Listed
below are definitions of some common summary statistics:
1)
Mean - The average of all
your data points. To obtain the mean,
simply add up all your values, and divide by the total number of data points.
2)
Standard Deviation - this is a estimate of deviation or probable error within your
measurements. This is an expression of
precision--how widely your experimental values vary. Standard deviation can be used not only to
express precision, but also the help determine whether an individual value from
your set of values is in error.
If
one could take an infinite set of measurements of something, then the
distribution graph of these values should look something like this:
This
is called a normal distribution, or Gaussian curve. At the center of the curve is the mean, which
reflects the highest number of measured values, and the best possible value for
your experimental measurement.
A standard deviation
is a percentage of the mean value of your data.
As the graph shows, most of the values cluster together around the mean,
and taper off with distance in either direction. The number of values falling
outside of two standard deviations above or below the mean is very small and
probability predicts that it will be less than or equal to 5 percent of the
total number of measurements (or, 95% of all data is within 2s, i.e. 2 standard deviations of the
mean). Therefore it is common practice
to reject data, as belonging to your population, if it falls outside of two
standard deviations (2s), as those
data fall within the outermost 5% of all possible values, and are probably in
error. Standard deviation is calculated
by:
s = Ö
S
(Xi - X)2
(N - 1)
which, in
English, means to take the differences between the individual values (Xi) and the mean (X),
square those differences, and then sum all the squares. Divide the sum of the squares by the total
number of values (N) minus one, and take the square root of your result.
The
easiest way to determine s (standard
deviation) is by making a table of your data, then putting each value into the s equation. A useful example would be the measurement of
stonefly larvae in Oak Creek. At one site six values (the Xi) were obtained for
individuals in an area of stream: 147,
140, 92, 142, 153, and 146. Here X (the
mean) is calculated by summing all the Xi values (820) and dividing by how many Xi values there are
(6). 820/6 = 137, so X = 137 ppm.
Xi Xi - X (Xi - X)2
147 10 100
140 3 9
92 -
45 2025
142 5 25
153 16 256
+
146 9
+ 81
S (Xi) = 820 S (Xi - X)2 =2496
So the Standard Deviation is:
s =Ö
S
(X1 - X)2 = Ö 2496 =
22.34, or 22 (utilizing correct
(N
- 1) 5 significant
figures)
Two standard
deviations would be 44. The X plus or
minus 2s = 93 to 181, and is the
range of accepted deviation to be included in the population. The value of 92 lies outside of 2s, and can be rejected as being in
error. The lower standard deviation
indicates greater precision in estimating the mean. Microsoft Excel provides a formula for
determining the standard deviation.
Many
times people use the Standard Error of the Mean (SE) to reflect the variation
in the sample. The SE is calculated as
the standard deviation divided by the square root of the samples size. It is a nice statistic to eye-ball whether
the mean of two samples are different.
If the mean + 1SE of each sample do not overlap, then in most
cases the samples will be statistically different using a t-test.
Once
you have collected your data you must determine if it is valuable (does it say
anything?), and you also need to find a way to present it so that it is clear
and easy to understand. Knowledge of
graphing and statistics is tremendously helpful in presenting your data and
determining if the results are meaningful.
All the graphing and statistics you will need to do can usually be accomplished
on a computer using Microsoft Excel.
Excel is a
powerful spreadsheet program which allows you to organize and manipulate data
you have collected in the field, as well as graph and run statistical tests on the
data. You can produce many different
types of graphs in Excel, depending on the nature and format of your data. The processes for producing different types
of graphs vary, so a general description of the procedure will be given
here. In general, you will highlight one
or more columns of your data with the mouse, then
click on the “Chart Wizard” icon located along the top of your screen (it looks
like a little bar graph). From here you
simply follow the directions to assign your x and y variables, label axes, and
give the graph a title. Once you hit
finish and your graph appears on the screen, you may change the dimensions by
clicking and dragging on the corners of the graph, and you may edit and modify
any aspect of the graph by double-clicking on the specific thing you want to
change. A sample Excel graph follows.
Points to Remember when creating figures
Figure titles are always
placed at the bottom of the figure.
The title should
be meaningful and explain what the figure illustrates.
The axes should be labeled and in general, abbreviations should never be
used, unless the abbreviation is explained in the figure title.
Font sizes should be consistent for axes, axes titles, and figure title.
Most
of the statistical tests we ask you to run can be conducted in Microsoft
Excel. Below are descriptions of the
tests you will be expected to use and directions for how to run them on the
computer.
Parametric vs Non-Parametric Statistics
There are two classes of statistical tests that are
typically used in biology.
Parametric statistical tests assume normal distributions and
equal variances, whereas non-parametric tests have fewer assumptions about the
distribution of the data.
Consequently, parametric tests are more powerful but
assumptions that are required for their usage are more easily violated than
non-parametric tests.
There is usually a nonparametric equivalent for every
parametric test.
t-test
~ Mann-Whitney U
ANOVA ~ Kruskall-Wallis
Linear Regression ~ Spearman Rank
Listed below are examples of the types of questions and corresponding
statistical tests used in ecology (Assuming parametric statistics assumptions
are met).
What is the difference in the mean between two
populations?
t-test or paired t-test
What is the difference in the mean of one trait among
more than two populations?
ANOVA
What is the difference in the mean of one trait among one
or more pairs or combinations of populations after one has determined that the
ANOVA test among populations is significant.
Post-hoc multiple comparison tests or A-priori contrast
tests
What is the difference in the mean of more than one trait
among more than two populations?
MANOVA
Are two or more variables related to each other, that is
do they tend to co-vary?
Correlation
(assuming no cause and effect) or Regression (assuming cause and effect)
t-Test
Explanation
t-Tests are used for determining the similarity or
difference between two sampled populations.
The value you get is called a t-statistic,
and should be used in its absolute value form.
The smaller the value, the more similar are your two samples. Other important values include t-critical value (two-tail) which is
calculated with an alpha level of 0.05 (remember the 5% on either side of 2
standard deviations from the mean). If
your t-statistic is greater than
this number, you can reject the null hypothesis (that the two samples are the
same), and say that there is a statistically significant difference between the
two samples. The P-value (two-tail) is important also, in that it gives the exact
probability of mistakenly rejecting the null hypothesis when you shouldn’t
have. A P-value less than 0.05 means the same thing as having a t-statistic greater than
your t-critical value, and thus allows you to reject the null hypothesis as
above.
Instructions for Excel*
1) Open Excel.
2) Under the file menu, select NEW, which will give
you a blank worksheet.
3) You will use two columns, each of which
represents one of the data
samples you collected.
4) At the top of each column, type in a label
that describes the sample.
5) Under each label, enter the data you have
collected for that sample.
6) At this point you should have your data in
two columns with a label at the top of
each.
Now it’s time to analyze your data.
7) Go to the options menu and select Tools.
Under Tools, select Data
Analysis.
8) Select the option called: t-test: two sample assuming equal variances.
9) Selecting this will bring up another window
that asks for information
regarding where Excel can find your
data. This is called the input range.
10) Enter the cell name and number where the data
begin and end, separated by a
colon
(example, a2:a6).
11) You need to do this for both columns of data,
separately.
12) You also need to tell Excel where to put the
results of the t-test. This is called
the output range.
13) It is easiest to specify the output range as
the row directly below the data.
(example, a10).
14) Press OK.
* from a handout developed by Dr.
Diane Ebert-May at NAU’s Science and Mathematics Learning Center.
STATISTICS USING MICROSOFT EXCEL
(continued)
Analysis of Variance
The
Analysis of Variance test is an extension of the t-test. T-test is used when you compare two
populations, Analysis of Variance tests are used when you want to compare more
than two populations to determine the similarity or difference among the
sampled populations. Analysis of
Variance tests are referred to by several names including F-tests, ANOVA, and
AOV. The value obtained is called a F-statistic,
and should be used in its absolute value form.
The smaller the value, the more similar are your samples. Other important values include the F-critical value, which is comparable
to the t-critical value, and the P-value
which is interpreted the same as it is in the t-test. The ANOVA only tells you if there are
differences between at least two populations in your set. If you want to examine differences between
population pairs you need to conduct multiple comparison tests. If you know beforehand what groups should be
different you can apply a priori contrast tests. Most people apply post-hoc multiple
comparison tests of which there are many to choose from and available in
statistical programs.
The
simple one-way ANOVA includes just one classification variable, where groups
differ in a single way (e.g. hi, med, lo fertilization). You may have an experiment where individuals
differ in more than one way (degrees of fertilization and watering). In these cases you apply 2-way on up to n-way
ANOVA’s to
include as many independent factors as you have in your design. In these analyses you can examine main
effects of single independent variables and interaction effects between main
effects.
ANOVA
tests come in many variations (nested, repeated measures, multivariate), which
can accommodate a wide variety of experimental and observational designs.
Instructions for Excel for a One-Way ANOVA*
1) Open Excel.
2) Under the file menu, select NEW, which will give
you a blank worksheet.
3) You will use three columns, each of which
represents one of the data
samples you collected.
4) At the top of each column, type in a label
that describes the sample.
5) Under each label, enter the data you have
collected for that sample.
6) At this point you should have your data in
three columns with a label at the top of
each.
Now it’s time to analyze your data.
7) Go to the options menu and select Tools.
Under Tools, select Data
Analysis.
8) Select the option called: Anova: Single Factor.
9) Selecting this will bring up another window
that asks for information
regarding where Excel can find your
data. This is called the input range.
10) Enter the cell name and number where the data
begin and end, separated by a
colon
(example, a2:c6).
11) You also need to tell Excel where to put the
results of the t-test. This is called
the output range.
12) It is easiest to specify the output range as
the row directly below the data.
(example, a10).
13) Press OK.
STATISTICS USING MICROSOFT EXCEL
(continued)
Correlation Analysis: Pearson
Correlation
Correlation Analysis: Pearson
Correlation
A correlation is a way to
statistically measure the association between two variables. A correlation produces an r-value, which tells you how closely
correlated the two variables are. r-Values range from 0-1, with 0 indicating completely
unrelated variables and 1 indicating a perfect (linear) relationship between
the two. If you square the r-value (r2) you get a number that can be expressed as a
percentage, telling you how much of the change in one variable can be explained
by the other.
For example, imagine you were trying to determine if sample temperature
or sample density more strongly affected the number of invertebrates
present. Let’s say you calculated an
r-value of .15
for the correlation between sample density and # of invertebrates present. That gives an r2 of .0225, or
2%. This means that the # of
invertebrates is essentially not influenced by sample density. Let’s say then for sample temperature and #
of invertebrates you got an r-value of .85.
This gives an r2
of .7225, or 72%. This means that 72% of
the variation in the number of invertebrates per sample is due to variations in
sample temperature.
Instructions for Excel*
1) Open Excel.
2) Under the file menu, select NEW, which will give
you a blank worksheet.
3) You will use two columns, each of which
represents one of the data
samples you collected.
4) At the top of each column, type in a label
that describes the sample.
5) Under each label, enter the data you have
collected for that sample.
6) At this point you should have your data in
two columns with a label at the top of
each.
Now it’s time to analyze your data.
7) Go to the options menu and select Tools.
Under Tools, select Data
Analysis.
8) Select the option called: Correlation
9) Selecting this will bring up another window
that asks for information
regarding where Excel can find your
data. This is called the input range.
10) Enter the cell names and numbers where the
data begin and end, separated by
a colon (example, a2:b6, where the data starts at a2 and ends
at b6) This
range should include both columns of data at one time
but should not include
the data labels.
11) You also need to tell Excel where to put the
results of the regression. This
is called the output
range.
12) It is easiest to specify the output range as
the row directly below the data.
(example, a7)
13) Press OK.
* from a handout developed by Dr. Diane Ebert-May at NAU’s Science and Mathematics Learning Center.
STATISTICS USING MICROSOFT EXCEL
(continued)
Regression Analysis: Linear
Regression
Linear regression is a
statistical method that allows you to predict one variable from another. If we graphed the data from our invertebrate
experiment (X-axis = temperature, y-axis = # of invertebrates), we would get a
scatter plot of points. The Excel
program can draw the line of best fit for us, which is a regression. The equation for the regression is in the
form Y=mx+b. Y
is the dependent variable, X is the independent variable, b is the y-intercept,
and m is the slope. Referring again to
our previous example, by using this equation you can predict how many
invertebrates you would find if the sample temperature were 30 oC. As
with the correlation you are also provided with an r2 value which
again tells you what percentage of the observed variation is explained by your
selected variable. The closer your r2
value is to 100% the more dependable your predictions will be.
Instructions for Excel
The following instructions assume that you have already
entered your data into an Excel worksheet for graphing or other statistical
tests.
1) Highlight both columns of your data with the
mouse.
2) Click on the “Chart Wizard” icon located
along the top of the screen.
3) Choose the XY (Scatter) chart type, and the option with the points not
connected by lines.
4) Follow the rest of the graphing
instructions, labeling your axes and giving the
chart a title.
5) Once your graph is displayed on the
worksheet with your data, click on one
of the data points with your right mouse button.
6) From the small menu which appears, choose
the “add Trendline” option.
7) You will be presented a series of regression
types to choose from, and the
linear option will
already be highlighted.
8) Before you hit OK, however, click on the
Options menu and check the boxes
entitled “Display equation on chart” and “Display R-squared
value on chart.”
9) Now click OK and your graph, complete with
the equation and r2-value, will
be displayed on the worksheet next to your data.
*Note: the regression
tool in Excel can be used to make regressions other than linear
STATISTICS USING MICROSOFT EXCEL
(continued)
Frequency Analysis: Chi-square
test
Explanation
The statistical procedures described above all deal with
testing hypotheses about continuous variables--those which involve direct
measurement of variables and the comparison of values obtained. Sometimes ecologists deal with data that are
grouped into classes, such as the number of individual organisms from each of
10 different species obtained in a soil sample.
You might want to know if the distribution of individuals among each
species differs from that determined in a previous study for a different soil
type. The chi-square test is a simple
way to test for differences in frequency data--variables distributed into two
or more classes (in this example, the classes are species). As with the t-test, you will use Excel to
calculate a test statistic, in this case the chi-square (X2)
statistic. The chi-square test statistic
is then compared with a critical value for X2. By comparing the test statistic with the
critical value obtained from a statistical table or software package, you will
decide whether you are able to reject your null hypothesis at the predetermined
probability (typically P = 0.05).
Instructions for Excel
1) Open
Excel.
2) Under
the file menu, select NEW, which
will give you a blank worksheet.
3) You
will use two columns. One of these represents
the data from the
samples
you collected. The other may be a second
set of sample data, or it
may
be an expected or hypothesized frequency suggested from ecological
theory,
past studies, or a well-known frequency distribution, such as a
normal
distribution. In all cases, you are
testing to see if the “observed” and
“expected”
frequencies differ.
4) At the
top of each column, type in a label that describes the sample
frequency
distribution.
5) Under
each label, enter the appropriate data you have collected.
6) At
this point you should have your data in
two columns with a label at the
top
of each. Now it’s time to analyze your
data.
7) Go to
the insert menu and select Function. Under Function,
select
Statistical.
8) In the
right-hand column select the option called: ChiTest.
9) Selecting
this will bring up another window that asks for information
regarding
where Excel can find your data. These
are the input ranges. In
this
test there are two: one is called the “observed,” and the other the
“expected range.” Note that even when you are comparing two
sample frequencies, you will have to
calculate the expected values. Think of
“expect” the distribution to exhibit a
random pattern. You will compare this
random pattern of "expected values" to the "observed" data.
10) Enter
the cell name and number where the first set of data begins and ends,
separated
by a colon. (Example a2:a6, where the
data starts at a2 and ends at
a6). Next enter the cell name and number where the
second set of data begins
and
ends, in the same manner.
11) After
you enter this second range, the p-value for your X2 test will
automatically
appear
in the upper right-hand corner of the box.
You may want to write this
down
now before hitting the “finish” button, as sometimes it can be difficult to
find
where
Excel chose to print out your p-value.