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.

load("more/bdims.RData")

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

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   1
## 2   17.0  23 71.8 175.3   1
## 3   16.9  28 80.7 193.5   1
## 4   16.6  23 72.6 186.5   1
## 5   18.0  22 78.8 187.2   1
## 6   16.9  21 74.8 181.5   1

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)

  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?

    Unimodal and symmetric Guassian curve.

hist(mdims$hgt, main = "Histogram of Male Height" , xlab = "Male Height (cm)")

hist(fdims$hgt, main = "Histogram of Female Height" ,xlab = "Female Height (cm)")

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.

#I was on my students' cases for years to always make sure axes were labeled with units. I had to fix that.
hist(fdims$hgt, probability = TRUE, main = "Histogram of Female Height" ,xlab = "Female Height (cm)")
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 fheight. 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).

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

The bins of the histogram raise and fall with the line closely. Yes, the distribution appears Normal.

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”.

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.

  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?
hist(sim_norm, probability = TRUE ,main = "Histogram of Simulated Female Height Data", xlab = "Female Height (cm)")

hist(fdims$hgt, probability = TRUE, main = "Histogram of Female Height" ,xlab = "Female Height (cm)")

Since the data are generated randomly you’d expect some variations in the two histograms, but peak and spread should be similar. In this case they are, further supporting our claim that height is Normally Distributed.

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(fdims$hgt)

  1. Does the normal probability plot for fdims$hgt look similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal?
qqnorm(fdims$hgt)
qqline(fdims$hgt)

These plots all look very similar. There are some variations along the y-axis, however non are so far off as to raise suspicion that the distributions are not normal.

  1. Using the same technique, determine whether or not female weights appear to come from a normal distribution.
fwgtmean <- mean(fdims$wgt)
fwgtsd   <- sd(fdims$wgt)
hist(fdims$wgt, probability = TRUE , main = "Histogram of Female Weight", xlab = "Female Weight (kg)")
x <- 40:90
y <- dnorm(x = x, mean = fwgtmean, sd = fwgtsd)
lines(x = x, y = y, col = "red")

sim_wgt <- rnorm(n = length(fdims$wgt), mean = fwgtmean, sd = fwgtsd)
hist(sim_wgt, probability = TRUE , main = "Histogram of Simulated Female Weight", xlab = "Female Weight (kg)")

qqnorm(fdims$wgt)
qqline(fdims$wgt)

qqnormsim(fdims$wgt)

It appears that we have evidence of the Obsity Epidemic, as the distribution for weight is right skewed with some outliers in the 100-110 kg range. Note that the Simulated data does not go far beyond 90 kg and the QQ plot for the real data diverges from the line.

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?

What is the probabilty a female is bewteen 140 cm and 150 cm?

pnorm(q = 150, mean = fhgtmean, sd = fhgtsd) - pnorm(q = 140, mean = fhgtmean, sd = fhgtsd)
## [1] 0.01145733
(sum(fdims$hgt < 150) - sum(fdims$hgt < 140)) / length(fdims$hgt)
## [1] 0.01153846

Very close agreement between the two methods because the Normal Distribution is a good fit for height.

What is the probablity that a female is less than 50 kg or greater than 70 kg?

(1- pnorm(q = 70, mean = fwgtmean, sd = fwgtsd)) + pnorm(q = 50, mean = fwgtmean, sd = fwgtsd)
## [1] 0.2992969
(sum(fdims$wgt > 70) + sum(fdims$wgt < 50)) / length(fdims$wgt)
## [1] 0.2615385

We can see that the weight question varies more from the Normal Distribution due to the data being right skewed.

On Your Own

a. The histogram for female biiliac (pelvic) diameter (“bii.di”) belongs to normal probability plot letter \(\textbf{B}\).

hist(fdims$bii.di,xlab = "Female Pelvic diameter in cm", main = "Histogram of Female Pelvic Diameter")

qqnorm(fdims$bii.di)
qqline(fdims$bii.di)

b. The histogram for female elbow diameter (elb.di) belongs to normal probability plot letter \(\textbf{C}\).

hist(bdims$elb.di,xlab = "Female elbow diameter in cm", main = "Histogram of Female Elbow Diameter")

qqnorm(fdims$elb.di)
qqline(fdims$elb.di)

c. The histogram for general age (age) belongs to normal probability plot letter \(\textbf{D}\).

hist(bdims$age, xlab = "Age in years", main = "Histogram of Sample Age in Years")

qqnorm(bdims$age)
qqline(bdims$age)

d. The histogram for female chest depth (che.de) belongs to normal probability plot letter \(\textbf{A}\).

hist(fdims$che.de,xlab = "Female chest diameter in cm", main = "Histogram of Female Chest Diameter")

qqnorm(fdims$che.de)
qqline(fdims$che.de)

qqnorm(fdims$kne.di)
qqline(fdims$kne.di)

This plot diverges on the right side like age and chest diameter, both of which are right skewed. Therefore, Knee diameter is also right skewed.

hist(fdims$kne.di, xlab = "Female knee diameter in cm", main = "Histogram of Female Knee Diameter")

Female Knee Diameter is right skewed as the QQ plot suggests.

histQQmatch

histQQmatch

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.