sampling distributions - github pages · every statistic has a sampling distribution, but not all...
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Sampling Distributions
David Gerard
2017-09-18
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Learning Objectives
• Statistics/parameters
• Sampling Distribution
• Sections 1.3.1, 1.3.2, 1.3.3, 4.1, 4.4 in DBC
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Population and Sample
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Recall: Population and Sample
population
A population is a set of cases (observational units) about which
information is wanted.
sample
A sample is a subset of the population.
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Examples
• We want to know demographic information of Americans so
we randomly select a group of 50 Americans and ask them a
bunch of questions. (sample? population?)
• We are interested in the quality of anchovies so we take 10
cans and taste them. (sample? population?)
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Why sample?
• It is expensive/impossible to collect information on the whole
population (when this is done it is called a census).
• Even when a census is performed, it is often less accurate
than a well-designed sample (hard to collect information on
everything, so this introduces biases into the observations you
see).
• With a large enough sample, we can be pretty sure of the
information we want on the population, making taking a
census unnecessary.
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Random Sampling
• Often, samples are collected randomly to remove bias.
• bias is where some cases are more likely to be in the sample
than other cases.
• E.g. some political pollsters mostly call landlines, which biases
the sample toward older individuals. What could be the issue
here?
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Statistics and Parameters
parameter
A parameter is a number that describes a population. It is
usually unknown and what we want information on. People
usually use greek letter µ, σ, ρ to represent parameters.
statistic
A statistic is a number that describes a sample. It is known and
is used to estimate a population parameter. People usually use
latin letters x̄ , s, r to represent statistics.
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Example
• We want to know the average height of U.S. males so we
measure the average height of a sample of 50 U.S. males and
came up with 5’11”. (parameter? statistic?)
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The sample mean
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Recall: NBA Data
Player statistics for the 2016-2017 season of the NBA
• player The name of the player.
• pts The total points for the season
• two pp Two point field goal percentage.
• three pp Three point field goal percentage.
• Many others ...
• Here, I only kept players that attempted at least 20 two-point
and 20 three-point field goals.
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Recall: NBA Data
library(tidyverse)
nba <- read_csv("../../data/nba2016.csv") %>%
filter(two_pa >= 20, three_pa >= 20) %>%
select(player, pts, two_pp, three_pp)
glimpse(nba)
Observations: 337
Variables: 4
$ player <chr> "Russell Westbrook", "James Harden", "...
$ pts <int> 2558, 2356, 2199, 2099, 2061, 2024, 20...
$ two_pp <dbl> 0.459, 0.530, 0.528, 0.524, 0.582, 0.4...
$ three_pp <dbl> 0.343, 0.347, 0.379, 0.299, 0.367, 0.3...
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The inference problem
• Suppose I want to know the average total points of NBA
players. However, I can only collect a sample of 5 players.
nsamp <- 5
samd <- sample(nba$pts, size = nsamp)
samd
[1] 709 479 130 1028 142
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Point Estimate
Of course, we know the actual mean number of points µ because
we have the entire population.
mean(nba$pts)
[1] 666.4
A good estimate might be the average of the sample x̄
mean(samd)
[1] 497.6
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Point Estimate
The sample average here is a point estimate of the population
mean.
point estimate
A point estimate is a single number used to estimate a
population parameter.
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Aside
• How would you estimate the population median?
• How would you estimate the population standard deviation?
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A different sample
However, since the sample was drawn at random, we could have
obtained a different sample, and so a different point estimate.
samd <- sample(nba$pts, size = nsamp)
samd
[1] 94 419 435 1742 1025
mean(samd)
[1] 743
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And another sample
samd <- sample(nba$pts, size = nsamp)
samd
[1] 1071 381 689 282 551
mean(samd)
[1] 594.8
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And another sample
samd <- sample(nba$pts, size = nsamp)
samd
[1] 327 59 700 281 107
mean(samd)
[1] 294.8
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And another sample
samd <- sample(nba$pts, size = nsamp)
samd
[1] 1002 425 1196 864 689
mean(samd)
[1] 835.2
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Sampling distribution
• With every sample we are getting a different x̄ .
• We can ask what possible values x̄ can take and how often it
takes those values.
• That is, we can ask about x̄ ’s distribution.
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Sampling distribution
sampling distribution
A sampling distribution is the distribution of a sample statistic.
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Repeat sample 1000 times.
itermax <- 1000
xbar_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
xbar_vec[index] <- mean(samd)
}
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Plot the results
hist(xbar_vec, main = "")
abline(v = mean(nba$pts), lty = 2, col = 2, lwd = 2)
legend("topright", "pop mean", lty = 2, col = 2, lwd = 2)
xbar_vec
Fre
quen
cy
500 1000 1500
050
100
150
pop mean
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The sampling distribution
• The sample mean has the correct center.
• There is a lot of variability about that center though.
sd(xbar_vec)
[1] 227.7
standard error
The standard deviation associated with a point estimate is called
a standard error.
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What if we have a bigger sample
nsamp <- 10
xbar10_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
xbar10_vec[index] <- mean(samd)
}sd(xbar10_vec)
[1] 156.3
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What if we have a bigger sample
nsamp <- 50
xbar50_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
xbar50_vec[index] <- mean(samd)
}sd(xbar50_vec)
[1] 66.51
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What if we have a bigger sample
nsamp <- 100
xbar100_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
xbar100_vec[index] <- mean(samd)
}sd(xbar100_vec)
[1] 42.61
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Standard error decreases with larger sample sizes!
400
800
1200
5 10 50 100
SampleSize
xbar
Dashed red line is population mean.
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Standard error
standard error
Given n independent observations from a population with
standard deviation σ, the standard error of the sample mean is
equal to
SE =σ√n.
• Since σ is generally unknown, we estimate SE with s/√n,
where s is the sample standard deviation.
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What happens as sample size increases?
Histogram of points
Points
Den
sity
0 500 1000 1500 2000 2500
0.00
000.
0006
0.00
12
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What happens as sample size increases?
Histogram of xbar
Mean Points, n = 5
Den
sity
200 400 600 800 1000 1200 1400 1600
0.00
000.
0010
0.00
20
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What happens as sample size increases?
Histogram of xbar
Mean Points, n = 5
Den
sity
500 600 700 800 900
0.00
00.
002
0.00
40.
006
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Wat happens as the sample size increases?
n = 1
qqnorm(nba$pts)
qqline(nba$pts)
−3 −2 −1 0 1 2 3
050
015
0025
00
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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Wat happens as the sample size increases?
n = 5
qqnorm(xbar_vec)
qqline(xbar_vec)
−3 −2 −1 0 1 2 3
200
600
1000
1400
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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Wat happens as the sample size increases?
n = 50
qqnorm(xbar50_vec)
qqline(xbar50_vec)
−3 −2 −1 0 1 2 3
500
600
700
800
900
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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General result
• In general, sample means converge to a normal distribution as
the sample size increases.
• Many other statistics do this as well (proportions, medians,
standard devaitions).
• We will provide a heuristic proof of this result later.
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Skewed distributions
For highly skewed distributions, it takes more samples for normality
to be a good approximation.
data(email, package = "openintro")
hist(email$num_char)
Histogram of email$num_char
email$num_char
Fre
quen
cy
0 50 100 150 200
010
0020
0030
00
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Skewed distributions, n = 5
Histogram of xvec
xvec
Fre
quen
cy
0 10 20 30 40 50
010
020
030
0
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Skewed distributions, n = 10
Histogram of xvec
xvec
Fre
quen
cy
5 10 15 20 25 30
050
100
150
200
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Skewed distributions, n = 50
Histogram of xvec
xvec
Fre
quen
cy
6 8 10 12 14 16 18
050
100
150
200
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Skewed distributions, n = 100
Histogram of xvec
xvec
Fre
quen
cy
6 8 10 12 14 16
050
150
250
40
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More sampling distributions
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Every statistic has a sampling distribution
nsamp <- 50
sd_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
sd_vec[index] <- sd(samd)
}
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Every statistic has a sampling distribution
hist(sd_vec, main = "Sampling distribution
of sample standard deviation",
xlab = "sd")
Sampling distribution of sample standard deviation
sd
Fre
quen
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300 400 500 600 700
050
150
250
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Every statistic has a sampling distribution
nsamp <- 50
med_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
med_vec[index] <- median(samd)
}
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Every statistic has a sampling distribution
hist(med_vec, main = "Sampling distribution
of sample median",
xlab = "median")
Sampling distribution of sample median
median
Fre
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300 400 500 600 700 800
010
020
030
0
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Every statistic has a sampling distribution, but not all sampling
distributions converge to a normal
nsamp <- 50
max_vec <- rep(NA, itermax)
for (index in 1:itermax) {samd <- sample(nba$pts, size = nsamp)
max_vec[index] <- max(samd)
}
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Every statistic has a sampling distribution, but not all sampling
distributions converge to a normal
hist(max_vec, main = "Sampling distribution
of sample maximum",
xlab = "max")
Sampling distribution of sample maximum
max
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1200 1400 1600 1800 2000 2200 2400 2600
050
150
250
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