Foundations for statistical inference - Sampling distributions
Olya Fomicheva
10/13/2017
In this lab, we investigate the ways in which the statistics from a random sample of data can serve as point estimates for population parameters. We’re interested in formulating a sampling distribution of our estimate in order to learn about the properties of the estimate, such as its distribution.
The data
We consider real estate data from the city of Ames, Iowa. The details of every real estate transaction in Ames is recorded by the City Assessor’s office. Our particular focus for this lab will be all residential home sales in Ames between 2006 and 2010. This collection represents our population of interest. In this lab we would like to learn about these home sales by taking smaller samples from the full population. Let’s load the data.
download.file("http://www.openintro.org/stat/data/ames.RData", destfile = "ames.RData")
load("ames.RData")We see that there are quite a few variables in the data set, enough to do a very in-depth analysis. For this lab, we’ll restrict our attention to just two of the variables: the above ground living area of the house in square feet (Gr.Liv.Area) and the sale price (SalePrice). To save some effort throughout the lab, create two variables with short names that represent these two variables.
area <- ames$Gr.Liv.Area
price <- ames$SalePriceLet’s look at the distribution of area in our population of home sales by calculating a few summary statistics and making a histogram.
summary(area)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 334 1126 1442 1500 1743 5642hist(area)
- Describe this population distribution.
The distribution of this population is unimodel since it has one clear peak. The distribution of area is not symmetric and it’s right skewed. Mean is greater than the median.
The unknown sampling distribution
In this lab we have access to the entire population, but this is rarely the case in real life. Gathering information on an entire population is often extremely costly or impossible. Because of this, we often take a sample of the population and use that to understand the properties of the population.
If we were interested in estimating the mean living area in Ames based on a sample, we can use the following command to survey the population.
samp1 <- sample(area, 50)This command collects a simple random sample of size 50 from the vector area, which is assigned to samp1. This is like going into the City Assessor’s database and pulling up the files on 50 random home sales. Working with these 50 files would be considerably simpler than working with all 2930 home sales.
- Describe the distribution of this sample. How does it compare to the distribution of the population?
summary(samp1)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 808 1116 1468 1474 1732 2646hist(samp1)
The distribution of this sample is unimodel too. The distribution of the sample is not symmetric and it’s right skewed too. Mean is greater than the median.
If we’re interested in estimating the average living area in homes in Ames using the sample, our best single guess is the sample mean.
mean(samp1)## [1] 1473.62Depending on which 50 homes you selected, your estimate could be a bit above or a bit below the true population mean of 1499.69 square feet. In general, though, the sample mean turns out to be a pretty good estimate of the average living area, and we were able to get it by sampling less than 3% of the population.
- Take a second sample, also of size 50, and call it
samp2. How does the mean ofsamp2compare with the mean ofsamp1?
samp2 <- sample(area, 50)
m1<-mean(samp1)
m2<-mean(samp2)
abs(m1-m2)/((m1+m2)/2)## [1] 0.0348196The different between the mean of samp2 and samp1 is around 5%. However, the code above produces different results.
Suppose we took two more samples, one of size 100 and one of size 1000. Which would you think would provide a more accurate estimate of the population mean?
I would assume that the larger sample size will produce more accurate estimate of the population mean.
Let’s check.
samp3 <- sample(area, 100)
samp4 <- sample(area, 1000)
m3<-mean(samp3)
m4<-mean(samp4)
m<-mean(area)
abs(m-m3)/((m+m3)/2)## [1] 0.007798859abs(m-m4)/((m+m4)/2)## [1] 0.003940739I ran the code a few times and got different results. However, most of the time the mean of the sample size of 1000 is closer to the population mean than the sample size of 100.
Not surprisingly, every time we take another random sample, we get a different sample mean. It’s useful to get a sense of just how much variability we should expect when estimating the population mean this way. The distribution of sample means, called the sampling distribution, can help us understand this variability. In this lab, because we have access to the population, we can build up the sampling distribution for the sample mean by repeating the above steps many times. Here we will generate 5000 samples and compute the sample mean of each.
sample_means50 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 50)
sample_means50[i] <- mean(samp)
}
hist(sample_means50)
If you would like to adjust the bin width of your histogram to show a little more detail, you can do so by changing the breaks argument.
hist(sample_means50, breaks = 25)
Here we use R to take 5000 samples of size 50 from the population, calculate the mean of each sample, and store each result in a vector called sample_means50. On the next page, we’ll review how this set of code works.
- How many elements are there in
sample_means50?
There are 5000 elements in sample_means50.
Describe the sampling distribution, and be sure to specifically note its center.
summary(sample_means50)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1259 1452 1498 1499 1545 1822summary(area)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 334 1126 1442 1500 1743 5642The mean and the median of sample_means50 is very close to the mean and the median of the population. Moreover, the mean and the median of sample_means50 are almost equal. Furthermore, the distribution of sample_means50 is bell-shaped and unimodal. It might be normally distributed.
Would you expect the distribution to change if we instead collected 50,000 sample means?
I expect that sample mean will not change a lot if we instead collect 50,000 sample means. Let’s check.
#initialize a vector of 50000 zeros
sample_means50_2 <- rep(NA, 50000)
#ran a loop that takes a sample of size 50 from `area`
for(i in 1:50000){
samp <- sample(area, 50)
sample_means50_2[i] <- mean(samp)
}hist(sample_means50_2, breaks = 25)
summary(sample_means50_2)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1242 1451 1498 1500 1546 1831summary(sample_means50)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1259 1452 1498 1499 1545 1822The mean and the media of 5000 sample means and 50000 sample means are almost equal.
Interlude: The for loop
Let’s take a break from the statistics for a moment to let that last block of code sink in. You have just run your first for loop, a cornerstone of computer programming. The idea behind the for loop is iteration: it allows you to execute code as many times as you want without having to type out every iteration. In the case above, we wanted to iterate the two lines of code inside the curly braces that take a random sample of size 50 from area then save the mean of that sample into the sample_means50 vector. Without the for loop, this would be painful:
sample_means50 <- rep(NA, 5000)
samp <- sample(area, 50)
sample_means50[1] <- mean(samp)
samp <- sample(area, 50)
sample_means50[2] <- mean(samp)
samp <- sample(area, 50)
sample_means50[3] <- mean(samp)
samp <- sample(area, 50)
sample_means50[4] <- mean(samp)and so on…
With the for loop, these thousands of lines of code are compressed into a handful of lines. We’ve added one extra line to the code below, which prints the variable i during each iteration of the for loop. Run this code.
sample_means50 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 50)
sample_means50[i] <- mean(samp)
print(i)
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## [1] 5000Let’s consider this code line by line to figure out what it does. In the first line we initialized a vector. In this case, we created a vector of 5000 zeros called sample_means50. This vector will will store values generated within the for loop.
The second line calls the for loop itself. The syntax can be loosely read as, “for every element i from 1 to 5000, run the following lines of code”. You can think of i as the counter that keeps track of which loop you’re on. Therefore, more precisely, the loop will run once when i = 1, then once when i = 2, and so on up to i = 5000.
The body of the for loop is the part inside the curly braces, and this set of code is run for each value of i. Here, on every loop, we take a random sample of size 50 from area, take its mean, and store it as the \(i\)th element of sample_means50.
In order to display that this is really happening, we asked R to print i at each iteration. This line of code is optional and is only used for displaying what’s going on while the for loop is running.
The for loop allows us to not just run the code 5000 times, but to neatly package the results, element by element, into the empty vector that we initialized at the outset.
- To make sure you understand what you’ve done in this loop, try running a smaller version. Initialize a vector of 100 zeros called
sample_means_small. Run a loop that takes a sample of size 50 fromareaand stores the sample mean insample_means_small, but only iterate from 1 to 100. Print the output to your screen (typesample_means_smallinto the console and press enter).
#initialize a vector of 100 zeros
sample_means_small <- rep(NA, 100)
#ran a loop that takes a sample of size 50 from `area`
for(i in 1:100){
sample <- sample(area, 50)
sample_means_small[i] <- mean(sample)
print(i)
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## [1] 100How many elements are there in this object called sample_means_small? What does each element represent?
The object sample_means_small contains 100 elements. Each element represents the mean of the sample of size of 50.
Sample size and the sampling distribution
Mechanics aside, let’s return to the reason we used a for loop: to compute a sampling distribution, specifically, this one.
hist(sample_means50)
The sampling distribution that we computed tells us much about estimating the average living area in homes in Ames. Because the sample mean is an unbiased estimator, the sampling distribution is centered at the true average living area of the the population, and the spread of the distribution indicates how much variability is induced by sampling only 50 home sales.
To get a sense of the effect that sample size has on our distribution, let’s build up two more sampling distributions: one based on a sample size of 10 and another based on a sample size of 100.
sample_means10 <- rep(NA, 5000)
sample_means100 <- rep(NA, 5000)
for(i in 1:5000){
samp <- sample(area, 10)
sample_means10[i] <- mean(samp)
samp <- sample(area, 100)
sample_means100[i] <- mean(samp)
}Here we’re able to use a single for loop to build two distributions by adding additional lines inside the curly braces. Don’t worry about the fact that samp is used for the name of two different objects. In the second command of the for loop, the mean of samp is saved to the relevant place in the vector sample_means10. With the mean saved, we’re now free to overwrite the object samp with a new sample, this time of size 100. In general, anytime you create an object using a name that is already in use, the old object will get replaced with the new one.
To see the effect that different sample sizes have on the sampling distribution, plot the three distributions on top of one another.
par(mfrow = c(3, 1))
xlimits <- range(sample_means10)
hist(sample_means10, breaks = 20, xlim = xlimits)
hist(sample_means50, breaks = 20, xlim = xlimits)
hist(sample_means100, breaks = 20, xlim = xlimits)
The first command specifies that you’d like to divide the plotting area into 3 rows and 1 column of plots (to return to the default setting of plotting one at a time, use par(mfrow = c(1, 1))). The breaks argument specifies the number of bins used in constructing the histogram. The xlim argument specifies the range of the x-axis of the histogram, and by setting it equal to xlimits for each histogram, we ensure that all three histograms will be plotted with the same limits on the x-axis.
- When the sample size is larger, what happens to the center? What about the spread?
summary(area)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 334 1126 1442 1500 1743 5642summary(sample_means10)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1004 1383 1487 1496 1595 2175summary(sample_means50)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1263 1449 1497 1499 1546 1750summary(sample_means100)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1333 1467 1500 1501 1534 1681When the sample size is larger the distribution is more symmetric and the mean is closer to the median. In the other words, the samples of larger sizes is more normally distributed than the samples of smaller sizes. Also, means of the samples of larger sizes is closer to the population mean.
On your own
So far, we have only focused on estimating the mean living area in homes in Ames. Now you’ll try to estimate the mean home price.
- Take a random sample of size 50 from
price. Using this sample, what is your best point estimate of the population mean?
sample <- rep(price, 50)
#the mean of the sample of size 50
mean(sample)## [1] 180796.1- Since you have access to the population, simulate the sampling distribution for \(\bar{x}_{price}\) by taking 5000 samples from the population of size 50 and computing 5000 sample means. Store these means in a vector called
sample_means50.
#initialize a vector of 5000 zeros
sample_means50 <- rep(NA, 5000)
#ran a loop that takes a sample of size 50 from `area`
for(i in 1:5000){
samp <- sample(price, 50)
sample_means50[i] <- mean(samp)
}Plot the data, then describe the shape of this sampling distribution. Based on this sampling distribution, what would you guess the mean home price of the population to be?
hist(sample_means50, breaks=25)
The distribution of sample_means50 is unimodel, pretty symmetric and bell-shaped. The distribution of sample_means50 might be normally distributed.
Finally, calculate and report the population mean.
#the mean of the sample of size 50
mean(sample_means50)## [1] 180792.7#the mean of the population
mean(price)## [1] 180796.1The sample mean is very close to the population mean.
- Change your sample size from 50 to 150, then compute the sampling distribution using the same method as above, and store these means in a new vector called
sample_means150.
#initialize a vector of 5000 zeros
sample_means150 <- rep(NA, 5000)
#ran a loop that takes a sample of size 50 from `area`
for(i in 1:5000){
samp <- sample(price, 150)
sample_means150[i] <- mean(samp)
}
hist(sample_means150, breaks=25)
Describe the shape of this sampling distribution, and compare it to the sampling distribution for a sample size of 50.
hist(sample_means150, breaks=25)
The distribution of sample_means150 is unimodel, pretty symmetric and bell-shaped. The distribution of sample_means150 might be normally distributed. It looks pretty much as the distribution of sample_means50
Based on this sampling distribution, what would you guess to be the mean sale price of homes in Ames?
Since we already found out that the samples of larger sizes provide more accurate mean estimate I would use the mean of sample_means150 to predict the mean of the population.
Let’s check.
mean(price)## [1] 180796.1mean(sample_means50)## [1] 180792.7mean(sample_means150)## [1] 180784.8The mean of sample_means150 is closer to the population mean than the mean of sample_means50.
- Of the sampling distributions from 2 and 3, which has a smaller spread? If we’re concerned with making estimates that are more often close to the true value, would we prefer a distribution with a large or small spread?
summary(sample_means50)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 146537 172980 180187 180793 188152 231049summary(sample_means150)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 158744 176345 180664 180785 185017 206257The distribution of sample_means50 is more spreaded than the distribution of sample_means150 because the distribution of sample_means50 has a greater difference between the lower and upper quartile and the difference between the maximum and the minimum.
Since we already figured out that the mean of sample_means150 estimates the mean of the population more accuratly I would say that the distribution with a smaller spread provides more accurate population mean estimates.
This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.