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.

The Data

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. Let’s take a quick peek at the first few rows of the data.

library(dplyr)
library(ggplot2)
library(oilabs)
data(bdims)
head(bdims)
##   bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi
## 1   42.9   26.0   31.5   17.7   28.0   13.1   10.4   18.8   14.1  106.2
## 2   43.7   28.5   33.5   16.9   30.8   14.0   11.8   20.6   15.1  110.5
## 3   40.1   28.2   33.3   20.9   31.7   13.9   10.9   19.7   14.1  115.1
## 4   44.3   29.9   34.0   18.4   28.2   13.9   11.2   20.9   15.0  104.5
## 5   42.5   29.9   34.0   21.5   29.4   15.2   11.6   20.7   14.9  107.5
## 6   43.3   27.0   31.5   19.6   31.3   14.0   11.5   18.8   13.9  119.8
##   che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi
## 1   89.5   71.5   74.5   93.5   51.5   32.5   26.0   34.5   36.5   23.5
## 2   97.0   79.0   86.5   94.8   51.5   34.4   28.0   36.5   37.5   24.5
## 3   97.5   83.2   82.9   95.0   57.3   33.4   28.8   37.0   37.3   21.9
## 4   97.0   77.8   78.8   94.0   53.0   31.0   26.2   37.0   34.8   23.0
## 5   97.5   80.0   82.5   98.5   55.4   32.0   28.4   37.7   38.6   24.4
## 6   99.9   82.5   80.1   95.3   57.5   33.0   28.0   36.6   36.1   23.5
##   wri.gi age  wgt   hgt sex
## 1   16.5  21 65.6 174.0   m
## 2   17.0  23 71.8 175.3   m
## 3   16.9  28 80.7 193.5   m
## 4   16.6  23 72.6 186.5   m
## 5   18.0  22 78.8 187.2   m
## 6   16.9  21 74.8 181.5   m

You’ll see that for every observation we have 25 measurements, many of which are either diameters or girths. You can learn about what the variable names mean by bringing up the help page.

?bdims

We’ll be focusing on just three columns to get started: weight in kg (wgt), height in cm (hgt), and sex (m indicates male, f 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 <- bdims %>%
  filter(sex == "m")
fdims <- bdims %>%
  filter(sex == "f")
  1. Make a histogram of men’s heights and a histogram of women’s heights. How would you compare the various aspects of the two distributions?
qplot(x = hgt, data = mdims, geom = "histogram", main = "Distribution of male heights")

qplot(x = hgt, data = fdims, geom = "histogram", main = "Distribution of female heights")

The shape of the two distributions are quite similar: both male and female heights show a symmetric and unimodal distribution (the multiple modes here are best thought of as a result of sampling variability and a small binwidth). The spread of the two distributions is also similar, with most of the observations falling within an interval spanning ~25 cm. They differ most notably in their centers, with a mean/median/mode of ~178 cm and ~165 cm for men and women, respectively.

The normal distribution

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.

qplot(x = hgt, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..)) +
  stat_function(fun = dnorm, args = c(mean = fhgtmean, sd = fhgtsd), col = "tomato")

After initializing a blank plot with the first command, the ggplot2 package allows us to add additional layers. The first layer is a density histogram. The second layer is a statistical function - the density of the normal curve, dnorm. We specify that we want the curve to have the same mean and standard deviation as the column of female heights. 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. Which is no fun.

  1. Based on the this plot, does it appear that the data follow a nearly normal distribution?

It’s difficult to tell. The histogram is vaguely bell-shaped, but it a higher concentration than the normal in the middle of the distribution. This is likely due to sampling variability, so the normal curve seems like a reasonable approximation.

Evaluating the normal distribution

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”.

qplot(sample = hgt, data = fdims, stat = "qq")

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 nrow() 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.

  1. Make a normal probability plot of sim_norm. Do all of the points fall on the line? How does this plot compare to the probability plot for the real data?
qplot(sample = sim_norm, stat = "qq")

The points don’t fall exactly on a line, but they’re quite close. The largest deviations come in the tail of the distribution. This simulated plot is more smoothly linear than the data, but that is likely due to a discretization in the data.

Even better than comparing the original plot to a single plot generated from a normal distribution is to compare it to many more plots using the following function. It may be helpful to click the zoom button in the plot window.

qqnormsim(sample = hgt, data = fdims)

  1. Does the normal probability plot for female heights look similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal?

The qq plot for female heights is strikingly similar to that from simulated normal data sets. In fact, several of the simulated plots show a greated deviation from linearity in the tails that does the original data. Again, the main difference is that the data has a stairstep shape.

  1. Using the same technique, determine whether or not female weights appear to come from a normal distribution.
qqnormsim(sample = wgt, data = fdims)

The normal approximation appears to be less appropriate for wgt than for hgt. This data shows some curvature in the shape of the qqplot that suggests a longer right tail that we’d expect from nearly normal data and also shows two notable outliers.

Normal probabilities

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)
## [1] 0.004434387

Note that the function pnorm() gives the area under the normal curve below a given value, q, with a given mean and standard deviation. Since we’re interested in the probability that someone is taller than 182 cm, we have to take one minus that probability.

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)
## [1] 0.003846154

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.

  1. Write out two probability questions that you would like to answer; one regarding female heights and one regarding female weights. Calculate the those probabilities using both the theoretical normal distribution as well as the empirical distribution (four probabilities in all). Which variable, height or weight, had a closer agreement between the two methods?

Answers will vary here. The important thing is to use pnorm to find the area under the curve to the Left of a value and use sum(data meeting condition) / length(data) to find the empirical probabilities.

On Your Own

This is likely due to the discrete scale on which the data was measured. When people report there age, they usually only provide integer values, not ages like 28.3746 years. This is what creates the step patter in the variable on the y-axis of the qqplot. The x-axis refers to the percentiles of the normal distribution, which is continuous, so the plots are continuous in their x-values.

qplot(sample = kne.di, data = fdims, stat = "qq")

It is very clear that the normal curve would be a poor approximation to female knee diameter. The qq plot shows strong deviations from linearity in the right tail, suggesting that it’s longer (right-skewed) sthan we’d expect under the normal distribution. We can verify this by looking at the histogram.

qplot(x = kne.di, data = fdims, geom = "histogram")

histQQmatchgg

This 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.