In this lab we’ll investigate the probability distribution that is most central to statistics: the normal distribution. If we are confident that our data are nearly normal, that opens the door to many powerful statistical methods. Here we’ll use the graphical tools of R to assess the normality of our data and also learn how to generate random numbers from a normal distribution.
This week we’ll be working with measurements of body dimensions. This data set contains measurements from 247 men and 260 women, most of whom were considered healthy young adults.
download.file("http://www.openintro.org/stat/data/bdims.RData", destfile = "bdims.RData")
load("bdims.RData")Be sure to put these commands in a code chunk at the top of your R notebook.
Let’s take a quick peek at the first few rows of the data.
head(bdims)You’ll see that for every observation we have 25 measurements, many
of which are either diameters or girths. A key to the variable names can
be found at http://www.openintro.org/stat/data/bdims.php,
but we’ll be focusing on just three columns to get started: weight in kg
(wgt), height in cm (hgt), and
sex (1 indicates male, 0
indicates female).
Since males and females tend to have different body dimensions, it will be useful to create two additional data sets: one with only men and another with only women.
mdims <- subset(bdims, sex == 1)
fdims <- subset(bdims, sex == 0)Now do Exercise 1.
In your description of the distributions, did you use words like bell-shaped or normal? It’s tempting to say so when faced with a unimodal symmetric distribution.
To see how accurate that description is, we can plot a normal distribution curve on top of a histogram to see how closely the data follow a normal distribution. This normal curve should have the same mean and standard deviation as the data. We’ll be working with women’s heights, so let’s store them as a separate object and then calculate some statistics that will be referenced later.
fhgtmean <- mean(fdims$hgt)
fhgtsd <- sd(fdims$hgt)Next we make a density histogram to use as the backdrop and use the
lines function to overlay a normal probability curve. The
difference between a frequency histogram and a density histogram is that
while in a frequency histogram the heights of the bars add up
to the total number of observations, in a density histogram the
areas of the bars add up to 1. The area of each bar can be
calculated as simply the height times the width of the bar.
Using a density histogram allows us to properly overlay a normal
distribution curve over the histogram since the curve is a normal
probability density function. Frequency and density histograms both
display the same exact shape; they only differ in their y-axis. You can
verify this by comparing the frequency histogram you constructed earlier
and the density histogram created by the commands below.
hist(fdims$hgt, probability = TRUE)
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y, col = "blue")After plotting the density histogram with the first command, we
create the x- and y-coordinates for the normal curve. We chose the
x range as 140 to 190 in order to span the entire range of
height. To create y, we use dnorm
to calculate the density of each of those x-values in a distribution
that is normal with mean fhgtmean and standard deviation
fhgtsd. The final command draws a curve on the existing
plot (the density histogram) by connecting each of the points specified
by x and y. The argument col
simply sets the color for the line to be drawn. If we left it out, the
line would be drawn in black.
The top of the curve is cut off because the limits of the x- and
y-axes are set to best fit the histogram. To adjust the y-axis you can
add a third argument to the histogram function:
ylim = c(0, 0.06).
Include a code chunk that creates the density histrogram (with the updated y-axis limits) and overlays the normal curve. Then complete the following exercise.
Now do Exercise 2.
Eyeballing the shape of the histogram is one way to determine if the data appear to be nearly normally distributed, but it can be frustrating to decide just how close the histogram is to the curve. An alternative approach involves constructing a normal probability plot, also called a normal Q-Q plot for “quantile-quantile”.
qqnorm(fdims$hgt)
qqline(fdims$hgt)A data set that is nearly normal will result in a probability plot where the points closely follow the line. Any deviations from normality leads to deviations of these points from the line. The plot for female heights shows points that tend to follow the line but with some errant points towards the tails. We’re left with the same problem that we encountered with the histogram above: how close is close enough?
A useful way to address this question is to rephrase it as: what do
probability plots look like for data that I know came from a
normal distribution? We can answer this by simulating data from a normal
distribution using rnorm.
sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)The first argument indicates how many numbers you’d like to generate,
which we specify to be the same number of heights in the
fdims data set using the length function. The
last two arguments determine the mean and standard deviation of the
normal distribution from which the simulated sample will be generated.
We can take a look at the shape of our simulated data set,
sim_norm, as well as its normal probability plot.
Now do Exercise 3.
Even better than comparing the original plot to a single plot generated from a normal distribution is to compare it to many plots. The following function creates multiple simulated data sets sampled from a normal distribution with the mean and standard deviation equal to the mean and standard deviation of the true female height data. It gives us an idea of the level of variability we might see in normal probability plots even when we can guarantee that the data come from a normal distribution. It may be helpful to click the zoom button in the plot window.
qqnormsim(fdims$hgt)Now do Exercise 4.
Okay, so now you have a slew of tools to judge whether or not a variable is normally distributed. Why should we care?
It turns out that statisticians know a lot about the normal distribution. Once we decide that a random variable is approximately normal, we can answer all sorts of questions about that variable related to probability. Take, for example, the question of, “What is the probability that a randomly chosen young adult female is taller than 6 feet (about 182 cm)?” (The study that published this data set is clear to point out that the sample was not random and therefore inference to a general population is not suggested. We do so here only as an exercise.)
If we assume that female heights are normally distributed (a very
close approximation is also okay), we can find this probability by
calculating a Z score and consulting a Z table (also called a normal
probability table). In R, this is done in one step with the function
pnorm.
1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)Note that the function pnorm gives the area under the
normal curve (with a given mean and standard deviation) and to the left
of the specific value q.
Since we’re interested in the probability that someone is taller than
182 cm, we actually want the area under the curve and to the right of
q. We have to take one minus the probability given by
pnorm.
Assuming a normal distribution has allowed us to calculate a theoretical probability. If we want to calculate the probability empirically, we simply need to determine how many observations fall above 182 then divide this number by the total sample size.
sum(fdims$hgt > 182) / length(fdims$hgt)Although the probabilities are not exactly the same, they are reasonably close. The closer that your distribution is to being normal, the more accurate the theoretical probabilities will be.
Now do Exercise 5.
So far, we have been looking at the female data. To help cement the ideas we have developed in this lab, let’s repeat some of the steps for the male height data. Complete the following exercises. Include code chunks to produce all of the requested plots and complete the requested analyses.
Now do Exercise 6.
Now do Exercise 7.
This lab is a modified is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by Mark Hansen of UCLA Statistics.