Download week 2 - SPIA UGA

Introduction to Data
Analysis
Sampling and Probability
Distributions
Today’s lecture

Sampling(A&F 2)
 Why
sample
 Sampling methods
Probability distributions (A&F 4)
 Normal distribution.

Sampling distributions = normal distributions.
 Standard errors (part 1).

2
Sampling introduction
Last week we were talking about populations
(albeit in some cases small ones, such as my
friends).
 Often when we see numbers used they are not
numbers relating to a population, but a sample
of that population.


Newspapers report the percentage of the electorate
thinking Tony Blair is trustworthy, but this is really the
percentage of their sample (say 1000 people) that they
asked about Blair’s trustworthiness.
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Samples and populations

For that statistic from the newspaper’s sample to be
useful, the sample has to be ‘representative’.


i.e. the % saying Blair is trustworthy in newspaper’s survey (the
sample of 1000 people) needs to be similar to the % in electorate
(the population of 40 million people).
An intuitively obvious way of doing this is to pick
1000 people at random.


For the survey, metaphorically (or literally with a big hat) put every
elector’s name into a hat and pull out 1000 names.
For a random sample of people in a large classroom, I could sample
every 10th person along each row.
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Why sample?

Cost.
We could ask all 40 million people that are eligible to
vote in Britain. This would prove somewhat expensive.
 The last British census cost £220 million…


Speed.


Equally the last British census took 5 years to process
the data…
Impossibility.

Consuming every bottle of wine from a vineyard to
assess its quality leaves no wine to sell…
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Why random?

Random sampling allows us to apply
probability theory to our samples.
This means that we can assess how likely it is (given
how big our sample is) that our sample is representative.
 Deal with this in more detail later on.


Intuitively, non-random sampling doesn’t
seem a very good idea.

Who’s heard of Alf Landon?
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Alf vs. FDR

In 1936 the Literary Digest magazine
predicted that the Republican Presidential
candidate (Landon) would beat FDR.
The LD sent 10 million questionnaires out, of the 2 ½
million that were sent back, a large majority claimed to
be voting Republican at the election.
 The LD wanted to estimate the % of voters for each
candidate (the parameter), and used the proportion from
their sample (the statistic) to estimate this.


But, FDR won…
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Why did the LD get it wrong?

LD’s sample was large, but unrepresentative.
They did not send questionnaires to randomly selected
people, but rather lists of people with club
memberships, lists of car /telephone owners.
 These people were wealthier and therefore more likely
to vote Republican; the sample was not representative of
the US electorate as a whole.


The LD’s sampling frame was not the
population (the electorate), but a wealthy
subset of the population.
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Non-probability sampling

The moral being…


If we don’t sample randomly, and instead use non-probability
sampling, then we are likely to get sample statistics that are not
similar to the population.
e.g. Newspapers and TV regularly invite readers/
viewers to ring up and ‘register their opinion’.


Scottish Daily Mirror ran a poll on who should be the new 1st
Minister in 2001. One of Jack McConnell’s fellow MSPs rang up
169 times to indicate he should take the position…
If the Daily Mail and Independent hold phone in-polls on the same
issue, the results will be different as the samples are different to one
another in a non-random way (social class, ideology, etc.).
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Experimental designs


Randomness is also useful in experimental sciences,
just as with observational data.
If we are giving one set of subjects a treatment and
one group nothing, then ideally we would randomly
select who is in each group.



e.g. psychiatrists studying a drug for manic depressives, would give
the drug to one group and a placebo to the other.
Their results are no good if the groups are initially different (say by
age, sex, etc.).
Random selection into a group makes these differences unlikely,
and allows us to test how likely it is that the drug has a real effect.
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Simple random sampling (SRS)

‘Names out of a hat’ sampling. Select the n of
the sample that we want, and then randomly
pick that n of observations from the
population.
Each member of the population is equally likely to be
sampled.
 e.g. if I wanted a sample from the room, then I might
give everyone a number, and then use a table of random
numbers to pick out 10 people. Any method that picks
people randomly is acceptable.

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Problems with SRS

A random sample may not include enough of a
particular interesting group for analysis.


Interested in experiences of racism, 100 random people will on
average include 85 whites, and an individual sample will
potentially have even fewer (maybe even zero) non-whites.
Can be costly and difficult.


A random sample of 50 school-children might include 49 in
England and Wales, and one in the Orkney’s.
A complete list of every school-child might be possible to obtain,
but what of every person living in Britain. A list of the population
of interest is not always available.
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Solutions (1)

Stratified random sampling.


Two stages: classify population members into groups,
then select by SRS within those groups.
e.g. ‘over-sample’ non-whites for our racism
study.
Once we had divided the population by race, we would
SRS within those racial groups.
 Might take 50 whites and 50 non-whites for our sample
if we were interested in comparing experiences of
racism.

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Solutions (2)

Cluster random sampling.


If population members are naturally clustered, then we
SRS those clusters and then SRS the population
members within those clusters.
Pupils in schools are naturally grouped by
school.
We may not have a list of every school-child, but we do
have a list of every school.
 Again two stages. We randomly pick 5 schools, and
then randomly pick 10 children in each school.

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One further problem
This is not to say that all problems with
random sampling are soluble.
 Non-response.

Not all members of our chosen sample may respond,
particularly when sanctions are nil and incentives are
low (or in fact usually negative…).
 This can matter if non-response is non-random. If
certain types of people tend to respond and others do
not.

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Non-response

In 1992 opinion polls predicted a Labour
victory, yet the Conservatives were returned
by a large majority of votes (if not seats).
One of the (many) factors that may have caused this
bias in the polls was that Conservative voters were less
likely to respond to surveys than other voters.
 If the members of the sample that choose to not respond
are different to those that do then we have a biased
sample. More on bias later on.
 Ultimately, tricky to deal with. Some more on this later
this semester.

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Sampling – a summary
Sampling is a easy way of collecting
information about a population.
 SRS means everyone in the population of
interest has the same chance of being selected.
 We often use slightly different methods to SRS
to overcome certain problems.
 Random sampling allows us to estimate the
probability of the sample being similar to the
population.

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Probability – an idiot’s guide
We’re interested in how likely, how probable,
it is that our sample is similar to the
population.
 In order to make this judgement, we need to
think about probability a little bit.
 In particular we need to think about probability
distributions.

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Probability





The proportion of times that an outcome would occur
in a long run of repeated observations.
Imagine tossing a coin, on any one flip the coin can
land heads or tails.
If we flip the coin lots of times then the number of
heads is likely to be similar to the number of tails
(law of large numbers).
Thus the probability of a coin landing heads on any
one flip is ½, or 0.5, or in bookmakers’ terms ‘evens’.
If the coin was double headed then the probability of
heads would be 1 – a certainty.
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Probability Distribution (1)





The mean of a probability distribution of a variable
is:
µ = ∑ y P(y) if y is discrete.
µ = ∫ y P(y)dy if y is continuous.
Also called the expected value: E(y)=Probability
times payoff
Standard Dev (σ) of prob dist measures variability.
 Larger
σ = more spread out distribution
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Probability distribution (2)




Lists the possible outcomes together with their
probabilities.
Now, let’s take a continuous-level variable, like hours
spent working by students per week.
The mean = 20, and standard deviation = 5 .
But what about the distribution…?


Assign probabilities to intervals of numbers, for example the
probability of students working between 0 and 10 hours is (let’s
say) 2½ per cent.
Can graph this, with the area under the curve for a certain interval
representing the probability of the variable taking that value.
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Probability distribution (3)
Area between 0 and 10
is 2.5 per cent of the total
Area beneath the curve
0
5
10
15
20
25
30
35
40
Time spent working (hours)
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Probability distribution (4)
Given this distribution, there is a 0.025
probability (2.5%, or 40-1 for the gamblers)
that if I picked a student they would have done
less than 10 hours work in a week.
 A lot of continuous variables have a certain
distribution – this is known as the normal
distribution.
 The student work distribution is ‘normal’.

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What is a ‘normal distribution’?

NDs are symmetrical.



The distribution higher than the mean is the same as the distribution
lower than the mean.
Unlike income, which has a skewed distribution.
For any normal distribution, the probability of falling
within z standard deviations of the mean is the same,
regardless of the distribution’s standard deviation.




The Empirical Rule tells us:
For 1 s.d. (or a z-value of 1) the probability is .68
For 2 s.d. (actually 1.96) the probability is .95
For 3 s.d. the probability is almost 1.
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Brief aside—what is Z?

The Z-score for a value Y on a variable is the number
of standard deviations that Y falls from µ.
z

y

We can use Z-scores to determine the probability in
the tail of a normal distribution that is beyond a
number Y.
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Normal distribution (1)
Area under the curve
here is 0.68 of the
total area under the
curve.
1 s.d. less than
the mean = 15
Hence the probability
of working between
15 and 25 hours is
0.68.
1 s.d. more than
the mean = 25
0
5
10
15
20
25
30
35
40
Time spent working (hours)
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Normal distribution (2)

For any value of z (i.e. not just whole numbers but
say 2.34 s.d.), there is a corresponding probability.



Most stats book have z tables in their front/back covers.
Thus if we were to pick a student out of our
population of known distribution we could work out
how likely it would be that she was a hard worker.
Even non-normal distributions can be transformed to
produce approximately normal distributions.

For example, incomes are not normally distributed, but we ‘log’
them to make a normal distribution (more on this later).
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What’s the point?
But, surely we don’t know the distribution or
mean of the population (that’s probably why
we’re sampling it after all), so what use is all
this…?
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Back to sampling
The reason that normal distributions are of
relevance to us, is that the distributions of
sample means are normally distributed.
 In order to understand what this means let’s
take an example of sampling.

I want to take a driving trip around the world, visit
every country and pay no attention to speed limits.
 I don’t particularly want to go to prison however, so
what to do…

29
Sampling example (1)



The plan is to bribe all
policemen when caught
speeding. Thus I want to
measure how much it costs on
average to bribe a policeman
to avoid a speeding ticket.
It’s costly to collect this
information, so I don’t want
to investigate every country
before I set off.
Therefore I sample the
countries to try and estimate
the average bribe I will need
to pay.
James’ car
Ryan’s car
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Sampling example (2)

I randomly sample 5 countries and measure the cost
of the bribe.

Imagine for the minute I know what the population distribution
looks like (it happens to be normal with a mean of $500).
Population distribution
Mean of population ($500)
Sample mean ($450)
One observation ($700)
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Sampling distributions (1)



If we took lots of samples we would get a distribution
of sample means, or the sampling distribution. It so
happens that this sampling distribution (the
distribution of sample means( or any statistic)) is
normally distributed.
Due to averaging the sample mean does not vary as
widely as the individual observations.
Moreover, if we took lots of samples then the
distribution of the sample means would be centred
around the population mean.
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Sampling distributions (2)

Imagine I took lots of samples. There would be a
normal distribution of their means, centred around
the population mean.
Mean of population
Mean of all sample means
Population distribution
Sampling distribution
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3 Very Important Things

If we have lots of sample means then the average will
be the same as the population mean.


If the sample size is large(ish), the distribution of
sample means (what is called the sampling
distribution) is approximately normal.


In technical language the sample mean is an unbiased estimator of
the population mean.
This is true regardless of the shape of the population distribution.
As n (the sample size) increases the sampling
distribution looks more and more like a normal
distribution.

This is called the central limit theorem.
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Sampling distributions (4)
Sampling distribution
Mean of population
Mean of all sample means
Population distribution
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Sampling distributions (5)
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‘Accurate’/‘inaccurate’ samples (1)


Some sampling
distributions are bigger
than others…
The top sampling
distribution is better for
estimating the
68% of distribution
population mean as
more of the sample
means lie near the
population mean.
68% of distribution
Mean of population
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‘Accurate’/‘inaccurate’ samples (2)

Sampling distributions that are tightly
clustered will give us a more accurate estimate
on average than those that are more dispersed.


Remember, high standard deviations give us a ‘short
and flabby’ distribution and low standard deviations
give us ‘tall and tight’ distribution.
We need to estimate what our sampling
distribution’s standard deviation is.

But how do we do this…?
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A (little) bit of math now…

Before we work out what the sampling distribution
looks like, some important terms.
Population mean  
Population standard deviation  
Sample observatio n  X
Sample mean  X
Sample standard deviation  s
Sample size  n
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Standard error (1)

For my bribery sample, we
know the following:
Sample mean
X  450

But, we want to know the
standard deviation of the
sampling distribution, so we
can see what the typical
deviation from the
population mean will be.
Sample standard deviation
s  150
Sample size
n5
40
Standard error (2)

Fortunately for us:
 
σ
n
We don' t know what σ is, but we do know s.
s
Standard error X 
n
Standard deviation X 
 

The standard error is an estimate of how far any sample mean
‘typically’ deviates from the population mean.
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Standard error (3)


For my bribery sample.
Thus, the ‘typical’
deviation of a sample
mean from the
population mean (of
$500) would be $64 , if
we repeatedly sampled
the population.
 
s
Standard error X 
n
150
Standard error X 
5
150

2.34
 64.10
 
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2 More Very Important Things

The formula for standard error means that as:
 …the
n of the sample increases the sampling
distribution is tighter.

This makes sense, the bigger the sample the better it is
at estimating the population mean.
 …the
distribution of the population becomes
tighter, the sampling distribution is also tighter.

This also makes sense. If a population is dispersed it
will be more unlikely to get observations near the mean.
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Binary variables

This works for binary variables too, where the mean is
just the proportion…
Population mean    population proportion  
Population standard deviation  σ   (1   )
Sample proportion  P.
Standard deviation P  
Standard error P  
 (1   )
n
P (1  P )
n

P (1  P )
n
44
Do we trust Blair?

Take the example of Blair’s trustworthiness.


1000 people in the sample, 30% trust him (i.e. the mean is 0.30).
Given this, we can work out the standard error.
Sample proportion  P  0.30.
Standard error P  

P(1  P)

n
0.3 * 0.7
 0.014
1000
The typical deviation from the proportion would be 1.4% if we
took lots of samples.
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And finally, standard error (4)

Don’t forget that we know the shape of the
distribution of the sample means (it’s normal).


We know the sample mean, the shape of the distribution
of all the sample means, and how dispersed the
distribution of sample means is.
So (at last) we can calculate the probability of
the sample mean being ‘near’ to the population
mean (i.e. calculate a z-score and look up
corresponding probability). But wait, there’s
more…
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