1. Sampling and Confidence Interval
Epidemiology/Biostatistics
Kenneth Kwan Ho Chui, PhD, MPH
Department of Public Health and Community Medicine

2. Learning objectives in the syllabus
Understand how a histogram can be read as a probability distribution
Understand the importance of random sampling in statistics
Understand how sample means can have distributions
Explain the behavior (distribution) of sample means and the Central Limit Theorem
Know how to interpret confidence intervals as seen in the medical literature
Know how to calculate a confidence interval for a mean

3. Population Distribution of sample means Know how to interpret and
calculate a confidence interval for statistical inference
Population
Parameter
Types of data
How to summarize data
Central tendency
Variability
How to evaluate graphs
Sample
Sample statistics

4. Assumed knowledge for today
Mean
Variance
Standard deviation
The rule

5. Central tendency: Mean
Consider a variable with data: 1, 2, 3, 3, 4, 4, 4, 5, 5, 6

6. Variance & Standard deviation2
Variance & Standard deviation
Values
Sum them up
Divide by (sample size – 1)
Variance
Observation #
SD = √Variance

7. The 68-95-99 rule 99% of samples are within ± 3SD
68% of sample are within ± 1SD
# # External package to generate four shades of blue
# library(RColorBrewer)
# cols <- rev(brewer.pal(4, "Blues"))
cols <- c("#2171B5", "#6BAED6", "#BDD7E7", "#EFF3FF")
# Sequence between -4 and 4 with 0.1 steps
x <- seq(-4, 4, 0.1)
png("c:\\temp\\normalcurve.png", width=2000, height=1000, res=250)
# Plot an empty chart with tight axis boundaries, and axis lines on bottom and left
plot(x, type="n", xaxs="i", yaxs="i", xlim=c(-4, 4), ylim=c(0, 0.4),
bty="l", xaxt="n", xlab="a variable", ylab="frequency")
# Function to plot each coloured portion of the curve, between "a" and "b" as a
# polygon; the function "dnorm" is the normal probability density function
polysection <- function(a, b, col, n=11){
dx <- seq(a, b, length.out=n)
polygon(c(a, dx, b), c(0, dnorm(dx), 0), col=col, border=NA)
# draw a white vertical line on "inside" side to separate each section
segments(a, 0, a, dnorm(a), col="white") }
# Build the four left and right portions of this bell curve
for(i in 0:3){
polysection( i, i+1, col=cols[i+1]) # Right side of 0
polysection(-i-1, -i, col=cols[i+1]) # Left right of 0
# Black outline of bell curve
lines(x, dnorm(x))
# Bottom axis values, where sigma represents standard deviation and mu is the mean
axis(1, at=-3:3, labels=expression(-3*SD, -2*SD, -1*SD, Mean,
1*SD, 2*SD, 3*SD))
# Add percent densities to each division, between x and x+1
pd <- sprintf("%.1f%%", 100*(pnorm(1:4) - pnorm(0:3)))
pd2 <- c("34%", "13.5%", "2%", "0.5%")
text(c((0:3)+0.5,(0:-3)-0.5), c(0.16, 0.05, 0.04, 0.02), pd2, col=c("white","white","black","black"))
segments(c(-2.5, -3.5, 2.5, 3.5), dnorm(c(2.5, 3.5)), c(-2.5, -3.5, 2.5, 3.5), c(0.03, 0.01))
dev.off()
# of SD:
50th
84th
97.5th
99.5th
16th
2.5th
0.5th
Percentile:

8. ? Population The true mean BMI of Boston, Massachusetts Parameter
Sample
Researcher
The mean BMI of
a sample from
Boston, Massachusetts
Sample statistics

9. Sample variation
The whole population ?
1, 2, 3, 4, 5, 6
Researcher 1
Researcher 2
Researcher 3
Researcher 4
Researchers
2, 4
4, 6
1, 2
1, 6
Samples
3.0
5.0
1.5
3.5
Means

10. Central
limit
theorem

11. Central limit theorem
The means obtained from many samplings from the same population have the following properties
The distribution of the means is always normal if the sample size is big enough (above 120 or so), regardless of the population’s distribution
The mean of the sample means is equal to the population mean
The standard deviation of the sample means, known as the standard error of the mean (SEM) is inversely related to the sample size: if we repeat the experiment with a bigger sample size, the resultant histogram will be “slimmer”

12. Understanding CLT through simulation
Population size: 10000
Possible values: 0 through 9, 1000 each
True population mean: 4.50

13. 10000 Simulation scheme Sample A population of 10000 n=500 Sample mean
Frequency
Sample mean

14. Sample size = 500; # of draws = 10000SE
99%
SD = 0.13
95%
68%
±1 SE:
67.95%
±2 SE:
95.04%
±3 SE:
99.10%
Frequency
x <- rep(0:9,rep(1000,10))
m.out <- vector()
lower <- vector()
upper <- vector()
while (length(m.out)<10000) {
samp01 <- sample(x,500)
m01 <- mean(samp01)
se01 <- sd(samp01)/sqrt(500-1)
lower01 <- m *se01
upper01 <- m *se01
m.out <- c(m.out, m01)
lower <- c(lower, lower01)
upper <- c(upper, upper01) }
my.data <- data.frame(m.out, lower, upper)
my.data.500 <- my.data[order(m.out),]
par(mfrow=c(2,1))
dotplot(my.data.200$m.out,pch=16,dot.col="#ff6600",main="",xmin=3,xmax=6)
dotplot(my.data.500$m.out,pch=16,dot.col="#ff660090",main="",xmin=4,xmax=5)
abline(v=mean(x),col="#336699",lwd=2)
abline(v=4.5)
plot(my.data.2$m.out, 1:800, xlim=c(min(my.data.2$lower), max(my.data.2$upper)))
segments(my.data.2$lower,1:200, my.data.2$upper, 1:200)
abline(v=mean(x))
plot(density(m.out))
varx <- my.data.500$m.out
my.m <- mean(my.data.500$m.out)
my.s <- sd(my.data.500$m.out)
sum(varx>(my.m-1.00*my.s)&varx<(my.m+1.00*my.s))
sum(varx>(my.m-1.96*my.s)&varx<(my.m+1.96*my.s))
sum(varx>(my.m-2.58*my.s)&varx<(my.m+2.58*my.s))
4.5
Sample means

15. Characteristics for the distribution of means
In the previous slide, the mean 4.5 is the true population parameter, for which we have a Greek name, μ (mu)
Similarly, the SD 0.13 is the true population parameter, called σ (sigma) in Greek. We call this SD of means “standard error of means” (SEM) or “standard error” (SE)
SE can be estimated using sample SD:

16. Why bigger sample sizes are often better
Sample means
Sample means
Sample means
Sample size = 200
Sample size = 500
Sample size = 1000
SE = 0.20
SE = 0.13
SE = 0.08

17. Confidence interval

18. I got CLT, so now what?
The histogram can be viewed as a “probability distribution”
The sample mean from a researcher can be any pixel under the bell curve
How should we define “acceptably close” to the population mean?
95%
# # External package to generate four shades of blue
# library(RColorBrewer)
# cols <- rev(brewer.pal(4, "Blues"))
cols <- c("#ce1256", "#df65b0", "#d7b5d8", "#f1eef6")
# Sequence between -4 and 4 with 0.1 steps
x <- seq(-4, 4, 0.1)
png("c:\\temp\\normalcurve.png", width=2000, height=1000, res=250)
# Plot an empty chart with tight axis boundaries, and axis lines on bottom and left
plot(x, type="n", xaxs="i", yaxs="i", xlim=c(-4, 4), ylim=c(0, 0.4),
bty="l", xaxt="n", xlab="sample means", ylab="probability")
# Function to plot each coloured portion of the curve, between "a" and "b" as a
# polygon; the function "dnorm" is the normal probability density function
polysection <- function(a, b, col, n=11){
dx <- seq(a, b, length.out=n)
polygon(c(a, dx, b), c(0, dnorm(dx), 0), col=col, border=NA)
# draw a white vertical line on "inside" side to separate each section
segments(a, 0, a, dnorm(a), col="white") }
# Build the four left and right portions of this bell curve
for(i in 0:3){
polysection( i, i+1, col=cols[i+1]) # Right side of 0
polysection(-i-1, -i, col=cols[i+1]) # Left right of 0
# Black outline of bell curve
lines(x, dnorm(x))
# Bottom axis values, where sigma represents standard deviation and mu is the mean
axis(1, at=-3:3, labels=expression(-3*SE, -2*SE, -1*SE, mu,
1*SE, 2*SE, 3*SE))
# Add percent densities to each division, between x and x+1
pd <- sprintf("%.1f%%", 100*(pnorm(1:4) - pnorm(0:3)))
pd2 <- c("34%", "13.5%", "2%", "0.5%")
text(c((0:3)+0.5,(0:-3)-0.5), c(0.16, 0.05, 0.04, 0.02), pd2, col=c("white","white","black","black"))
segments(c(-2.5, -3.5, 2.5, 3.5), dnorm(c(2.5, 3.5)), c(-2.5, -3.5, 2.5, 3.5), c(0.03, 0.01))
dev.off()

19. The confidence interval
95%
# # External package to generate four shades of blue
# library(RColorBrewer)
# cols <- rev(brewer.pal(4, "Blues"))
cols <- c("#ce1256", "#df65b0", "#d7b5d8", "#f1eef6")
# Sequence between -4 and 4 with 0.1 steps
x <- seq(-4, 4, 0.1)
png("c:\\temp\\normalcurve.png", width=2000, height=1000, res=250)
# Plot an empty chart with tight axis boundaries, and axis lines on bottom and left
plot(x, type="n", xaxs="i", yaxs="i", xlim=c(-4, 4), ylim=c(0, 0.4),
bty="l", xaxt="n", xlab="sample means", ylab="probability")
# Function to plot each coloured portion of the curve, between "a" and "b" as a
# polygon; the function "dnorm" is the normal probability density function
polysection <- function(a, b, col, n=11){
dx <- seq(a, b, length.out=n)
polygon(c(a, dx, b), c(0, dnorm(dx), 0), col=col, border=NA)
# draw a white vertical line on "inside" side to separate each section
segments(a, 0, a, dnorm(a), col="white") }
# Build the four left and right portions of this bell curve
for(i in 0:3){
polysection( i, i+1, col=cols[i+1]) # Right side of 0
polysection(-i-1, -i, col=cols[i+1]) # Left right of 0
# Black outline of bell curve
lines(x, dnorm(x))
# Bottom axis values, where sigma represents standard deviation and mu is the mean
axis(1, at=-3:3, labels=expression(-3*SE, -2*SE, -1*SE, mu,
1*SE, 2*SE, 3*SE))
# Add percent densities to each division, between x and x+1
pd <- sprintf("%.1f%%", 100*(pnorm(1:4) - pnorm(0:3)))
pd2 <- c("34%", "13.5%", "2%", "0.5%")
text(c((0:3)+0.5,(0:-3)-0.5), c(0.16, 0.05, 0.04, 0.02), pd2, col=c("white","white","black","black"))
segments(c(-2.5, -3.5, 2.5, 3.5), dnorm(c(2.5, 3.5)), c(-2.5, -3.5, 2.5, 3.5), c(0.03, 0.01))
dev.off()

20. If we put a CI on every sample
mean, about 95% of them would include the true mean.
The two red ones are the “unlucky” samples which do not include the true mean.
True mean

21. Interpretation of a confidence interval
The mean and 95% confidence interval (CI) of the blood glucose of a sample is: 140 mg/dl (95%CI: 120, 160)
We are 95% confident that the interval 120 and 160 mg/dl includes the true population mean. Our best estimate is 140 mg/dl (i.e. the sample mean)
Why only 95% certain? Because the sample mean can be, unfortunately, an extreme one beyond ± 2 SE (the blue zones)

22. Some common CIs and their z-score multipliers
There are two numbers in a confidence interval: the lower and upper confidence limits
90%CI:
Mean ± 1.65  SE
95%CI:
Mean ± 1.96  SE
2.00 is an approximation, 1.96 is recommended
The most commonly used criterion
99%CI:
Mean ± 2.58  SE
The more certain we want the interval to include the true mean, the wider the CI becomes
“I am 100% certain that the true mean is between –∞ and ∞.”

23. How to narrow down confidence interval?
Lower our certainty by opting for, say, a 90%CI instead of a 95%CI
Decrease sample standard deviation (for instance, using a more accurate measurement device)
Increase sample size

24. Are confidence intervals always symmetric?
Not in all occasions. CIs for untransformed continuous variables are symmetric
However, CIs for other statistics such as odds ratios and relative risks are calculated on logarithmic scale. When back-transformed to the ratios, the interval will be asymmetric
“Multivariable analysis revealed a more than 2-fold increase in the risk of total stroke among men with job strain (combination of high job demand and low job control) (hazard ratio, 2.73; 95% confidence interval, )”