5.05 Three distributions

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Do curso por University of Amsterdam

Basic Statistics

2059 classificações

University of Amsterdam

Basic Statistics

2059 classificações

Curso 3 de 5 em Specialization Métodos e Estatística Aplicados às Ciências Sociais

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.

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Sampling Distributions

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.

5.03 The sampling distribution7:10

5.04 The central limit theorem7:24

5.05 Three distributions7:16

Conheça os instrutores

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

Many social, political, and religious groups have their own sacred text.

Hipsters, also have an almost holy canonical text.

It's a book called On the Road.

We are interested in the question,

how much time hipsters in New York have spent reading On the Road.

Assume we know that in the entire population,

the mean time hipsters have spent reading the book is 943 minutes.

You also know, that the population standard deviation equals 212 minutes.

You have drawn a simple random sample of 200 respondents from this population.

The mean reading time in this sample, equals 867 minutes.

The standard deviation in this sample is 188 minutes.

In this video,

I will talk about three distributions that are important for our research project.

The population distribution, the data or sample distribution, and

the sampling distribution.

I will also show you, how you can compute the probabilities of selecting individuals

with particular scores.

And samples with particular sample means from this population.

As said, three distributions are of importants here.

The first one, is the population distribution.

It looks like this.

It's approximately bell-shaped.

I've already told you what the mean is, 943.

The subjects in the population are all New York hipsters.

So, the mean means that if we add up the On the Road reading times of all New York

hipsters and that divide that, by the total number of New York hipsters.

We get a value of 943 minutes.

The population mean is symbolized by mu.

The standard deviation in the population, symbolized by sigma, measures

the variability of all the individual reading times in the population,

around the mean reading time.

It equals 212 minutes.

The second distribution is the data or sample distribution.

It is the distribution of the sample data.

This is what it looks like.

It is, just like the population distribution, approximately bell-shaped.

The cases here, are the 200 sampled respondents.

The mean of this distribution is symbolized by x-bar.

It equals 867 minutes.

As you can see,

this mean is not very far removed from the population mean of 943 minutes.

The standard deviation in the sample is 188 minutes.

The mean and standard deviation are symbolized by Roman characters,

because they are sample statistics.

The mean and standard deviation in the population are symbolized by

Greek symbols, because they are population parameters.

The third distribution, is the sampling distribution of the sampling mean.

And this is what it looks like.

Because of the central limit theorem, it is normally distributed.

The cases in this distribution are not individuals, but an indefinite number of

samples of 200 respondents from a population of New York hipsters.

The mean of the sampling distribution of the sample mean,

is the mean of these infinite sample means.

The value of the mean of the sampling distribution,

equals the mean of the population distribution.

Which was 943 minutes.

To show that we are talking about the mean of the sampling distribution, we add x-bar

to indicate that it is a mean of sample means and not a mean of individual scores.

The standard deviation of the sampling distribution, equals the standard

deviation of the population, divided by the square root of n.

That makes 212, divided by the square root of 200, equals 15.

You need to remember that this third distribution,

the sampling distribution, is a theoretical distribution.

We don't actually collect an infinite number of samples.

That's not even possible.

It's not necessary either, because we know what the sampling distribution looks like,

as long as we know the values of the mean and

the standard deviation in the population.

The very nice thing about normal distributions,

is that we can find probabilities by changing original scores into z scores.

And by then employing the z table.

Now, if we would like to know what the probabilities are,

of selecting random samples or subjects from a population,

we can apply this logic to sampling distributions

and as long as they are normally distributed to population distributions.

Now image you select a random hipster from the population.

What is the probability that this hipster has a reading time of 1,000

minutes or more?

Well, first, we want to know how many standard deviations,

a person with a reading time of 1000 is removed form the mean.

So, we have to compute this person's z score in the population.

This is the formula.

And so the z score is 1000, minus 943,

divided by 212, equals 0.27.

We're interested in the area, to the right of this value.

If we look it up in our z table, we find that the chance of selecting a person with

a reading time of 1,000 minutes or more, is 39%.

Now, imagine we draw a simple random sample of n is 200 from the population.

What is the probability that the sample mean is 1,000 minutes or higher?

Now, pay close attention.

This is a completely different question.

We're not talking about selecting a specific person from the population.

Instead, we're talking about a statistic

based on a specific sample from the population.

Therefore, we don't make use of the population distribution, but

of the sampling distribution of the sample mean.

In general, the procedure is the same, but

now we make use of other means and standard deviations.

So, this is how we compute that z score now.

We subtract the mean of the sampling distribution of the sample mean,

from the same mean we're interested in.

That makes 1000 minus 943.

We divide, by the standard deviation of the sampling distribution.

That sigma, divided by the square root of n.

That is 212, divided by the square root of 200.

That's 15.

So, 1000, minus 943, divided by 15.

That gives a z score of 3.8.

If we look it up in our z table, we find that the chance of drawing a sample with

the mean reading time of one thousand minutes or more is 0.01 percent.

So, you need to be very careful, when deciding which distribution to use.

If you're interested in selecting individual subjects,

you should use the population distribution, but

if you're interested in selecting samples, you should use the sampling distribution.

In the actual research practice, confusion between a population and

a sampling distribution, will almost never occur.

Because you wouldn't know, what your population looks like.

The only thing you know, is what your sample looks like.

Later on, you'll see how we can make use of the sampling distribution without

information, about the population distribution.

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