Level I CFA Quant: Sampling and Estimation-Lecture 1 - YouTube

Sampling and Estimation. Here are the sections in

this reading. We'll talk about how to use sample data

to estimate population parameters and our main focus

will be on the population mean generally denoted

by mu. Let us understand the basic concept of sampling.

Say you have a large population such as the returns

on all stocks in the United States for last year, and

you are interested in the mean return or the average

return, that is an example of a parameter, generally

denoted by the symbol, mu, so a parameter such as the

mean return is used to describe a population. Population

here being the return on all stocks in the United

States last year. You might not have the time to go

through every single stock and come up with the average

return. So what you could do is pull a sample from

the population. So let's say you pull out a sample

of 100 stocks and then you find the average of these

100, that average is generally denoted by X bar,

and this is called a statistic so a statistic is used to describe

the sample. Simple random sampling a simple random

sample is a subset of a larger population such that

each element has an equal probability of being selected

to the subset. So let's say that this population

has a thousand items. So simplistically, let's

say that there are a thousand stocks in our overall

population. When we create a sample of hundred, if

every stock in this population has an equal probability

of being selected when we say that we have a simple

random sample. The next concept you need to know

is that of Sampling Error. Clearly this population

has a certain mean called mu, and when you come up

with a sample and you compute the sample mean, this

sample mean is not necessarily going to be exactly

the same as the population mean. The difference between

the sample mean and the population mean is called

the sampling error. So to put this formally, sampling

error is the difference between the observed value

of the statistic and the quantity it is intended

to estimate. Sample Distribution of a Statistic.

Sample Distribution of a Statistic is the distribution

of all the distinct possible values that the statistic

can assume when computed from a randomly drawn sample.

Let's continue with our example. So we have this

large population which consists of all stocks in

the United States and you are concerned about the

population mean. Let's say you draw the first sample

with a sample size equal to 100 and you come up with the

mean for sample 1 and let's say that that number

is 10%. Then you draw another sample, obviously the

mean here will not be exactly the same as the mean

for the first sample. Assume that you are drawing

the same sample size, let's say that the mean for

sample 2 turns out to be 11% then you draw a third sample,

and here the mean might be 9% and so on. You keep drawing

samples, and you will notice that there is a certain

distribution of the sample means and that is called

the sample distribution of a statistic. So now this

definition will make more sense. Sample distribution

of a statistic is the distribution of all the distinct

possible values that the statistic can assume when

computed from a randomly drawn sample. Let us now

take a look at Stratified Random Sampling. Say you

want to become a little more sophisticated in terms

of picking your sample from this large population

of American stocks and you recognize the fact that

they are different exchanges and you want to make

sure that your sample has a representation from

each exchange. So what you can do is divide the population

into strata based on one or more classification

criteria, in my example, the classification criteria

is the different exchanges so you can say there is

exchanged 1 exchange 2 exchange 3 and exchange

4 so now you have the different sub-populations

or strata next you pull a sample from each sub-population,

then you draw a simple random sample from each stratum

in sizes proportional to the relative size of each

stratum in the population. In other words, if exchange

2 is two times bigger than exchange 1 then clearly

the random sample that you draw from exchange 2

must be two times bigger than exchange 1 so here

you might have 20 stocks, here, you would pull a sample

of 10 and so on. In my simple example exchange 3

is the same size as exchange 2 so here you will

also have 20 and exchange 4 is the same size as exchange

1 so here you would have 10. And then you pool these

samples to form a stratified random sample so hear

your sample of 60 has representation from all four

sub-populations, and this would be called a Stratified

Random Sample. Let us look at a simple practice

question. Paul wants to categorize publicly listed

stocks for his research project. He first divides

the stocks into 15 industries. Then from each industry,

he categorizes companies into three groups.

Finally, he divides these into value vs growth

stocks. How many cells of strata does the sampling

plan entail? Now this is a little more complicated

than what I just talked about and here we need to recognize

that there are 15 Industries, for each there are three groups,

and then we have value vs growth so we multiply it by 2 so the correct

answer is simply a product of these three numbers,

which is 90. Time Series and Cross-Sectional Data. Time

Series is a sample of observations taken at specific

and equally spaced points in time. For example,

the monthly returns on Microsoft stock from January

1995 to January 2005 that is fairly self-explanatory.

Cross-sectional data is a sample of observations

taken at a single point in time. So for example, the

sample of reported earnings per share for all NASDAQ

companies for 2005, so t notice that here, the data

is for 2005 for a range of companies and this would

be an example of cross-sectional data. For both

time series and cross-sectional data the random

sample must be representative of the populations

we wish to study. Consider this practice question.

A researcher needs to make use of the 2012 household

budget data for Scandinavian countries. Is this

cross-sectional data, time series or panel data? Based

on what we've just talked about, this is cross-sectional

data. Distribution of the Sample Mean. Let's say

you draw several samples from the population and

for every sample you compute the mean. Clearly, there

will be a certain distribution for the sample means

and that is what we are going to talk about. For a population

with mean, mu, and variance, sigma squared, the sampling

distribution of the sample means so that is the

distribution of the X bars. Of all possible samples

of size n, each of these samples needs to be of size n

and let's say that n is 100 in our example,

so the distribution of the sample mean will be approximately

normal with a mean equal to mu and a variance equal

two sigma squared over n. So there are 3 core statements

being made. The first one is that the distribution

of x bar or the distribution of the sample mean is

going to be normal. The second point is that the mean

is mu which is essentially the mean of the population,

and the variance of this distribution is sigma squared

over n. Hopefully it is fairly obvious that the mean

of this distribution is mu because you are drawing

these samples from the population. So the expected

value of x bar or the sample mean should be the population

mean. In terms of the variance think of it this way, if the population has a very

high variance, so the population data is spread

out a lot, then you would expect x bar to also be spread

out. In other words, the distribution of x bar would

also have a high variance. That is why we have sigma

squared in the numerator, on the other hand if the

size of the sample is large then the distribution

of x bar become smaller because largest samples

mean that you are more likely to be close to the population

mean. In other words, larger sample means that you

are more likely to have X bar or a sample mean that is

closer to the population mean that's why n is in

the numerator. What we have just learned is the Central

Limit Theorem and I will repeat exactly what is

stated in the curriculum. Given a population described

by any probability distribution and this is critical

because the central limit theorem applies to a distribution

whether or not it is normal. So the population has

a mean mu and a finite variance sigma squared, this

is what we saw on the last slide, the sampling distribution

of the sample mean, the sample mean is the X bar and

we are talking about the distribution of x bar assuming

that you pull several samples from the mean computed

from samples of size n. Again, I'm repeating because

this is important, the sample size always has to be

the same. The distribution of x bar will be approximately

normal with a mean equal to mu and variance equal

to sigma squared over n. When the sample size is

large, and large means a sample size of 30 or more.

The standard error of the sample mean is the standard

deviation of the distribution of the sample means.

So there is a new piece of terminology but the concept

of straightforward. We just said that the variance

of the distribution of x bar is sigma squared over

n. The standard deviation, which is simply the square

root of this expression is essentially sigma over

root n, that is the standard error of the sample mean.

If we know the population variance then we simply

plug that here and we compute the standard deviation

of X bar or the sample mean using this expression.

If the population variance is not known then the

standard deviation for the distribution of x bar

is equal to s, this is the standard deviation of the

sample. Clearly, if the population variance is not known

then we use the variance or the standard deviation

of the sample as a proxy. And again divide that by root n so both

these expressions are similar, here we are using

the population standard deviation because it's

known. Here we are using the sample standard deviation

because we don't know the population standard deviation.

Let's look at this simple question. You need to use

the formula that we just discussed which is sigma

over root n, sigma is 3, n is 64 so 3 over root of 64 should

give you A, which is 0.375