library(knitr)
load(url("http://www.openintro.org/stat/data/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)
  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?

Men histogram has a peak column which looks higher than the women’s, it can be assumed from here that men are taller than women.

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

It’s a normal distribution.

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

fhgtmean <- sum(fdims$hgt[1:length(fdims$hgt)])/length(fdims$hgt)      
fhgtsd   <- sd(fdims$hgt)
sim_norm <- rnorm(n = length(fdims$hgt), mean = fhgtmean, sd = fhgtsd)
qqnorm(sim_norm)
qqline(sim_norm)

  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?
qqnormsim(fdims$hgt)

The output results differ every time simulated but most of the points fall on or very close to the line every time more than the real data.

  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?

The normal probability and the output results of the simulated data for fdims$hgt are very similar with slight diffrerences at the edges of the line. fdims$hgt is a normal distribution.

  1. Using the same technique, determine whether or not female weights appear to come from a normal distribution.
fwgtmean <- sum(fdims$wgt[1:length(fdims$wgt)])/length(fdims$wgt)      
fwgtsd   <- sd(fdims$wgt)

sim_normw <- rnorm(n = length(fdims$wgt), mean = fwgtmean, sd = fwgtsd)
qqnorm(sim_normw)
qqline(sim_normw)

qqnormsim(fdims$wgt)

It’s almost close to be called normal distribution like the fdims$hgt, but fdims$wgt clearly shows at the tail of the line that it’s an almost normal distribution. The plot shows that it is a right skewed.

1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)
## [1] 0.004434387
sum(fdims$hgt > 182) / length(fdims$hgt)
## [1] 0.003846154
  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 percent of women are under 150 cm?

## [1] "Theoretical for height: %(Z= -2.27245406145884 )= 1.15 %"
## [1] "Empirical for height: %= 1.15 %"

What percent of women are over 90Kg?

## [1] "Theoretical for weight: %(Z= 3.05746015192308 )= 99.89 %"
## [1] "Empirical for weight: %= 0.77 %"
height weight
Theoritical = Empirical Theoritical != Empirical

- The height had a closer agreement than the weight.

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?
    Normal probability plots C and D are for age and elbow respectively. Discrete (age) or rounded values close to discrete (elbow) can output such a stepwise pettern.

  • 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.



Right skewed

histQQmatch