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PYPR1 lecture 2 :
Populations and Samples
Dr David Field
General Information
This lecture contains material that is crucial for
understanding the rest of the course
– read the text book
• important sections are indicated by, e.g.
– go to the workshop
– download the lecture
– make use of the university maths support service
• Specialist statistics tutor available every Wednesday
afternoon in term time from 2.00pm-4.00pm
• Alternatively, in a form with your question on the website
and get a reply by email
• http://www.reading.ac.uk/mathssupport/
Populations and samples
• At the end of this lecture we will be able to make
statements of the form
– “We measured the number of hours slept per night of a
sample of 50 students in the UK. The mean number of
hours slept was 7.2. We can be 95% confident that the
population mean lies between 6.8 and 7.6 hours of
sleep per night.”
• We will aim to understand the logic underlying the
confidence interval around the mean in the above
statement
• This is based on the properties of normal
distributions and sampling distributions
Populations and samples
• If we weighed every adult domestic cat in the UK we
would be able to calculate the population mean weight
and the population SD
– other populations of interest might be engineering students or
amateur cricketers or cars
• Measuring the whole population is expensive and
impractical, and so scientists invariably measure only
a fraction of the population of interest, using sampling
• The aim of the sampling procedure is to obtain as
good an estimate of the unknown true population
statistics as possible
• This requires the relationship between the sample you
have and the unknown population to be quantified
– If I weigh 100 cats, how confident can I be that my observed
mean is close to the unobservable population mean?
Representative and unrepresentative samples
• We can only assess the relationship between a sample
and an unobservable population if the sample is
representative of the target population
• This is an issue of study design, but it determines how
broadly we can interpret our numeric statistics
• If a sample of engineering students was selected
exclusively from Oxford University then measures obtained
from it might not be an accurate reflection of engineering
students in general
• There are a number of ways to obtain a representative
sample, the simplest case being random selection from the
entire population
What does random mean?
• Each time you sample a single case, every member of the
underlying population had an equal chance of being
selected
• The classic case is rolling an unbiased dice
• Each time the die is rolled you have an equal chance of
the result being 1,2,3,4,5, or 6
• There are no history effects!
• If you rolled 600,000 dice and recorded the results you
would end up with very close to 100,000 occurrences of
each outcome
– What would the frequency histogram of this data look like?
• In Psychology we often use opportunity samples and treat
them as if they were random samples from a target
population
Key concept: the normal distribution
• Values close to the mean of a variable are often more
frequent than values far from the mean
– This is true of the height of adults
– It is not true of rolling a dice repeatedly
• When true, and if you sample randomly from the
population, this produces a bell shaped frequency
histogram
• Many psychological variable are normally distributed, e.g.
IQ
– in the case of the IQ test, it is designed to be like that
• Normal distributions can be visualised using frequency
histograms just like the ones from Lecture 1
UK cats
Mean 5 Kg
SD 0.8 Kg
Carl Frederick Gaus
(1777-1855)
Curve shape is
independent of
sample size
UK cats
Mean 5 Kg
SD 0.8 Kg
The standard deviation - revision
scores
(pints)
deviations
squared
deviations
1
4
5
6
9
11
-5
-2
-1
0
3
5
25
4
1
0
9
25
• The sum of the squared deviations is 64
• The mean deviation (variance) is therefore
64 /(6 – 1) = 12.8
• Square rooting the variance returns it to the original
measurement units of the variable
• Therefore the SD is 3.57 pints
Greek cats
Mean 3.55 Kg
SD 0.4 Kg
UK cats
Mean 5.0 Kg
SD 0.8 Kg
• A Greek cat weighing 3.95 Kg and a British cat
weighing 5.8 Kg are clearly different from each
other in important ways, including their weight
• But, to a statistician, they are identical in one
important respect:
– they occupy the same position in their respective
sample distributions
– relative to their sample means they are both equally
“unusual” occurrences
– Both can be described by “Mean + 1SD”
– If you randomly select 1 cat from the 10,000 Greek and
the 10,000 British cats then the probability of sampling
a 5.8 Kg British cat is equal to the probability of
sampling a 3.95 Kg Greek cat
6800 Greek
cats
UK cats
Mean 5.0 Kg
SD 0.8 Kg
• Recall the student sleep example from the start of
the lecture, and the 95% confidence interval
around the mean number of hours slept?
• For any population that is normally distributed
95% of all scores will fall within 1.96 SD either
side of the mean
• This means that if you randomly select one case,
it’s score is 95% likely to fall within 1.96 SD of the
mean
• The confidence interval is based on the properties
of normal distributions, but there are some
intermediate steps to understand
The standard normal distribution and
z scores
• The family of normal distributions has an
infinite number of members, each defined
by a unique combination of mean and SD
• There is one particular normal distribution,
called the standard normal distribution,
which has a special status
– It has a mean of 0
– It has a SD of 1
The standard normal distribution and z scores
• One useful thing about the standard normal
distribution is that scores from any other normal
distribution can be converted into scores on the
standard normal distribution
• The converted scores are called z scores
• The new scores loose their original units (e.g.
Kg), and are now expressed in units of SD
• This is useful for comparing between samples
– If you know the z score for a Greek cat and for a
British cat you see directly which cat is a relatively
heavier example of it’s own population
Calculating z scores
•
•
•
•
•
•
•
z = (score – sample mean) / SD of sample
z score for a British cat weighing 4.7 Kg
(4.7 – 5.0) / 0.8 = -0.62
z score for a Greek cat weighing 4.7 Kg
(4.7 – 3.5) / 0.4 = 2.88
The Greek cat is a very large cat
The British cat is fairly typical, perhaps slightly on
the small side
What happens when the sample is small?
• In Psychology we usually work with small
samples, and we often have little idea of the
underlying population parameters
• With a small sample, you can still calculate a
mean and an SD, although the sample might be
too small to assess whether the underlying
population is normally distributed
• Lying at the heart of statistics is the question of
the relationship between populations and samples
• To explore this, we can use examples where
population parameters are known
Population:
Mean 5 Kg
SD 0.8 Kg
Sample of 10:
Mean 4.6 Kg
SD 0.7 Kg
Confidence in the sample mean
• Given a sample, I can produce a sample mean
• Statistics is about describing the relationship between
measured samples and underlying populations
– A statistician will ask how good the sample mean is as an indicator
of the population mean
• If you have a large and representative sample, then the
sample mean is such a good estimate of the population
mean that it can be used interchangeably
• But often we have a small sample, and no population data
(or large sample) to compare it with
• We can use the cats example, treating the full sample of
10,000 as the population, to explore the relationship
between small samples and the population
• A key point is that the mean of a large sample is more
likely to lie close to the population mean than the mean of
a small sample
Quantifying confidence in the sample mean
• The sample mean is a point estimate of the population
mean
– If we collected a second random sample we would have two
different point estimates of the population mean
– By definition, they can’t both be correct
– This situation implies an underlying continuum on which point
estimates can lie
– A continuum can be thought of as a curve plotted on a graph, like
the normal distributions you saw earlier
• In statistics, we aim to convert the sample point estimate
into an interval estimate
– This is a range on the underlying continuum
– We want to be able to say that given the sample, the population
mean lies somewhere between X and Y
– We will have to be content to say that we are 95% sure
Where each black
line crosses the x
axis represents a
separate point
estimate of the
population mean
The horizontal line
is a visually
judged interval
estimate of the
population mean
given the 10
samples
Mean 5 Kg
SD 0.8 Kg
Sampling distributions
• Each sub-sample of 10 cats has a mean
weight and SD that is slightly different from
the full population of 10,000
• Individual samples of 10 are often not
normally distributed
• Imagine collecting 100 separate sub-samples
of 10 cats, and producing 100 sample means
• The mean of the 100 sample means would be
an excellent estimate of the population mean
• It is possible to plot a frequency histogram of
the 100 obtained sample means
Sampling distributions
• The frequency histogram of the 100 samples of 10
will itself be normally distributed
– This means the distribution can be defined by its mean
and SD
• More generally, if you collect a sample, then
theoretically speaking there is an underlying
population of samples, of which yours is just one
case
– The population of sample means is normally distributed
Sampling distributions
• In the frequency histogram of the original sample,
each case was a single cat
• In the corresponding sampling distribution, each
case is the mean weight of (10) cats selected
randomly from the cat population
• A key property of sampling distributions is that
provided the individual samples have N >= 30
they are ALWAYS normally distributed
– even if the actual population is skewed (e.g. reaction
time)
– or bimodal
– if population is normal sampling distribution will also be
normal for small samples < 30
• The black curves are frequency histograms of the
means of samples randomly selected from the
pink population distribution
Populations and samples
• If we had enough samples to plot a sampling distribution,
then for one sample we could assess exactly how close it
is to the population mean (mean of the sampling
distribution)
• But what if we only have one sample?
– With only one sample, because sampling distributions are normally
distributed we still know that the mean of the single sample is 95%
likely to fall within 1.96 SD either side of the mean of a theoretical
sampling distribution
• Avoid confusion:
– the mean of a sampling distribution IS equal to the population
mean
– the SD of a sampling distribution is NOT equal to the SD of the
population
Standard error (SE)
• In statistics, the standard deviation of a sampling
distribution is given a different name to distinguish
it from the standard deviation of a single sample
or the standard deviation of a population
• It is called the standard error (SE)
• Its name contains the word “error” because
statisticians use it to estimate measurement error
Standard error (SE)
• We can be 95% sure that a sample mean will lie within + / - 1.96 SE of
the mean of the distribution of sample means
– this provides the 95% confidence interval from the example about how
long students sleep given at the start of the lecture
– 7.2 hours – 0.4 hours (1.96 * SE) = 6.8 hours is the lower bound
– 7.2 hours + 0.4 hours (1.96 * SE) = 7.6 hours is the upper bound
• But at the moment we have only seen how to calculate the SE if you
have collected a huge number of samples of a specific size
• Therefore, the problem to solve is how to find the SE from a single
sample
• The starting point for solving this problem is the fact that the SE of the
sampling distribution shrinks as the individual samples making up the
distribution increase in size
– This in turn is because the mean of a large sample is more likely to be
close to the population mean than the mean of a small sample
• The SE of the sampling distribution is smaller
when the individual samples making up the
distribution are larger
Standard error and confidence intervals
• The previous slide illustrates that the confidence interval
around a sample mean will be smaller if the sample size is
bigger
– smaller confidence intervals imply more accurate and therefore
more useful measurements
• There is a lawful relationship between the sample size and
the SE of the resulting sampling distribution
– the SE is halved when the sample size is quadrupled
• The SE is also dependent upon the SD of the population
the samples were drawn from
– the SE is smaller when the population SD is smaller
– As you don’t know the SD of the population, the sample SD is used
as an estimate of the population SD
Standard error and confidence intervals
• This relationship between the SE, sample size,
and the SD is captured in the following formula
SE =
SD of the sample
Sample size
Standard error and confidence intervals
• Standard error (SE) = sample SD / square root of
sample size
• For the small sample of 10 UK cats we observed a
SD of 0.7 kg
• 0.7 / square root of 10 (3.16) = 0.22 Kg
• One more step is required to arrive at a 95%
confidence interval
• The standard error is 1 SD of the sampling
distribution
• What proportion of samples have means that lie
within 1SE of the mean of the sampling
distribution?
It is 68% likely that the
population mean falls
within 1 standard error
above or below the
sample mean
Standard error and confidence intervals
• By convention, we usually want to make the
statement that we are 95% certain that the
population mean lies between X and Y
• Therefore, we use the properties of normal
distributions, which tells us that 95% of all
samples in the sampling distribution of the mean
fall within 1.96 SE of the mean
• The confidence interval we give around the
sample mean is 1.96 * SE either side of the mean
Standard error and confidence intervals
• The mean of the 10 cat small sample was 4.6Kg
(SD 0.7). This gives a standard error of 0.22 Kg
• Confidence interval = 1.96 * 0.22 = 0.43 Kg either
side of the mean
– “We measured the weight of 10 adult cats in the UK.
The mean weight of the sample was 4.6 Kg. We can be
95% confident that the population mean weight of adult
cats in the UK lies between 4.16 and 5.03 Kg.”
• The mean of the population of 10,000 cats was
5.0 Kg, and we can see that the above statement
is true (but only just in this case!)
Standard error and confidence intervals
• It is important to remember that we accept a 5% risk that
the interval estimate is wrong, and the population mean
does not lie within the range of values it defines
• This is the risk that the sample we have is located in one of
the two tails of the underlying sampling distribution
• Imagine we draw a second sample, this time of 40 UK
adult cats, from the population of 10,000
• This time, we obtain a mean weight of 4.87 Kg, with a SD
of 0.84 Kg
• This SD is bigger than the SD of the small sample, which
was 0.7, so in a sense this sample is showing greater
variation
Standard error and confidence intervals
• You might expect that the SE and confidence
interval will also be larger for this sample
• But, because the SD formula involves dividing by
the square root of the sample size it turns out that
this is not the case
• Standard error is 0.835 / square root 40 (6.32) =
0.13
• Confidence interval is 0.13 * 1.96 = 0.258 Kg
either side of the mean
• Confidence interval for the small sample was 0.43
Kg either side of the mean
List of statistical terms for revision
• Note that the technical meaning of terms in statistics
is not always the same as the everyday meaning of the
words. If you understand each of these concepts then
you are well on the way to understanding statistics!
• population
• sample
• random
• normal distribution (also known as a bell curve or
Gaussian distribution)
• frequency
• standard normal distribution
• z score
• sampling distribution
• standard error
• confidence interval