download.file("http://www.openintro.org/stat/data/bdims.RData",destfile="bdims.RData")
load("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
mdims <- subset(bdims, sex == 1)
fdims <- subset(bdims, sex == 0)
Ex 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?
fhgtmean <- mean(fdims$hgt)
fhgtsd <- sd(fdims$hgt)
hist(fdims$hgt, probability = TRUE)
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y, col = "blue")
Ex 2, Based on the this plot, does it appear that the data follow a nearly normal distribution?
qqnorm(fdims$hgt)
qqline(fdims$hgt)
sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)
Ex 3, 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?
qqnormsim(fdims$hgt)
Ex 4, Does the normal probability plot for fdims$hgt look similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal?
Answer: Yes they look similar - all the real data (white circles) were within the gray area of the Q-Q plot - the gray area representing the 500 simulations done. The higher and lower quantilevalues are farther away (although still within the gray area) from the q-q line indicating a slight deviation from the normal distribution. The result of this plot simulation (for normal distribution) does support the statement that female heights for this sample follow a near normal distribution
Ex 5, Using the same technique, determine whether or not female weights appear to come from a normal distribution.
1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)
## [1] 0.004434387
sum(fdims$hgt > 182) / length(fdims$hgt)
## [1] 0.003846154
Ex 6, 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 percent of women are under 152.4 cm. (5 feet)?
pnorm(q = 152.4, mean = fhgtmean, sd = fhgtsd)
## [1] 0.028342
sum(fdims$hgt < 152.4) / length(fdims$hgt)
## [1] 0.02692308
2, What percent of women are over 90.72 kg. (200 lbs.)?
fwgtmean <- mean(fdims$wgt)
fwgtsd <- sd(fdims$wgt)
pnorm(q = 90.72, mean = fwgtmean, sd = fwgtsd)
## [1] 0.9991329
sum(fdims$wgt > 90.72) / length(fdims$wgt)
## [1] 0.007692308
On Your Own
1, 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
Answer: B
b, The histogram for female elbow diameter (elb.di) belongs to normal probability plot letter
Answer: C
c, The histogram for general age (age) belongs to normal probability plot letter
Answer: D
d, The histogram for female chest depth (che.de) belongs to normal probability plot letter
Answer: A
2, Note that normal probability plots C and D have a slight stepwise pattern.
Why do you think this is the case?
Answer: stepwise probalility plot patterns are usually produced by discrete variables such as age whereas continous patterns (such as a normal curve) are produced by continuous variables.
3, 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.
qqnorm(fdims$kne.di)
qqline(fdims$kne.di)
