Statistics – A Survival Manual
Part 1 - Measures Of Central Tendency
"What's the price of bacon?" "How is the weather in Southern California
this time of year?" "How tall does the Douglas fir grow?" These are examples of
questions that are really looking for a figure that would be representative of your
total population, that would give a fairly good general impression, a measure of
central tendency. Some form of "average" is generally used to provide the answer.
The mean and median are the most common measures of central tendency.
Mean
The mean, occasionally called the arithmetic mean, is the most common measure
of central tendency. This measure of central tendency allows the value (magnitude)
of each score in a sample to have an influence on the final measure. To obtain a
mean, all of the observations (X) in a sample are added together, then that sum is
divided by the total number (n) of observations in the sample.
Mean =  X/n
Median
The median is a second measure of central tendency. Unless you have a
particularly large group of numbers, this is a rather simple measure. To determine
the median, sort your data so the observations either increase or decrease in value.
The median is then simply the middle observation in your data set.
If the number of observations in your data set is odd, the middle observation
can be found by dividing the total number of observations by two, then rounding the
resulting number up to the nearest whole number. For example, if you have 25
observations in your data set, the median would be obtained by dividing 25 by 2,
giving 12.5. Rounding up to the nearest whole number gives 13, making observation
13 in your ordered data set the median value for your sample.
If the number of observations in the data set is even, the median is obtained by
averaging the values for the middle two observations after the data set has been
ordered. For example, in a data set with 24 values, the middle two observations
would be observations 12 and 13. If the value for observation 12 was " 100" and the
value for observation 13 was " 102", the median for the data set would be "101 ".
Part 2 - Measures Of Variability Or Dispersion
Means and medians are expressions of central tendency. They provide a
general picture of what a group of numbers is like, however, measures of central
tendency can be misleading. For example, let us say the average height of two
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basketball teams is 180 cm. On team A, all of the players are exactly 180 cm, but
on team B, one is 180 cm, two are 160 cm and two are 200 cm. Knowledge of this
variation, or dispersion from the mean would be meaningful to the coach of team A.
A number of measures of dispersion are in common use; the range and the
interquartile range (used in conjunction with the median) and the variance, standard
deviation and coefficient of variance (used in conjunction with the mean) being the
most common.
The Range
When the nature of the distribution is not known, or when the distribution is
known to be other than normal (e.g., skewed), the range will give a rough idea of the
dispersion. The range is the difference between the high and low scores. The
range does not tell us anything about the nature of the distribution, only its extent.
If our sample observations were 18, 17, 17, 12, 11, 11, the range would be 7, (18 - 11
= 7).
The Interguartile Range
Quartiles, like the median, represent points in an ordered set of observations.
The first quartile (Ql) is a point below which 25% of the observations fall; 50% of
the cases fall below Q2 and 75% below Q3. The interquartile range includes those
50% of the scores that fall between Q I and Q3. When the distribution of
numbers is not normal because of the extremes on either or both ends, an
inspection of the middle 50% may prove to be most revealing in describing a group
of numbers. Inspect the following set of ordered data. In this data set, the first
quartile includes observations 1 through 5, the second quartile includes
observations 6 through 10 the third quartile includes observations 11 through 15,
and the interquartile range extends from 17 to 61.
Obs. #
Value
Obs. #
Value
1
2
3
4
5
6
7
8
9
10
10
10
11
13
15
17
21
23
26
30
11
12
13
14
15
16
17
18
19
20
35
39
42
50
61
73
95
102
113
140
The Variance
Another group of measures of dispersion is based on the distance of each
measurement from the center of the distribution (i.e. the mean or median). The
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first of these is the variance. To obtain the variance of a set of numbers, follow
these steps:
1.
2.
3.
4.
5.
Subtract each observation from the measure of central tendency (usually the
mean) to obtain the "deviation" of each observation.
Square each of these deviations.
Add together all of your squared deviations.
Subtract 1 from the total number of observations in your sample (n- 1)
Divide your "sum of squared deviations" by this number.
Bingo! You've got the variance of your data set. Just as a side note, dividing by "n1 " gives what is technically know as the sample variance.
The Standard Deviation
With a normal distribution, the mean is the most accurate and descriptive
measure of central tendency because it considers the magnitude (or value) of each
score. In like manner the standard deviation considers the magnitude of each score
and therefore is the preferred measure of dispersion if the distribution is normal.
Although the variance is a useful starting point in describing in the variability of a
normal data set, the standard deviation is more commonly used (for reasons we will
not get into now). Once you have the variance of a set of scores, getting the
standard deviation is simple. All you have to do is take the square root of the
variance.
Coefficient of Variance
The final measure of dispersion is the coefficient of variation (CV for short).
The CV is a useful measure of dispersion because it allows us to compare the
magnitude of variation between two sets of data when the means (and the ranges)
differ significantly. To obtain the CV, simply divide the standard deviation by the
mean, and multiply the result by 100.
Part 3 - A Test Of Location
Perhaps one of the most basic questions we ask in science is "Do these two
[fill in the blank with your favorite research topic] differ?" In most cases, we begin
answering that question by using a statistical test that looks for differences in the
location of the measure of central tendency for the two "things". The most
commonly used test for differences in central tendency between two samples is the
t-test.
The t-test should only be applied to independent samples. This means the
selection of a member of one group has no influence on the selection of any member
of the second group. If you do not understand what this means, you need to ask me
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about it, because this concept of independence between or among samples is a very
basic assumption in many of the statistics commonly used in the biological sciences.
The t-test incorporates not only a measure of central tendency (the mean)
for the samples, it also incorporates measures of dispersion (the standard
deviation) and the total number of observations in each sample (the sample size).
Our ability to detect a true difference between two samples depends not just on
the absolute difference between the means, but on the amount of dispersion and
the size of our sample. In general, for the same difference between means, as
variability decreases, our ability to detect a statistically significant difference
increases. Conversely, as sample size increases , our ability to detect a difference
increases.
Once our arithmetic manipulations are completed, the result is applied to a
table of values (often called a "values of t" table, see the attached sheet). The
table has several columns of values or levels of significance. The researcher must
decide how much flexibility can be allowed. This will depend on the degree of
accuracy of the measuring instrument. Most scientific research sets the
significance level at .05. This means there is only a 5 percent chance an error has
been made in saying the means of the two groups were not exactly alike.
The t-test
The formula for the t-test was developed by W.S. Gossett in the early
1900's. As a brewery employee he developed a system of sampling the brew as a
periodic check on quality. He published his works under the name Student which
explains why you hear this formula called the Student t from time to time.
In order to proceed with a t test, the "standard error of the difference"
between the means of the two groups must be determined. Just as the name
implies, this is a measure of variability in our estimate of the difference between
the two means. To illustrate the t-test, look at the following table. The numbers
represent average fish lengths from an urban and rural stream.
Mean Length
Urban Stream
Rural Stream
80
84
Standard
Deviation
9.16
7.83
Sample Size
30
30
At first glance this would seem to suggest fish in the rural steam are larger.
But, we must ask the question, is the 4 mm difference a chance difference or does
it really represent a true difference. If we were to take another sample from each
stream and do it again, would we get the same results?
To answer this question, you must first compute the standard error of the
difference (sdiff). The following formula is appropriate for independent samples, if
the distributions are normal. Once you compute the standard error of the
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difference you can determine if a true (statistically significant) difference exists
with the aid of the t-test and tables or an appropriate computer program. If we
plug our numbers in from the data table, we find that the "Standard Error of the
Difference" for our 4 mm length difference is 2.29 mm. The next step in the
process is to use our value for sdiff to calculate a t-score. The formula for the tscore is given on the next page.
As mentioned above, in order to determine if there is a true difference
between the means, you need a table of values of t. You also need to know the
degrees of freedom used in the problem. Degrees of freedom are determined by
adding the number of scores in each group and subtracting 2, [n1 + n2 - 2].
Looking at the attached sheet, the left hand column is labeled df (for degree
of freedom), Look down that column until you find the appropriate df for your
problem and read across to the column labeled .05 for two tailed tests. If the
figure you found in the t-test is larger than the .05 figure there is a true
difference between the means.
For the example above, looking down the column for degrees of freedom we
pick row 50. The degrees of freedom for this problem would be 58, (n, + n2 - 2)
but in that there is no row 58 we would drop to row 50, the more conservative
alternative. The two tailed values in this row are .05 = 2.009, .01 = 2.678 and .001 =
3.496. Our t value was 1.746 which is smaller than 2.009 so we can say there is not
a significant difference between the means at the .05 level or 5% level. It should
be noted that this only tells us if there is a significant difference, not the
direction of any difference; to conclude the latter requires more than the intent of
this survival manual.
Formula to Calculate the Standard Error of the Difference
5
Formula to Calculate t-scores
Part 4 - Correlation
Even though we often think of statistical techniques as a way of telling if two
things, groups or sets of data are different, we can also use statistical analysis to
ask if two things are related. For example, we might want to know whether the
rate of growth in a bacterial colony is related to temperature. Alternatively, we
might be interested in the relationship between the density of two types of plants
that use similar types of resources. These and many other questions can be tested
using a statistical process called correlation.
There are several different formulas to determine correlation. The most
frequently used is the Pearson product moment correlation, named after Karl
Pearson. This procedure requires that the numbers be at least interval in nature.
It also requires that the data be paired.
We can use the bacterial colony question as an example of what is meant by
paired data. Each item of the pair must have a common denominator. In this
instance, the number of colonies counted and the temperature both come from the
same petri dish. The temperature in a given dish is one of the paired numbers and
the number of bacterial colonies is the other member of the pair. The pair of
scores for dish #1 (see the next table) are 100 (Number of colonies) and 85
(temperature).
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Dish Number
Number of Colonies
Temperature
1
2
3
4
5
6
100
70
85
75
65
60
42
34
34
32
30
29
From the data above it can easily be seen that the warmer the dish, the
greater the number of colonies. Knowing this may be adequate for your purposes,
or you may wish to have a more precise measure of the degree of relationship. You
may even be interested in graphically showing this phenomenon.
Scattergram
The graphical procedure to show correlation is called a scattergram. To
develop a scattergram you first place one set of the numbers in a column in
descending order and label it X. In our example of temperature and bacterial
growth, we can label the number of colonies as the X column. The column of the
paired figures (temperature) is listed next to the X column and labeled the Y
column. Do not break up the pair. Except in the instance of a perfect positive
correlation the Y column will not be in perfect descending order.
On graph paper, a horizontal scale is marked off for the X column near the
bottom of the page. The values increase from left to right. A vertical scale is
drawn to intersect one line to the left of the lowest X value. The vertical scale
represents the temperature on the Y column. The horizontal scale is the X axis and
the vertical scale the Y axis. The length of the scale on the X axis should be about
equal to the length of the scale on the Y axis. There is no rule that requires this,
however the procedure helps space the scattergram which makes it easier to
interpret. Each pair of figures is plotted on the graph. A dot is placed where the
X and Y values intersect for each pair of figures. The dots are not connected.
Graph the data presented above on a sheet of graph paper. If you have done
this correctly, the dots should form nearly a straight line from lower left to upper
right. This configuration would be considered a positive correlation, and the more
nearly the dots form a straight line the stronger the relationship between the two
variable.
If the dots had formed a pattern from upper left to lower right, the
correlation would have been regarded as negative; that is to say there was indeed a
relationship, but the conclusion would be that the variables plotted had an inverse
affect on each other. If we had continued to increase the temperature above 50,
we might have seen a decline in the number of colonies as we exceeded the
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temperature tolerance of the bacterial species we were testing. If we had only
graphed the response to temperatures above 50, we might have observed a
negative relationship. If the dots appeared randomly over the scattergam, you
would interpret this as no relationship between the X and Y column measurements.
Computation
As mentioned above, there are several formulas for determining correlation.
At this point we shall only be concerned with the raw score method. The formula
is:
Correlation coefficients should always fall between (+1.0) and (-1.0); it can not
be anything else. The closer to 1.0, the more perfect the relationship is between
the two sets of numbers in a positive way, that is, when one figure goes up, the
corresponding paired number also goes up and in like manner when one of the paired
numbers goes down, so will the other. The negative correlation supports the
opposite conclusion, when the first number goes up, its pair goes down or vice versa.
The closer to zero, the less likelihood of any relationship.
Part 5 – Contingency Table Analysis
Introduction
The types of statistics you have seen so far allow you to describe measures
of central tendency and dispersion (means, medians, ranges, standard deviations,
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etc.), test for differences among populations (the t-test), or look for relationships
between two measures (correlation). One additional group of statistical techniques
that is often useful in ecological studies are tests that allow us to compare the
frequency of observations in various categories between two different sets of
observations. Our comparisons may be between a set of real world observations
and some expected distribution of observations, or we may want to compare
frequency of counts between two sets of real world observations drawn from
different areas or populations.
You have already encountered the first type of comparison in your
introductory biology courses when you compared observed phenotypes in a
population to those that would be expected if the population were in HardyWeinberg equilibrium. This general class of tests is known as a "goodness-of-fit"
test since we are attempting to determine if our observations "fit" our
expectations. The most common goodness-of-fit test is the chi-square goodnessof-fit test.
The second type of comparison is more common in ecological field studies.
In this case, we usually have two variables that we have used to classify our
observations. Each variable has two or more categories. Based on these categories
we can set up an table, with n columns (where n = the number of categories for the
first variable) and p rows (where p = the number of categories for the second
variable). Each observation can then be assigned to one of the cells in our table.
The question we hope to answer is whether the value for the column variable has
any influence on the values of the row variables. Statistically, we are asking the
following question:
"Is the value of our second variable independent of the value for our first
variable?"
This type of test is known as a chi-square contingency table analysis. The
null hypothesis for contingency table tests is "The effect of our first variable is
not influenced by our second variable." If this seems a little fuzzy, hang on while
we run through an example. If it still seems fuzzy after that, come and talk to me
about it!
Frequency Distributions
Before we get too far along on the contingency table test, we should take a
moment to understand the idea of frequency distributions. Let's start with a
simple idea that everybody can grasp, gender. Assume we have a class that has 50
students in it. We can classify each person in the class as male or female. If the
class is split evenly according to gender, then the frequency of males in the class is
25 and the frequency of females is 25. Gender is easy because it is what is
commonly called a discrete or "nominal scale" variable. There is little doubt how
many categories we have.
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But what if we wanted to categorize the class not by gender, but by height?
Height is a "continuous" variable, meaning that if we had a measuring device that
was accurate enough, there are an infinite number of categories into which we could
place the members of our class. What we normally do with continuous data when we
want to determine frequencies is establish intervals of equal "width" or magnitude,
and assign our observations to those categories. In fact, we do this as a matter of
course every day. For example, we might hear a person claim that they are 175
centimeters tall (or we would if we used metrics like the rest of the world), but if
we were to measure their true height, we would find that in fact they were actually
174.67 cm in height. The "width" of our categories depends both on the number of
observations we have and the range of our data.
Contingency Table Analysis
To illustrate the idea of contingency table analysis, let's continue with the
same example. We can ask the question, does the gender of an individual have an
effect on their height? As a null hypothesis, we could state this question as "The
gender of an individual has no effect on their height". Assume that we measure the
height and record the gender of all the members of our hypothetical 50 member
class. The raw data is given in the table below.
Height
150
151
152
152
153
154
155
155
156
157
Gende
r
F
M
M
M
F
F
F
M
M
M
Height
158
158
159
160
161
161
162
163
164
164
Gende
r
F
F
F
M
M
M
F
F
F
F
Height
165
166
167
167
168
169
170
170
171
172
Gende
r
M
M
F
M
F
F
F
F
F
M
Height
173
173
174
175
176
176
177
178
179
179
Gende
r
F
M
F
M
M
F
F
M
M
F
Height
Gende
r
M
M
M
M
M
F
F
F
M
M
180
181
182
182
183
184
185
185
186
187
When setting up a contingency table, we want to have at least 5 observation
in 80% of our cells (one of those statistical rules that you should just take as a
given), so when we are deciding about the "width" of our intervals, we should keep
this in mind. For the data above, an interval width of 10 cm works pretty well. That
gives us four classes, individuals less than or equal to 160 cm, individuals from 161170 cm, from 171-180 cm, and over 180 cm. We can then assign all of our
observations to one of the cells of a 2  4 contingency table, where the columns
represent gender and the rows represent the height interval.
10
Height
Interval (cm)
Gender
Male
Female
< = 160
161 - 170
171 - 180
> 180
7
5
7
6
7
9
6
3
This table represents our observed frequencies. What do we compare it to?
We need to create a second table that represents what our data would look like if
gender and height were totally independent of one another. To do this, we use
our observations to estimate what our cell frequencies would be if this were true.
The first step in this process is get total counts for each row, column, and for the
overall table, as illustrated below.
< = 160
161 - 170
171 - 180
> 180
Column Total
Male
Female
Row Total
7
5
7
6
25
7
9
6
3
25
14
14
13
9
Grand Total
50
The next step is to use the row total, the column total, and the grand total to
determine the expected value for each cell in the table. To do this, you multiply
the column total by the row total for each cell, then divide that number by the
grand total. The table below illustrates the process.
< = 160
161 - 170
171 - 180
> 180
Column Total
Male
Female
Row Total
(14*25)/50 = 7
(14*25)/50 = 7
(13*25)/50 = 6.5
(9*25)/50 = 4.5
25
(14*25)/50 = 7
(14*25)/50 = 7
(13*25)/50 = 6.5
(9*25)/50 = 4.5
25
14
14
13
9
Grand Total
50
Finally, now that we have our expected distribution, we can compare that to our
observations to determine if there is a difference. To accomplish this, we begin by
subtracting our expected values from our observed values for each cell in our table.
We square that number, then divide by our expected value.
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Step 1
Male
< = 160
161 - 170
171 - 180
> 180
7-7 = 0
5-7 = -2
7-6.5 = 0.5
6-4.5 = 1.5
Step 2
Female
7-7 = 0
9-7 = 2
6-6.5 = -0.5
3-4.5 = -1.5
Male
Female
(0) /7 = 0.00
(-2)2/7 = 0.57
(0.5)2/6.5 = 0.04
(1.5)2/4.5 = 0.50
(0)2/7 = 0.00
(2)2/7 = 0.57
(-0.5)2/6.5 = 0.04
(-1.5)2/4.5 = 0.50
2
Once we have the values from step 2, we sum these values to obtain our chi-square
value. In this case, our calculated value is 2.22. We then take this value to our
handy chi-square table (attached to this handout) to determine if we should reject
or fail to reject our null hypothesis. To get your p-value, you need to know (as
always) the degrees of freedom you have. In contingency tables, your degrees of
freedom are equal to (number of rows - 1)  (number of columns -1) = (4-1)  (2-1) =
3. Looking at our table, we find that with 3 degrees of freedom, the chi-square
value at the 0.05 level of significance is 7.815. To reject our null hypothesis, our
calculated chi-square value must be greater than 7.815. Since our calculated value
is less than this, we cannot reject our null hypothesis. Based on this data, the
height of an individual does not depend on his or her gender. This does not surprise
me since I used a random number generator to assign heights to males and females!
Practice, practice, practice!
To be sure that you can do a contingency table analysis, play with the data set
below on your own time. The data represent the color patterns of a species of
tiger beetle during different times of the year. Your null hypothesis is that season
does not affect color. If you carry out the test correctly, you should obtain a chisquare value near (given rounding errors) 27.7.
Color
Season
Bright Red
Pale Red
Early Spring
Late Spring
Early Summer
Late Summer
29
273
8
64
11
191
31
64
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The information given in this survival manual is drawn from the following three
sources:
Ott, L, R. F. Larson, and W. Mendenhall. 1983. Statistics: A tool for the social
sciences.
Roys, K. B. 1989. Research and Statistics: An introduction
Sokal, R. R. and F. J. Rohlf 1981. Biometry
Additional references that are useful for sampling and statistical information are
listed below.
Cochran, W. G. 1977. Sampling techniques.
Green, R. H. 1979. Sampling design and statistical methods.for environmental
biologists.
Krebs, C. J. 1989. Ecological Methodology.
Southwood, T. R. E. 1979. Ecological methods.
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