The normal distribution

The Data

load("more/bdims.RData")

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?

hist(mdims$hgt, main = "Male Heights", xlab = "(cm)")

hist(fdims$hgt, main = "Female Heights", xlab = "(cm)")

Answer: Females have a more uniform, slightly right-skewed distribution of heights, with a mean around 165 cm. Males have an approximately normal distribution with a higher mean, around 178 cm.

The normal distribution

fhgtmean <- mean(fdims$hgt)
fhgtsd   <- sd(fdims$hgt)

hist(fdims$hgt, 
     probability = TRUE, 
     ylim = c(0, .06), 
     main = "Female Height vs. Normal Distribution",
     xlab = "(cm)")
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y=y, col = "blue")

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

Answer: Yes, it appears nearly normal.

qqnorm(fdims$hgt)
qqline(fdims$hgt)

sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)

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?

qqnorm(sim_norm)
qqline(sim_norm)

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?

Answer: Yes

Using the same technique, determine whether or not female weights appear to come from a normal distribution.

qqnormsim(fdims$wgt)

Answer: It’s not that far off, but the real data appears to be more right-skewed than the simulated, normal data. In other words, there are more very high outliers in the real data.

Normal probabilities

1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)

## [1] 0.004434387

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?

Answer: 1. What is the probability that a woman is taller than 172 cm?

Normal:

1 - pnorm(q = 172, mean = fhgtmean, sd = fhgtsd)

## [1] 0.138056

1 - pnorm(q = 75, mean = mean(fdims$wgt), sd = sd(fdims$wgt))

## [1] 0.06713016

Empirical:

sum(fdims$hgt > 172) / length(fdims$hgt)

## [1] 0.1615385

sum(fdims$wgt > 75) / length(fdims$wgt)

## [1] 0.07307692

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.

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

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

d. The histogram for female chest depth (che.de) belongs to normal probability plot letter A.
Note that normal probability plots C and D have a slight stepwise pattern.
Why do you think this is the case? Because people round to the nearest whole (or at least half) unit measurement
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.

Answer: It’s right-skewed

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

hist(fdims$kne.di, probability = T)

histQQmatch