Biol 131 Intro to Evolution
Fall 2001
Quantitative Variation
in
Helianthus annus (Sunflower) Seed Stripes
Quantitative variation within and among groups is at the base of all studies of
evolutionary change. In order to discuss changes in populations we need to develop a
language and a set of concepts that allow us to describe and compare characteristics of
populations. These skills involves understanding the distribution of characters in two
broad categories: measures of central tendency; and, ways of describing the
distribution of values around these measures of centrality. In this lab we will measure
the extent of variation in a natural population within and among groups using the
common sunflower (Helianthus annus) seed as the subject of our investigation. The
language and concepts that we begin to explore today are the basis for making
comparisons between different populations that exist at the same time and for examining
changes in a population at different times.
We will use histograms to represent the distribution of a populations with respect to a
particular character. In a histogram the character being discussed is represented on one
axis while the number of individuals in the population with that value of the character are
represented on the other axis.
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Biol 131 Intro to Evolution
Fall 2001
In this graph we see a representation of the following hypothetical data:
#Stripes
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
#Seeds
5
3
25
65
68
24
74
32
25
12
8
14
10
5
2
Part 1 – Collecting data
Each group will get a pile of sunflower seeds. First, take a few minutes to spread them
out and look at the ways they vary. In this first part of the activity you will divide up your
sunflower seeds based on the number of stripes that they have. We will probably have
some class discussion of how to score stripes. Keep them in separate piles based on the
number of stripes and score as many as you can in 30 minutes.
Part 2 – Representing your data
Record the number of seeds in each category (number of stripes). Then place them in test
tubes arranged in a rack to create a histogram-like representation of your data. Compare
your population to several other group’s populations. Then write 2-3 sentences describing
your population with respect to the number of stripes.
Part 3 – Measures of central tendency
Calculate the mean, median and mode for your population. Show where each of these
occurs by laying a piece of paper at the base of your test tubes and marking where each of
these measures occur. Compare your results with a couple of other groups and write a
sentence or 2 describing your population.
mean = (∑ x)/n
where x is a value and n is the number of values
This statistic is also commonly referred to as the average. It is computed by adding each
of the values in the distribution and dividing by the number of values in the distribution.
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Biol 131 Intro to Evolution
Fall 2001
Median: the middle value
The median of a distribution can be found by putting the numbers in ascending order and
taking the number in the middle, e.g. if the distribution has five values ( 5, 5, 4, 2, 1),
then the third value is the median (4). If there is more than one middle value, then the
mean of the two values is used.
Mode: the most frequent value
The mode is simply the value which occurs more often, e.g. the mode of this distribution,
8, 7 ,5 ,5, 5, 3, 1 is five.
Mode
Rat Weight
Media n
7.5
5
Number of rats
10
Mean
2.5
1
2
3
4
6
5
7
8
9
10
Weight (g)
Part 4 – Measures of distribution
Enter your data into the JMP statistics program and have it calculate the standard
deviation, skewness, and kurtosis values for your population distribution.
Standard Deviation = √[∑ (x - "frequency mean")2/(n - 1) ]
where x is a value and n is the number of values
The standard deviation is a measure of the spread of a distribution. It is calculated by
first calculating the mean of the distribution. Then, the difference between each value
and the mean should be squared and all of those numbers added up. That value is divided
by the total number of values minus one in the distribution and the square root of that
number is the standard deviation.
The standard deviation of a distribution tells you where most of the values are found. If
you look at the values between the mean and one standard deviation above the mean 32%
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Biol 131 Intro to Evolution
Fall 2001
of the population values are represented (assuming that the population is normally
distributed). We can extend this understanding of how the standard deviation describes a
distribution to arrive at the following conclusions:
68% of the population falls within ±1 SD of the mean
95% of the population falls within ±2 SD of the mean
99.7% of the population falls within ±3 SD of the mean
This is shown graphically on the figure below.
Shell Length
MEAN
40
20
NUMBER
30
10
10
20
30
40
Shell Length
± 1-
68%
± 2-
95%
± 3-
99.7%
Skewness = (1/ns3)∑(x - "frequency mean")3
where s is the std. deviation, x and n as before
Skewness is a statistic that describes the relative sizes of the tails of the distribution. A
negative skewness value implies that the left tail of the distribution is longer. A positive
skewness value implies that the right tail of the distribution is longer.
The calculation is similar to calculations for the standard deviation. After finding the
frequency mean, the cubes of the differences between the each of the values and the mean
are summed. Then, that number is divided by the number of values in the distribution
and the cube of the standard deviation.
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Biol 131 Intro to Evolution
Fall 2001
Fruit Weight
30
NUMBER
20
10
1
2
3
4
5
6
7
8
9
10
Fruit Weight
A positively skewed distribution. The red line represents a symmetrical distribution for
reference. Note that in a positively skewed distribution the mean will be greater than the
median population value.
Kurtosis = (1/ns4)∑(x - "frequency mean")4 - 3
s, n and x as before
Kurtosis is a statistic that describes how sharp the peak of the distribution is. A negative
score indicates platykurtosis (a relatively flat peak), while a positive score indicates
leptokurtosis (a relatively sharp peak).
mesokurtic
"normal"
platykurtic
300
100
leptokurtic
10
20
30
X
5
Number
200
Biol 131 Intro to Evolution
Fall 2001
Part 5 – Multidimensional variation
Using the seeds from within one category of stripe number, take one additional sets of
measurements on your seeds. For example, if you looked at all 19 seeds that had 11
stripes (hypothetical data), you could measure seed weight, buoyancy, seed length, etc. to
examine intraclass variation.
Lab Write up
Your write up should include:



Your definition for the character stripe number.

A clearly labeled graph of the distribution of your population on which you
have indicated the mean, median, mode (with reference to the histogram) and
the calculated values for standard deviation, skew and kurtosis.

A clearly labeled graph of the distribution of another groups population.

A short description of your data (not simply listing the statistics but describing
it as you might to a friend who was not familiar with statistics) and
comparison of your data to the other group’s data. Do you think they are both
samples from the same population? Why or why not?

A definition of the second character you measured. A description of your
findings on interclass variation including a histogram of your results.

A brief discussion of how this detailed information about variation is relevant
to evolution.
Your raw data for variation with respect to stripe number in your population.
This is a modified version of a lab developed by John Jungck at Beloit College. Several of
the figures were taken from the Biometrics module by Daniel Hornbach published the
BioQUEST Library.
See http://bioquest.org/biostat for additional information about these statistics.
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Biol 131 Intro to Evolution
Fall 2001
A more detailed explanation of how to calculate these statistics can be found in Biometry
by Sokal and Rohlf, which is the text for the Beloit College Biometrics course. Chapter
4, Desriptive Statistics, gives a good explanation of the first four statistics. Chapter 2 is
on frequency distributions and may be of some help.
Range
Standard deviation mean ±1 SD is 68% of population; ±2SD is 95% of population; ±3SD
is 99.7% of pop.
Skewedness +skew is shifted left or pulled right (mean is greater than the median) –skew
is shifted right or pulled left (mean is less that the median)
Kurtosis – is platykurtic (flat); + is leptokurtic (pointy); and 0 is mesokurtic (normal).
[Intro to JMP: columns, variables, distributions, changing axes, looking at moments.
Enter your data into JMP and generate a graph of the distribution of your population.
Print it. [print preview}
putting it on paper in front of test tubes
10-15 minutes intro to measures of distribution
10-15 minute intro to JMP
They enter data do caluculations print graphs
[put up on board some of the means and see if that is a useful way to describe populations
for comparisons. Discuss why central tendency may not be such an important factor for
natural historian with a darwinian perspective. ]
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