Univariate data analysis

Learning outcomes

• The study of variability important to help answer: "what happened?"
• Univariate tools such as the histogram, median, MAD, standard deviation, quartiles will be reviewed from prior courses (as a refresher)
• The normal and t-distribution will be important in our work: what are they, how to interpret them, and use tables of these distributions
• The central limit theorem will be explained conceptually: you cannot finish a course on stats without knowing the key result from this theorem.
• Using and interpreting confidence intervals will be crucial in all the modules that follow.

Extended readings

• New Boeing planes will generate 0.5 TB of data per flight. Read about this, and other sources of data: "every piece of that plane has an internet connection, from the engines to the flaps to the landing gear".
• All students, but especially the 600-level students should read the article by Peter J. Rousseeuw, Tutorial to Robust Statistics it is easy to read, and contains so much useful content.

Resources

• Class notes 2015
• Class notes 2014
• Textbook, chapter 2
• Quizzes (with solutions): attempt these after you have watched the videos

Tasks to do first	Quiz	Solution
Complete steps 10, 11, 12 and 13 of the software tutorial (also steps 1 through 9)	Quiz	Solution
Watch videos 1, 2, 3, 4, and 5	Quiz	Solution
Watch videos 6, 7, and 8	Quiz	Solution
Watch videos 9 and 10	Quiz	Solution
Watch videos 11, 12, and 13	Quiz	Solution
Watch videos 14, 15, and 16	Quiz	Solution

• A demonstration of R
• Using tables of the normal distribution

Class videos from prior years

Videos from 2015

1. Introduction [05:59]
2. Histograms [04:50]
3. Basic terminology [06:41]
4. Outliers, medians and MAD [04:42]
5. The central limit theorem [06:56]
6. The normal distribution, and standardizing variables [05:54]
7. Normal distribution notation and using tables and R [05:48]
8. Checking if data are normally distributed [05:57]
9. Introducing the idea of a confidence interval [covered in class]
10. Confidence intervals when we don't know the variance [07:59]
11. Interpreting the confidence interval [07:52]
12. A worked example: calculating and interpreting the CI [03:37]
13. A motivating example to see why tests for differences are important [08:29]
14. The mathematical derivation for a confidence interval for differences [covered in class]
15. Using the confidence interval to test for differences to solve the motivating example [covered in class]
16. Confidence intervals for paired tests: theory and an example [11:59]

Covered in class | No video | Script

Audio only | No video | Script

Audio only | No video | Script

Videos from 2014

See the webpage from 2014

Videos from 2013

See the webpage from 2013

Software codes for this section

Understanding the central limit theorem with the rolling dice example

Web-based version of the code

N = 500
m <- t(matrix(seq(1,6), 3, 2))
layout(m)
s1 <- as.integer(runif(N, 1, 7))
s2 <- as.integer(runif(N, 1, 7))
s3 <- as.integer(runif(N, 1, 7))
s4 <- as.integer(runif(N, 1, 7))
s5 <- as.integer(runif(N, 1, 7))
s6 <- as.integer(runif(N, 1, 7))
s7 <- as.integer(runif(N, 1, 7))
s8 <- as.integer(runif(N, 1, 7))
s9 <- as.integer(runif(N, 1, 7))
s10 <- as.integer(runif(N, 1, 7))

hist(s1, main="", xlab="One throw", breaks=seq(0,6)+0.5)
bins = 8
hist((s1+s2)/2, breaks=bins, main="", xlab="Average of two throws")
hist((s1+s2+s3+s4)/4, breaks=bins, main="", xlab="Average of 4 throws")
hist((s1+s2+s3+s4+s5+s6)/6, breaks=bins, main="", xlab="Average of 6 throws")
bins=12
hist((s1+s2+s3+s4+s5+s6+s7+s8)/8,  breaks=bins, main="", xlab="Average of 8 throws")
hist((s1+s2+s3+s4+s5+s6+s7+s8+s9+s10)/10, breaks=bins, main="", xlab="Average of 10 throws")

Code used to illustrate how the q-q plot is constructed

Web-based version of the code

N <- 10

# What are the quantiles from the theoretical normal distribution?
index <- seq(1, N)
P <- (index - 0.5) / N
theoretical.quantity <- qnorm(P)

# Our sampled data:
yields <- c(86.2, 85.7, 71.9, 95.3, 77.1, 71.4, 68.9, 78.9, 86.9, 78.4)
mean.yield <- mean(yields)       # 80.0
sd.yield <- sd(yields)           # 8.35

# What are the quantiles for the sampled data?
yields.z <- (yields - mean.yield)/sd.yield
yields.z
 
yields.z.sorted <- sort(yields.z)

# Compare the values in text:
yields.z.sorted 
theoretical.quantity  

# Compare them graphically:
plot(theoretical.quantity, yields.z.sorted, asp=1)
abline(a=0, b=1)

# Built-in R function to do all the above for you:
qqnorm(yields)
qqline(yields)

# A better function: see http://learnche.mcmaster.ca/4C3/Software_tutorial/Extending_R_with_packages
library(car)
qqPlot(yields)

Code to illustrate the central limit theorem's reduction in variance

Web-based version of the code

# Show the 3 plots side by side
layout(matrix(c(1,2,3), 1, 3))

# Sample the population:
N <- 100
x <- rnorm(N, mean=80, sd=5)
mean(x)
sd(x)

# Plot the raw data
x.range <- range(x)
plot(x, ylim=x.range, main='Raw data')

# Subgroups of 2
subsize <- 2
x.2 <- numeric(N/subsize)
for (i in 1:(N/subsize))
{
    x.2[i] <- mean(x[((i-1)*subsize+1):(i*subsize)])
}
plot(x.2, ylim=x.range, main='Subgroups of 2')

# Subgroups of 4
subsize <- 4
x.4 <- numeric(N/subsize)
for (i in 1:(N/subsize))
{
    x.4[i] <- mean(x[((i-1)*subsize+1):(i*subsize)])
}
plot(x.4, ylim=x.range, main='Subgroups of 4')

Paired test example

Web-based version of the code

# Analysis of the data here:
dilution <-   c(11, 26, 18, 16, 20, 12,  8, 26, 12, 17, 14)
manometric <- c(25,  3, 27, 30, 33, 16, 28, 27, 12, 32, 16)
N <- length(dilution)

mean(manometric)
mean(dilution)

plot(c(dilution, manometric), ylab="BOD values", xaxt='n')
text(5.5,3, "Dilution")
text(18,3, "Manometric")
abline(v=11.5)

par(mar=c(4.2, 4.2, 0.2, 0.2))  # (bottom, left, top, right); defaults are par(mar=c(5, 4, 4, 2) + 0.1)
plot(dilution, type="p", pch=4, 
    cex=2, cex.lab=1.5, cex.main=1.8, cex.sub=1.8, cex.axis=1.8, 
    ylab="BOD values", xlab="Sample number",
    ylim=c(0,35), xlim=c(0,11.5), col="darkgreen")
lines(manometric, type="p", pch=16, cex=2, col="blue")
lines(rep(0, N), dilution, type="p", pch=4, cex=2, col="darkgreen")
lines(rep(0, N), manometric, type="p", pch=16, cex=2, col="blue")

abline(v=0.5)

legend(8, 5, pch=c(4, 16), c("Dilution", "Manometric"), col=c("darkgreen", "blue"), pt.cex=2)

par(mar=c(4.2, 4.2, 0.2, 0.2))  # (bottom, left, top, right); defaults are par(mar=c(5, 4, 4, 2) + 0.1)
plot(dilution-manometric, type="p", ylab="Dilution - Manometric", xlab="Sample number", 
     cex.lab=1.5, cex.main=1.8, cex.sub=1.8, cex.axis=1.8, cex=2)
abline(h=0, col="grey60")