Lab 3 - The Normal Distribution

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)

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?
```
library(ggplot2)
library(gridExtra)
gginit.male <- ggplot(mdims,aes(x=hgt))
plottype.male <- geom_histogram(binwidth=3,color='red',fill='pink',alpha=0.5)
gginit.female <- ggplot(fdims,aes(x=hgt))
plottype.female <- geom_histogram(binwidth=3,color='blue',fill='blue',alpha=0.5)
plottheme <- theme_bw()
male.plot <- gginit.male + plottype.male + plottheme + xlab('Height (cm)') + ggtitle('Male') + xlim(140,200)
female.plot <- gginit.female + plottype.female + plottheme + xlab('Height (cm)') + ggtitle('Female') + xlim(140,200)
grid.arrange(male.plot, female.plot, ncol = 2)
```
It appears that both the male and female histograms are approximately normal. I can see an argument where we could say the male histogram is very slightly left-skewed, and the female histogram is a marginally more symmetric. However, from the histograms alone, these appear to both be symmetric.

The main difference lies in where the means are centered. As expected, the mean for males is higher than the female mean. There is also a slightly wider distribution for males (155 to 200 cm) than the females (145 to 185 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.

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 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).

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

From this plot, I would say the data is nearly normal. It is marginally right-skewed with a little more density on the left than the right. However, the peak of the curve is aligned with the most dense areas, and the edges of the curve align with areas of less density. Therefore, this is approximately 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.

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?
```
sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)
qqnorm(sim_norm)
qqline(sim_norm)
```
It appears that from -1 to 1 theoretical quantiles, the data falls on the line. Further past that, some of the points vary off the line, but not much at all. The real data is very comparable to the simulation and nearly matches.

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)

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?

The real data plot is very similar to all of the simulated plots. Therefore, we can say that the real data for female heights is approximately normal.
Using the same technique, determine whether or not female weights appear to come from a normal distribution.
```
qqnormsim(fdims$wgt)
```
In the case of female heights, the distribution for the real data does not quite match the simulated data. In fact, the real data displays a positive U-shape which indicated a right-skewed distribution. Therefore, we can guess that female heights do not come from a normal distribution.

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.

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 probability that a female’s height is between 155 and 175 cm?
The results between the normal distribution and the empricial calculation differ by less than 3%.
```
pnorm(q=175,mean=fhgtmean,sd=fhgtsd) - pnorm(q=155,mean=fhgtmean,sd=fhgtsd)
```
```
## [1] 0.8734095
```
```
sum(fdims$hgt < 175 & fdims$hgt > 155)/ length(fdims$hgt)
```
```
## [1] 0.8461538
```
- What is the probability that a female’s height is between 65 and 75 kg?
The results between the normal distribution and the emprical calculation differ by less than 7%.
```
fwgtmean <- mean(fdims$wgt)
fwgtsd   <- sd(fdims$wgt)
```
```
pnorm(q=75,mean=fwgtmean,sd=fwgtsd) - pnorm(q=65,mean=fwgtmean,sd=fwgtsd)
```
```
## [1] 0.2565095
```
```
sum(fdims$wgt < 75 & fdims$wgt > 65)/ length(fdims$wgt)
```
```
## [1] 0.1884615
```
Since the height empricial data matches much more closely with the normal data than the weight emprical data matches the normal data, then we can say that the height data is more likely to follow a normal distribution than the weight. With the additional Q-Q plot information from previous exercises, we can be fairly certain of this claim.

On Your Own

Now let’s consider some of the other variables in the body dimensions data set. Using the figures at the end of the exercises, match the histogram to its normal probability plot. All of the variables have been standardized (first subtract the mean, then divide by the standard deviation), so the units won’t be of any help. If you are uncertain based on these figures, generate the plots in R to check.

a. The histogram for female biiliac (pelvic) diameter (bii.di) belongs to normal probability plot letter B.

This histogram appeared to be slightly left-skewed, and the only normal plot with concave down curvature was plot B.

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

The histogram appeared to be the closest to a normal distribution.

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

The third and fourth histograms are both right-skewed. However, if you look at the axis on the third plot, the frequency is much larger, showing that this has a greater skew. Between normal plots A and D, the D plot appears to be more concave up then A.

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

For the same reasoning, as part (c).
Note that normal probability plots C and D have a slight stepwise pattern.
Why do you think this is the case?

Normal probability plots C and D correspond to elbow diameter and age, respectively. The elbow diameter data shows close to normal distributions, but you can see small deviations from how a normal distribution would fit on it. The same goes for age and how it would fit with a skewed curve. These small steps represent these small deviations from the normal distribution. Histogram plots one and four show more of a smoothness in their histograms, which results in less deviations from the qq-line.
As you can see, normal probability plots can be used both to assess normality and visualize skewness. Make a normal probability plot for female knee diameter (kne.di). Based on this normal probability plot, is this variable left skewed, symmetric, or right skewed? Use a histogram to confirm your findings.

Below shows the normal plot for female knee diameter. The concave up curvature, typically means that this variable is right-skewed.
```
qqnorm(fdims$kne.di)
qqline(fdims$kne.di)
```
The below histogram also confirms a right-skew.
```
gginit.knee <- ggplot(fdims,aes(x=kne.di))
plottype.knee <- geom_histogram(binwidth=1,color='red',fill='pink',alpha=0.5)
plottheme <- theme_bw()
gginit.knee + plottype.knee + plottheme + xlab('Knee Diameter (cm)') + ylab('Count')
```

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.