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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?
library(ggplot2)

qplot(x = hgt, data = mdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), bindwidth = 3, fill = "light blue") +
  ylim(0, 0.08) +
  xlim(140, 200) +
  stat_function(fun = dnorm, args = c(mean = mean(mdims$hgt), sd = sd(mdims$hgt)), col = "blue") + 
  labs(title = "Men's height distribution")

summary(mdims$hgt)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   157.2   172.9   177.8   177.7   182.7   198.1
IQR(mdims$hgt)
## [1] 9.75
sd(mdims$hgt)
## [1] 7.183629
qplot(x = hgt, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 3, fill = "pink") +
  ylim(0, 0.08) +
  xlim(140, 200) +
  stat_function(fun = dnorm, args = c(mean = mean(fdims$hgt), sd = sd(fdims$hgt)), col = "red") + 
  labs(title = "Women's height distribution")

summary(fdims$hgt)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   147.2   160.0   164.5   164.9   169.5   182.9
IQR(fdims$hgt)
## [1] 9.5
sd(fdims$hgt)
## [1] 6.544602

The distribution of Men’s height shows more symmmetric distribution than that of Women’s.
Men’s height distribution looks more normal and fits the bell-shaped curve closely.
The multiple modes are better considered as a result of sampling variability and small binwidth.
The distributions differ the most in their center.

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).

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

The histogram appears to be normally distributed. It shows higher concentration in the middle than it should in a normal distribution; It could be due to sampling variability, so the normal distribution is a reasonable conclusion in this case.

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?
qqnorm(sim_norm)
qqline(sim_norm)

Compared to the real data(female heights distribution), Not all points fall on the line. While most points appears to align with the normal line, there are some errant points towards the tails. The original dataset seems to have more variations than the simulated 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(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?

Yes, Both original and simulated data look nearly normal. Each of their values show some variance at the far ends, but it is too small to represent the trend.

  1. Using the same technique, determine whether or not female weights appear to come from a normal distribution.
qplot(x = wgt, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 3, fill = "pink") +
  stat_function(fun = dnorm, args = c(mean = mean(fdims$wgt), sd = sd(fdims$wgt)), col = "red") + 
  labs(title = "Women's weight distribution")

summary(fdims$wgt)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    42.0    54.5    59.0    60.6    65.6   105.2
IQR(fdims$wgt) # 11.1
## [1] 11.1
sd(fdims$wgt) # 9.6156
## [1] 9.615699
sim_norm_wgt <- rnorm(n = length(fdims$wgt), mean = mean(fdims$wgt), sd = sd(fdims$wgt))
qqnorm(sim_norm_wgt)
qqline(sim_norm_wgt)

qqnormsim(fdims$wgt)

The female weight distribution does not appear to come from a normal distribution. The actual dataset has errant points in each far ends with a tendancy of right skewness. The weight seems to show big difference from median(59) on each far ends.

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 probability that a randomly chosen female is taller than 180cm?
What is the probability that a randomly chosen female weighs more 70kgs?

# What is the probability that a randomly chosen female is taller than 180cm?

emp.hgt <- 1 - pnorm(q = 180, mean = mean(fdims$hgt), sd = sd(fdims$hgt))
the.hgt <- sum(fdims$hgt > 180) / length(fdims$hgt)
emp.hgt
## [1] 0.01040328
the.hgt
## [1] 0.007692308
emp.hgt - the.hgt
## [1] 0.002710974
# What is the probability that a randomly chosen female weighs more 70kgs?
emp.wgt <- 1 - pnorm(q = 70, mean = mean(fdims$wgt), sd = sd(fdims$wgt))
the.wgt <- sum(fdims$wgt > 70) / length(fdims$wgt)
emp.wgt
## [1] 0.1641539
the.wgt
## [1] 0.1576923
emp.wgt - the.wgt
## [1] 0.006461585

The height data had a closer agreement between the theoretical prob calculation and empirical prob calculation.
This seems right as the distribution of the female height shows more normal than that of the weight.

On Your Own

B

a <- fdims$bii.di
hist(a)

#standardize the data
a <- (a - mean(a)) / sd(a)

qplot(x = a, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 1, fill = "pink") +
  stat_function(fun = dnorm, args = c(mean = mean(a), sd = sd(a)), col = "red") + 
  labs(title = "Women's biiliac diameter distribution - standardized")

a.sim <- rnorm(n = length(a), mean = mean(a), sd = sd(a))
qqnorm(a.sim)
qqline(a.sim)

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

C

b <- fdims$elb.di
hist(b)

#standardize the data
b <- (b - mean(b)) / sd(b)

qplot(x = b, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 1, fill = "pink") +
  stat_function(fun = dnorm, args = c(mean = mean(b), sd = sd(b)), col = "red") + 
  labs(title = "Women's elbow diameter distribution - standardized")

b.sim <- rnorm(n = length(b), mean = mean(b), sd = sd(b))
qqnorm(b.sim)
qqline(b.sim)

**c.** The histogram for general age (`age`) belongs to normal probability 
plot letter ____.

D

c <- bdims$age
hist(c)

#standardize the data
c <- (c - mean(c)) / sd(c)

qplot(x = c, data = bdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 1, fill = "pink") +
  stat_function(fun = dnorm, args = c(mean = mean(c), sd = sd(c)), col = "red") + 
  labs(title = "general age distribution - standardized")

c.sim <- rnorm(n = length(c), mean = mean(c), sd = sd(c))
qqnorm(c.sim)
qqline(c.sim)

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

A

d <- fdims$che.de
hist(d)

#standardize the data
d <- (d - mean(d)) / sd(d)

qplot(x = d, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..), binwidth = 1, fill = "pink") +
  stat_function(fun = dnorm, args = c(mean = mean(d), sd = sd(d)), col = "red") + 
  labs(title = "general age distribution - standardized")

d.sim <- rnorm(n = length(d), mean = mean(d), sd = sd(d))
qqnorm(d.sim)
qqline(d.sim)

Plot C = female elbow diameter, Plot D = general age
This is because there are many repeated values / how data was measured.
General age is discrete numeric data, thus integer, not double precision. This is how the step pattern shows on the plot.
We can also spot how often repeated values show below the table.

library(mosaic)

table(bdims$age)
## 
## 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 
##  4 22 34 25 28 39 28 34 23 21 17 22 15  8 20 13 15 10  9 16  7  5 11  9  7 
## 43 44 45 46 47 48 49 50 51 52 53 54 55 56 60 62 64 65 67 
##  8 10 10  6  1  4  3  1  3  4  2  1  3  2  1  3  1  1  1
favstats(bdims$age)
##  min Q1 median Q3 max     mean       sd   n missing
##   18 23     27 36  67 30.18146 9.608472 507       0
#age 23 was observed 39 times.

table(fdims$elb.di)
## 
##  9.9 10.1 10.3 10.4 10.6 10.7 10.8 10.9   11 11.1 11.2 11.3 11.4 11.5 11.6 
##    1    1    1    1    3    1    1    2    4    3    8    6    1   13   11 
## 11.7 11.8 11.9   12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9   13 13.1 
##    3   12    4   17    6   10    6   29    3   15    7   14   13   10    9 
## 13.2 13.3 13.4 13.6 13.7 13.8 13.9   14 14.1 14.2 14.3   15 
##   11    3   14    3    1    4    1    4    1    1    1    1
favstats(fdims$elb.di)
##  min   Q1 median   Q3 max     mean        sd   n missing
##  9.9 11.8   12.4 12.9  15 12.36692 0.8363728 260       0
#elb.di 12.4 was observed 29 times.

The variable for female kne.di shows right-skewness.
it’s q-q plot is convex with the end points directing upwards.

knee <- fdims$kne.di
hist(knee)

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

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.