Jun Pan DATA 606 Lab 3
Jun Pan
October 7, 2018
Lab 3: The normal distribution 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("C:/Users/tbao/Desktop/CUNY MSDS notes/606/chapter 3 wk 5 to wk 6/lab 3/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 1tail(bdims)## bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi
## 502 36.5 29.7 34.0 20.2 28.5 13.3 9.8 18.9 12.4 104.0
## 503 38.0 30.4 32.9 17.0 27.1 12.9 10.4 19.5 14.4 108.4
## 504 35.3 28.7 30.4 17.7 25.6 12.4 9.8 17.3 13.6 99.3
## 505 34.7 24.9 24.7 17.3 24.2 12.0 10.2 18.0 13.6 91.9
## 506 38.5 29.0 32.9 15.3 25.6 12.0 9.8 18.6 13.3 107.1
## 507 35.6 29.0 29.0 20.4 26.8 13.4 10.8 18.7 13.8 100.5
## che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi
## 502 95.4 86.0 107.1 112.1 64.4 32.3 26.4 35.7 37.0 21.4
## 503 91.8 69.9 90.4 101.0 60.6 30.3 25.4 37.7 37.9 22.4
## 504 87.3 63.5 79.2 89.5 55.2 30.1 23.6 35.6 33.3 22.4
## 505 78.1 57.9 75.1 86.9 51.8 27.4 24.0 34.4 34.1 21.2
## 506 90.9 72.2 89.4 98.6 59.0 30.6 24.9 38.4 36.6 22.0
## 507 97.1 80.4 100.8 102.2 57.4 33.2 25.5 39.6 35.9 23.0
## wri.gi age wgt hgt sex
## 502 15.8 40 76.8 157.5 0
## 503 15.4 29 71.8 176.5 0
## 504 15.2 21 55.5 164.4 0
## 505 15.5 33 48.6 160.7 0
## 506 15.5 33 66.4 174.0 0
## 507 16.4 38 67.3 163.8 0dim(bdims)## [1] 507 25You’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).
Description
Body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender, are given for 507 physically active individuals - 247 men and 260 women. These data can be used to provide statistics students practice in the art of data analysis. Such analyses range from simple descriptive displays to more complicated multivariate analyses such as multiple regression and discriminant analysis.
Source: Heinz G, Peterson LJ, Johnson RW, Kerk CJ. 2003. Exploring Relationships in Body Dimensions. Journal of Statistics Education 11(2).
Format A data frame with 507 observations on the following 25 variables.
bia.di: a numerical vector, respondent’s biacromial diameter in centimeters bii.di: a numerical vector, respondent’s biiliac diameter (pelvic breadth) in centimeters bit.di: a numerical vector, respondent’s bitrochanteric diameter in centimeters che.de: a numerical vector, respondent’s chest depth in centimeters, measured between spine and sternum at nipple level, mid-expiration che.di: a numerical vector, respondent’s chest diameter in centimeters, measured at nipple level, mid-expiration elb.di: a numerical vector, respondent’s elbow diameter in centimeters, measured as sum of two elbows wri.di: a numerical vector, respondent’s wrist diameter in centimeters, measured as sum of two wrists kne.di: a numerical vector, respondent’s knee diameter in centimeters, measured as sum of two knees ank.di: a numerical vector, respondent’s ankle diameter in centimeters, measured as sum of two ankles sho.gi: a numerical vector, respondent’s shoulder girth in centimeters, measured over deltoid muscles che.gi: a numerical vector, respondent’s chest girth in centimeters, measured at nipple line in males and just above breast tissue in females, mid-expiration wai.gi: a numerical vector, respondent’s waist girth in centimeters, measured at the narrowest part of torso below the rib cage as average of contracted and relaxed position nav.gi: a numerical vector, respondent’s navel (abdominal) girth in centimeters, measured at umbilicus and iliac crest using iliac crest as a landmark hip.gi: a numerical vector, respondent’s hip girth in centimeters, measured at at level of bitrochanteric diameter thi.gi: a numerical vector, respondent’s thigh girth in centimeters, measured below gluteal fold as the average of right and left girths bic.gi: a numerical vector, respondent’s bicep girth in centimeters, measured when flexed as the average of right and left girths for.gi: a numerical vector, respondent’s forearm girth in centimeters, measured when extended, palm up as the average of right and left girths kne.gi: a numerical vector, respondent’s knee diameter in centimeters, measured as sum of two knees cal.gi: a numerical vector, respondent’s calf maximum girth in centimeters, measured as average of right and left girths ank.gi: a numerical vector, respondent’s ankle minimum girth in centimeters, measured as average of right and left girths wri.gi: a numerical vector, respondent’s wrist minimum girth in centimeters, measured as average of right and left girths age: a numerical vector, respondent’s age in years wgt: a numerical vector, respondent’s weight in kilograms hgt: a numerical vector, respondent’s height in centimeters sex: a categorical vector, 1 if the respondent is male, 0 if 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)head(mdims)## 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 1head(fdims)## bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi
## 248 37.6 25.0 31.3 16.2 24.9 11.2 9.2 17.0 12.3 95.0
## 249 36.7 26.4 31.0 16.8 24.5 12.1 9.9 19.3 12.8 99.5
## 250 34.8 25.9 30.2 16.4 24.2 11.3 8.9 17.0 12.2 88.0
## 251 36.6 27.9 31.8 19.3 24.9 12.3 9.5 18.6 13.0 97.0
## 252 35.5 28.2 31.0 18.2 26.2 11.5 9.1 17.2 12.4 103.3
## 253 37.0 28.0 32.0 15.1 25.7 12.5 10.0 17.2 13.2 93.5
## che.gi wai.gi nav.gi hip.gi thi.gi bic.gi for.gi kne.gi cal.gi ank.gi
## 248 83.0 66.5 79.0 92.0 53.5 24.3 20.5 32.0 32.2 21.0
## 249 78.5 61.5 70.5 90.5 57.7 27.8 24.0 38.5 38.5 22.5
## 250 75.0 61.2 66.5 91.0 53.0 24.0 22.0 32.5 32.5 19.0
## 251 86.5 78.0 91.0 99.5 61.5 28.0 24.0 35.2 36.7 23.0
## 252 91.0 70.5 80.5 91.5 55.0 26.9 22.7 33.0 33.3 19.9
## 253 79.5 66.5 78.5 94.0 54.0 26.5 22.5 34.0 35.0 23.0
## wri.gi age wgt hgt sex
## 248 13.5 22 51.6 161.2 0
## 249 15.0 20 59.0 167.5 0
## 250 14.0 19 49.2 159.5 0
## 251 15.0 25 63.0 157.0 0
## 252 14.5 21 53.6 155.8 0
## 253 14.5 23 59.0 170.0 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? 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.
for female
fhgtmean <- mean(fdims$hgt)
fhgtsd <- sd(fdims$hgt)mean of female height
fhgtmean ## [1] 164.8723standard deviation of female height
fhgtsd ## [1] 6.544602for male
mhgtmean <- mean(mdims$hgt)
mhgtsd <- sd(mdims$hgt)mean of male height
mhgtmean## [1] 177.7453standard deviation of male height
mhgtsd## [1] 7.183629Next 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.
for female
hist(fdims$hgt, probability = TRUE)
x <- 140:190
y <- dnorm(x = x, mean = fhgtmean, sd = fhgtsd)
lines(x = x, y = y, col = "blue")
for male
hist(mdims$hgt, probability = TRUE)
x <- 160:210
y <- dnorm(x = x, mean = mhgtmean, sd =mhgtsd)
lines(x = x, y = y, col = "red")
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).
- Based on the this plot, does it appear that the data follow a nearly normal distribution? 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”.
for female
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$hgt), mean = fhgtmean, sd = fhgtsd)The first argument indicates how many numbers you’d like to generate, which we specify to be the same number of heights in the fdims data set using the length function. The last two arguments determine the mean and standard deviation of the normal distribution from which the simulated sample will be generated. We can take a look at the shape of our simulated data set, sim_norm, as well as its normal probability plot.
for male
qqnorm(mdims$hgt)
qqline(mdims$hgt)
- Make a normal probability plot of sim_norm. Do all of the points fall on the line? How does this plot compare to the probability plot for the real data? Even better than comparing the original plot to a single plot generated from a normal distribution is to compare it to many more plots using the following function. It may be helpful to click the zoom button in the plot window.
qqnormsim(fdims$hgt)
Does the normal probability plot for fdims$hgt look similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal? Answer: Yes, simulated plots looks similar to real data. Some of the simulated plots show observations at tail ends are also closer to line. Overall visuals of simulated and real data plots look similar.
Using the same technique, determine whether or not female weights appear to come from a normal distribution. Normal probabilities
Calulate mean and standard deviation of female weight
fwgtmean <- mean(fdims$wgt)
fwgtmean## [1] 60.60038fwgtsd <- sd(fdims$wgt)
fwgtsd## [1] 9.615699Actual normal distribution plot for real data
qqnorm(fdims$wgt)
qqline(fdims$wgt)
Simulated normal distribution plot for weight
sim_wgt <- rnorm(n = length(fdims$wgt), mean = fwgtmean, sd = fwgtsd)
qqnorm(sim_wgt)
qqline(sim_wgt)
qqnormsim(fdims$wgt)
Answer: Above normal distribution plots are for female weights. First one show real data and second one shows simulated data. Both plots are visually similar. Observations show smooth line hugging pattern in the middle and higher deviation at tail ends. Curves appears to be right sweked with 2 or more outliers in both cases.
Okay, so now you have a slew of tools to judge whether or not a variable is normally distributed. Why should we care? It turns out that statisticians know a lot about the normal distribution. Once we decide that a random variable is approximately normal, we can answer all sorts of questions about that variable related to probability. Take, for example, the question of, “What is the probability that a randomly chosen young adult female is taller than 6 feet (about 182 cm)?”
(The study that published this data set is clear to point out that the sample was not random and therefore inference to a general population is not suggested. We do so here only as an exercise.) If we assume that female heights are normally distributed (a very close approximation is also okay), we can find this probability by calculating a Z score and consulting a Z table (also called a normal probability table). In R, this is done in one step with the function pnorm.
1 - pnorm(q = 182, mean = fhgtmean, sd = fhgtsd)## [1] 0.004434387Note that the function pnorm gives the area under the normal curve below a given value, q, with a given mean and standard deviation. Since we’re interested in the probability that someone is taller than 182 cm, we have to take one minus that probability. Assuming a normal distribution has allowed us to calculate a theoretical probability. If we want to calculate the probability empirically, we simply need to determine how many observations fall above 182 then divide this number by the total sample size.
sum(fdims$hgt > 182) / length(fdims$hgt)## [1] 0.003846154Although the probabilities are not exactly the same, they are reasonably close. The closer that your distribution is to being normal, the more accurate the theoretical probabilities will be.
- Write out two probability questions that you would like to answer; one regarding female heights and one regarding female weights. Calculate the those probabilities using both the theoretical normal distribution as well as the empirical distribution (four probabilities in all). Which variable, height or weight, had a closer agreement between the two methods?
Question 1: What is the probability that a randomly chosen young adult female is atleast 165 cm tall?
pnorm(q = 165, mean = fhgtmean, sd = fhgtsd)## [1] 0.5077833sum(fdims$hgt <= 165) / length(fdims$hgt)## [1] 0.5115385Question 2: What is the probability that a randomly chosen young adult female weight is over 80 kgs?
1 - pnorm(q = 80, mean = fwgtmean, sd = fwgtsd)## [1] 0.02182199sum(fdims$wgt > 80) / length(fdims$wgt)## [1] 0.04230769On Your Own
- Now let’s consider some of the other variables in the body dimensions data set. Using the figures at the end of the exercises, match the histogram to its normal probability plot. All of the variables have been standardized (first subtract the mean, then divide by the standard deviation), so the units won’t be of any help. If you are uncertain based on these figures, generate the plots in R to check.
The histogram for female biiliac (pelvic) diameter (bii.di) belongs to normal probability plot letter B.
The histogram for female elbow diameter (elb.di) belongs to normal probability plot letter C.
The histogram for general age (age) belongs to normal probability plot letter D.
The histogram for female chest depth (che.de) belongs to normal probability plot letter A__.
Note that normal probability plots C and D have a slight stepwise pattern. Why do you think this is the case? Answer: It’s likely that these values (elbow diameter and age) were rounded, and reported as discrete variables. If they were continuous variables, they’d more resemble a normal curve.
As you can see, normal probability plots can be used both to assess normality and visualize skewness. Make a normal probability plot for female knee diameter (kne.di). Based on this normal probability plot, is this variable left skewed, symmetric, or right skewed? Use a histogram to confirm your findings.
qqnorm(fdims$kne.di, main = "Female Knee Diameter")
qqline(fdims$kne.di)
hist(fdims$kne.di)
The normal plot has a stepwise pattern which suggests the data is right skewed. This is verified in the histogram.