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5.03 The sampling distribution

Methods for summarizing sample data are called descriptive statistics. However, in most studies we’re not interested in samples, but in underlying populations. If we employ data obtained from a sample to draw conclusions about a wider population, we are using methods of inferential statistics. It is therefore of essential importance that you know how you should draw samples. In this module we’ll pay attention to good sampling methods as well as some poor practices. To draw conclusions about the population a sample is from, researchers make use of a probability distribution that is very important in the world of statistics: the sampling distribution. We’ll discuss sampling distributions in great detail and compare them to data distributions and population distributions. We’ll look at the sampling distribution of the sample mean and the sampling distribution of the sample proportion.

统计基础

阿姆斯特丹大学

课程 3（共 5 门，社会科学的方法和统计专项课程）

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

查看授课大纲

您将学习的技能

Statistics, Confidence Interval, Statistical Hypothesis Testing, R Programming

教学方

Matthijs Rooduijn
Dr.
Emiel van Loon
Assistant Professor

脚本

Researchers often use a sample to draw inferences about a population that sample

is from.

To do that, they make use of a probability distribution that is

very important in the world of statistics, the sampling distribution.

In this video, I will explain what the sampling distribution is.

Pay close attention because the sampling distribution is the link

that helps researchers to draw conclusions about a population

on the basis of only one sample.

One note before we start, in this video,

we'll pretend that we know what a population looks like.

In the actual research practice, you'll never know that, but

this step is necessary to understand inferential statistics later on.

Okay, here we go.

Let's assume that a group of Scandinavian hipsters organizes the Festival

of the Beards on one of the small islands near Oslo, the capital of Norway.

The audience of the festival is, as you might have expected, men with beards.

The organization has sold about 5,000 tickets and

offers free transportation to the festival island.

All people with a ticket gather in the Harbour of Oslo.

The organization distributes all of them randomly over boats that

carry the passengers to the island.

Every boat carries 30 festival goers.

Now, one of the boats gets lost in the maze of tiny Norwegian islands.

To make matters even worse, the mobile phone network has crashed, so

it is impossible for the organization to reach the captain of the boat, or

for the passengers to reach the organization.

So the hipster organization decides to send some employees to find the lost boat.

You are one of those employees.

After half an hour, you see a crashed boat with about 30 people, yes,

you think, finally I've found them.

You're about to send a message by walkie talkie

to the organization that the lost boat has been found.

Then, however, you take a look at the people in the boat one more time.

You see that they're comprised of families with young children.

This is strange because you'd expect that a boat with randomly selected people who

are going to the Festival of the Beards would be, well, men with beards,

not young families.

You decide that it is very unlikely that this is the boat you're looking for and

decide to continue your search.

Later on, it turned out that this was a good decision.

The boat you encountered was a boat carrying people to a family park on

one of the other islands.

Why did I tell you this story?

Well, if you understand this line of reasoning,

you'll understand the basic idea behind the sampling distribution.

The reasoning goes like this.

If you draw a simple random sample from a population, it is very unlikely that

the sample will strongly differ from the population from which it is drawn.

In our case, the people going to the Festival of the Beards,

they form our population.

A boat with 30 randomly selected people from this population

forms a simple random sample.

In fact, all of the boats traveling between the Harbour of Oslo and

the festival island could be seen as simple random samples.

Of course, every boat is different from the next, but

a large majority of them will contain a large proportion of bearded men.

It is very unlikely that a boat will contain various young families.

Of course, it's possible there will be some families attending the Festival of

the Beards, but finding a randomly composed boat

completely consisting of young families is highly improbable.

Suppose you decide to measure the mean beard length for every boat.

In every boat, there are 30 people going to the Festival of the Beards.

Now, imagine you know that the mean beard length in the population of 5,000

festival goers is 10.3 millimeters, so mu is 10.3 millimeters.

You also know that beard length has a bell-shaped distribution

in the population.

In one boat, you'll find a mean beard length of,

say, 9.4 millimeters, and in another one say 10.8 millimeters.

However, it is very unlikely that you will encounter a boat with a mean beard length

as low as 3.4 millimeters, or

a boat with a mean beard length as high as 19.2 millimeters.

Because these boat means could be seen as sample means, we symbolize them by x-bars.

Now imagine you look at three boats.

This is what the probability distribution could look like.

In one boat, the mean beard length is 9.9 millimeters, in one 10.7, and in one 10.2.

We have three boats, so every mean has a probability of 0.33.

Now imagine you look at 17 boats, that would look something like this.

Now you look at 40 boats, and now at 100 boats.

You can see that the distribution of these beard length means

increasingly looks like a bell-shape distribution.

Moreover, you can see that the mean of the distribution is about 10.3 millimeters,

exactly the population mean.

This is not strange if you think about it.

You would expect that in most cases the mean in a boat is close to

population mean.

In one boat, the mean will be a little higher, and in another a little lower.

However, if you look at many boats, you would expect that the mean of

all these different boat-means would be be the population mean.

Now imagine your population consists of all Norwegian men.

You know that the mean beard length in this population is 1.22 milimeters, and

that the variable has a bell-shaped distribution in the population.

If you draw a simple random sample of 30 respondents,

you will find the mean close to this value, say, 1.34 milimeters.

If you draw another random sample,

the mean will be close to the population value of 1.22 as well.

Say it is 1.19 milimeters.

If you do that five times, you will get five different values,

which are all close to the population value.

If we draw 20 samples,

the distribution of the sample means will be approximately bell-shaped.

If we do it 100 times, it will be even more strongly bell-shaped.

And if we draw an infinite number of samples,

the distribution will be perfectly bell-shaped, and

the mean will be exactly 1.22 millimeters, which is the population value.

We call this distribution the sampling distribution of the sample mean.

It is a distribution that you get if you draw an infinite number of samples

from your population and compute the mean of all the collected sample means.

For now, you need to realize that in the actual research practice you will never,

never collect an infinite number of samples from a certain population.

However, it is important that you understand that, if you would do that,

the mean of the distribution would be equal to the mean in the population, and

that we would call this distribution the sampling distribution of the sample mean.

Don't confuse this with the sample or data distribution,

which is just the distribution of scores in the one sample that was actually drawn,

and for which data were actually collected.

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