Data 606 Lab4- “The normal distribution” by Sufian
Rpubs:
http://rpubs.com/ssufian/533410
Github:
https://github.com/ssufian/Data_606
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.
Let’s take a quick peek at the first few rows of the data.
## 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 1You’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.
## 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## 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?
# histogram men & women
library(ggplot2)
mdims <- subset(bdims, sex == 1)
fdims <- subset(bdims, sex == 0)
ggplot(bdims) + geom_histogram(mapping = aes(x = hgt, fill = sex),color = "blue", binwidth = 3) + facet_wrap(~sex)
Ans: note: sex =1 is male and sex = 0 is female
The histograms are nearly normal. The female plot appears to be a little more symmetric.
The male histogram contains left skew, with a mode of just a little under 180. The male heights has a few
outliers (fatter tails), resulting in higher variance.
They both have different means, but the male’s being higher.
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)
#female weights
fwgtmean <- mean(fdims$wgt)
fwgtsd <- sd(fdims$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 = "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?
ans:
Based on “eye-ball” judgement, yes it still appear to the naked eye that its normal
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”.
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.
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.
- 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?
ans:
Most of the interior points fall on the 45 degree line like the original plot. However, the exterior
points are falling off the line, albeit, less than the original one.
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.
- Does the normal probability plot for
fdims$hgtlook similar to the plots created for the simulated data? That is, do plots provide evidence that the female heights are nearly normal?
ans:
Yes, the simulated plots still look similar to the original plot with the higher data points
on the fringes deviating off the 45 degree line. This suggested that female heights comes from a
normally distrbuted set
- Using the same technique, determine whether or not female weights appear to come from a normal distribution.
Ans:
The first simulated plot really showed more weights data point was off the line. This suggested that
female weights have more outlier data points on the high end; resulting in right skewness. Data
transformation comes to mind to make it more normal looking, take the log of the data. The reason is
since you have more outliers, the logs “squeeze” (bunching up) extreme data points closer together,
minimizing outliers in the process. This logging process almost always make it more normal looking but
one has to be carefulbecause, this can be use as a crutch and hide true outliers that may be important
that needs further in-depth investigation.
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?
ans:
If we can ascertain for sure that our data sets are normal, we can then perform further analytics via
parametric analysis. Parametric studies comes with powerful testing tool kits such as hypothesis testing
interval estimators and inferential studies to name a few. Its not only powerful but also very simple to
implement. Once we know for sure our data is normal, we only need two statistic; mean and standard
deviation to know everything about your data.
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] 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.
## [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?
ans 1:
What is the probability that female height falls between approximately 5’ and 6’, inclusive?
## [1] 0.9672236## [1] 0.9692308ans 1:
What is the probability that female weight falls between 50 and 82 kilos, inclusive?
## [1] 0.9472567## [1] 0.8807692ans:
Female heights empirically calculated value is closer to the theoretical norm. Therefore, female height
has a more normal distribution than weights.
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 (left skewness).b. The histogram for female elbow diameter (
elb.di) belongs to normal probability plot letter __C_(nearly normal)_.c. The histogram for general age (
age) belongs to normal probability plot letter __A right skewness_.d. The histogram for female chest depth (
che.de) belongs to normal probability plot letter __D more outliers, heavier tails.Note that normal probability plots C and D have a slight stepwise pattern.
Why do you think this is the case?
ans:
Local behaviour: this differences in local concentrated data points like a spike of (at higher elevated)
from the surrounding data points, made values aligned horizontally, resulting in step-wise pattern
- 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.
## [1] 18.09692## [1] 18ans:
Looking at the normal and Q-Q plots, knee diameters appears to be right skewed. One telling sign is when
the mean is higher than the median; because mean are more susceptible to outlier distortions than median.
histQQmatch