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The Sampling Distribution of a Statistic
As we move toward inferential statistics (making inferences about a population using
data + statistics + haruspicy) we will be focusing on the idea of the sampling distribution
of a statistic. The sampling distributions of statistics are probability distributions based
on mathematical models involving:
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

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Random sampling (for us, simple random sampling)
The distribution of the characteristic in a population
Probabilities of outcomes associated with sampling events
(In experiments) Probabilities associated with assignment to treatments
Additional assumptions as needed (for “testing hypotheses.”)
Sampling distributions are never actually observed, and our knowledge of them comes
from two sources:
1. Very seriously involved mathematics (“mathematical statistics”)
2. Simulation (e.g. actually roll a die or flip a spoon lots of times)
In today’s activity we will use simulation as our method of choice and get an estimate of
what the sampling distribution of our statistic looks like. We will not be “real close”
because we can’t do very many trials of our simulation. (In real life a statistician might
do tens of thousands of trials.
Another concession we will make to our time constraints is the order of events. The
usual way a simulation would be run is to:
1. Run the mathematical (on computer) or physical (by hand) simulation to get results
from many repetitions.
2. Run our experiment and get our statistic
3. Compare our result to the simulated sampling distribution.
We will do these steps in the order 2, 3, 1 so that we can enjoy the “fruits” of our labor in
#2, while trudging through #1.
Species Diversity
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Species Diversity Institute Laboratory
Team ________________________
________________________
Biodiversity (“biological diversity”) is one measure
of the health of an ecosystem. Scientists generally
believe the more diverse an ecological system is,
the healthier it is. Two common measures of
species diversity are Shannon’s Diversity Index and
Shannon’s Equitability Index.
Recently a search-and-capture operation was staged
in the Wild and Wooly section of the Wilderness
region of a local grocery store. A representative
sample of the Wild and Wooly animals was
captured and the raiding party (one member shown
at right in his cool red hat) returned safely, leaving
the individual samples for analysis and potential
predation. Four species are believed to be
indigenous to the area, but it is possible that some
interloper species were just passing through and
were swept up in the sampling. We will calculate diversity indices for the Wild and
Wooly population.
Before analyzing your sample statistically, you will need to identify (a) each species of
animal and (b) count the number of each species in your sample. Since many species
have similar characteristics you will need to provide a graphic representation of the
distinguishing features of each species. Tranquilize one representative of each species
and trace its body outline on the thoughtfully provided taxonomy paper.
We will assume that the most frequently occurring 4 species are the indigenous ones,
and any others are just passing through. Sometimes genetic variations of the 4 main
species occur. These genetic variations sometimes take the form of what can appear to be
“pieces” of the full animal. In order to maintain the purity of our measurement, and thus
our analysis, we will not consider these aberrant creatures. In fact, they are available for
predation on sight! We do know that the genetic variations, as well as interloper species
are FDA approved for the cure of statisticitus (Derivation: statistic- + itis = inflammation,
caused by abrasion between statistics students and homework problems.
Species Diversity
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Our first task, great naturalists that we are, will be to identify the species in the classical
naturalist manner – sketch them in the space below. Below each sketch, indicate how
many of these creatures are in your sample. (You will use this information later for
problem #2).
Grinnell Species Diversity Institute Laboratory
Taxonomy Paper for Species rendering
1.
_______________________
2. _______________________
# in your sample:
3.
_______________________
# in your sample:
4. _______________________
# in your sample:
# in your sample:
Species Diversity
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Shannon’s Diversity Index (H)
To measure species diversity a wildlife biologist counts the number of individuals of all
species of interest in a sample from an ecological system. One measure of species
diversity is the Shannon Diversity Index, calculated thus:
Let ni  number of individuals of species i
pi  proportion represented by species i
N  total number of individuals of all species
S  number of species
then,
ni
N
and the Shannon Diversity Index is:
pi 
S
H   pi ln  pi 
i 1
(Math students take note: It seems odd that there should be that negative in the formula;
it is actually there to make H positive. The pi 's are proportions, and each ln  pi  is
negative since the natural logarithms of the pi 's are negative, so the products are
negative.)
(Physics students take note: Shannon’s Diversity Index is derived from the Second Law
of Thermodynamics and is, in effect, a measure of entropy.)
The highest possible number for Shannon’s Diversity Index is reached when all the
species are distributed evenly. In our case, distributed evenly would mean that all the pi
are equal. For S species, the maximum value is ln S, the “natural log” of S. Here are the
calculations for 4 species:
S
H max   pi ln pi
i 1
1 1
1
1 1 
  ln    4  ln    ln
4
4
4 4
i 1 4
4
  ln1  ln 4
 ln 4
 1.3863
Species Diversity
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Shannon’s Equitability Index (E)
Shannon’s Equitability Index is calculated by dividing Shannon’s Diversity Index by its
maximum value, H max . This results in a number between 0 and 1. For our simulation
with the “known” number of species, 4, Shannon’s Equitability Index would be:
E
H
H
H


H max ln 4 1.3863
To help with the arithmetic drudgery when doing the simulation with cards,
here is a handy dandy table of values for the natural logs of the proportions:
pi
pi ln  pi 
pi
pi ln  pi 
pi
pi ln  pi 
0.1
-0.2303
0.4
-0.3665
0.7
-0.2497
0.2
-0.3219
0.5
-0.3466
0.8
-0.1785
0.3
-0.3612
0.6
-0.3065
0.9
-0.0948
Should you need assistance with the calculations for your samples, please
consult a colleague or one of the highly mathematically literate adults in the
room – who will know how to do this on the calculator.
Species Diversity
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1. A population with 4 species that actually exhibits maximum entropy is a deck of
common playing cards. We will illustrate the concept of the sampling distribution of
a statistic by sampling from this population. For this part of the activity you will need
two (2) decks of common playing cards. You may work singly or in teams of 2.
Take ten samples per team member, and fill in the table below with your results.
Shuffle the decks thoroughly EVERY time! Careful shuffling is what produces
the randomness needed for our sampling procedure. Your samples should be of
size 10 for easy calculation of the sample proportions. (A sample calculation is
shown below.)
# hearts
# spades
# diamonds
# clubs
H
E
2. Calculate Shannon’s Diversity Index and Shannon’s Equitability Index for YOUR
sample of creatures from the Wild and Wooly. Your work should clearly
communicate what you are calculating, what formula you are using, what numbers
you substitute into that formula, and – of course – your answers!
Species Diversity
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Sample Calculation for Shannon’s Indices
# hearts
# spades
# diamonds
# clubs
H
E
3
4
2
1
1.28
0.9233
S
H max   pi ln pi
i 1
   0.3ln(0.3)  0.4 ln(0.4)  0.2 ln(0.2)  0.1ln(0.1) 
  0.3  0.3612   0.4  0.3665   0.2  0.3219   0.1 0.2303 
 1.280
E
H
1.280 1.280


 0.9233
H max
ln 4 1.3863
Species Diversity
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