootstrap distribution
# Add a curve to the plot
curve(dnorm(x,
            mean = mean(coasterBoot),
            sd   =   sd(coasterBoot)),
      col = "purple", lwd=3,
      add = TRUE)

… looks like an even better fit to the normal model, despite being from the same data.

Now that we can see the distribution, we will start using these together for a few additional approaches to statistical testing.