library(ggplot2)
library(dplyr)
## Warning: package 'dplyr' was built under R version 3.4.2
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

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.

setwd("~/R/Lab3")
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? Histogram of men’s heighs
hist(mdims$hgt, freq = F, breaks = 30, col = "green", xlab= "Sampled values", main = "Distribution of Men Height")

hist(fdims$hgt, freq = F, breaks = 30, col = "green", xlab= "Sampled values", main = "Distribution of Women Height - Breaks = 30")

hist(fdims$hgt, freq = F, breaks = 5, col = "green", xlab= "Sampled values", main = "Distribution of Women Height - Breaks = 5")

summary(mdims$hgt)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   157.2   172.9   177.8   177.7   182.7   198.1
summary(fdims$hgt)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   147.2   160.0   164.5   164.9   169.5   182.9

Both distributions are similar. They are normal distribute, symetric, bell-shaped and unimodal; but they have a different mean, the distribution of a men hight has a mean of 177.7 and the distribution of a women height has a mean of 164.9

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).

hist(fdims$hgt, probability = TRUE, ylim = c(0, 0.06), main = "Histogram of Women height", , col = "green")
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y , col = "blue")

qplot(x = hgt, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..)) +
  stat_function(fun = dnorm, args = c(mean = fhgtmean, sd = fhgtsd), col = "blue")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

2. Based on the this plot, does it appear that the data follow a nearly normal distribution? The distribution of Women’s heigth is well approximated by the normal model, the histogram above fit the normal curve overlaid in the plot and the sample mean fhgtmean and standard deviation fhgtsd are used as the parameters of the best fitting normal curve.

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 end 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?

Most of the points fall on the line but there is errant point above or under the line.

This plot seems more accurate to conclude that the data follow the normal model because most of the sample’s points are closer to the straight line.

qqnorm(sim_norm, ylab = "Women Height", main = "Normal Q-Q plot")
qqline(sim_norm, col = "blue")

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)

4. 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 simulated data seems to have more concentrations of dots closer to the line, it could be the sample sized increases and the normal probability plots becomes straighter and more stable compare to the normal probibilty plot fdims$hgt. The normal probability plots shows that women height are nearly normal.

The normal probability plot is nearly a perfect straight line

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

The Normal Distribution

fwgtmean <- mean(fdims$wgt)
fwgtsd   <- sd(fdims$wgt)
max(fdims$wgt)
## [1] 105.2
min(fdims$wgt)
## [1] 42
hist(fdims$wgt, probability = TRUE, main = "Histogram of Women Weight",  col = "red")
x <- 40:110
y <- dnorm(x = x, mean = fwgtmean, sd = fwgtsd)
lines(x = x, y = y , col = "blue")

qplot(x = wgt, data = fdims, geom = "blank") +
  geom_histogram(aes(y = ..density..)) +
  stat_function(fun = dnorm, args = c(mean = fwgtmean, sd = fwgtsd), col = "blue")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

##Evaluating the normal distribution

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

#Simulating data from a normal distribution using the mean fwgtmean and the standard deviation fwgtsd as parameters

sim_norm_w <- rnorm(n = length(fdims$wgt), mean = fwgtmean, sd = fwgtsd)
qqnorm(sim_norm_w, ylab = "Women Weight", main = "Normal Q-Q plot")
qqline(sim_norm_w, col = "blue")

qqnormsim(fdims$wgt)

The normal approximation of the women weight reflects more deviations from the line than the normal approximation of the women height. It seems that the Normal Distribution of Women height is following the normal model.

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?

    1.What is the probability that women’s hight is greater than 155 cm tall?

pnorm(q = 155, mean = fhgtmean, sd = fhgtsd)
## [1] 0.06571769
sum(fdims$hgt >155)/length(fdims$hgt)
## [1] 0.9307692
  1. What is the probability that women’s weigh less than 55 kg? The probability that someone’s weight is less than 55 kg
1 - pnorm(q = 55, mean = fwgtmean, sd = fwgtsd)
## [1] 0.7198584
sum(fdims$wgt < 55)/length(fdims$wgt)
## [1] 0.2884615

Based on these questionses, the results shows a highest probabilty for weight than height.

On Your Own

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

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

R/: *Q-Q plot B*
    #qqnorm(fdims$bii.di)
    #qqline(fdims$bii.di)

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

 R/: *Q-Q plot C*
    #qqnorm(fdims$elb.di)
    #qqline(fdims$elb.di)


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

 R/: *Q-Q plot D*
    #qqnorm(fdims$age)
    #qqline(fdims$age)

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

R/: *Q-Q plot A*
    #qqnorm(fdims$che.de)
    #qqline(fdims$che.de)
qqnorm(fdims$kne.di)
qqline(fdims$kne.di)

qplot(sample = kne.di, data = fdims)

hist(fdims$kne.di, probability = TRUE, main = "Histogram of Women's Knee Diameter in cms", , col = "brown")