Description
Attached is my assignment that has to be done on excel and read the directions.
Table 3. Effects of pesticide pollution on hatching success of fish eggs
Group
1
2
3
4
5
6
7
8
9
10
Proportion eggs hatching
No Pesticide
0,80
0,76
0,81
0,90
0,95
0,84
0,88
0,99
0,86
0,93
Pesticide
0,50
0,45
0,68
0,77
0,64
0,60
0,54
0,57
0,62
0,57
Let’s assume that a researcher is attempting to
determine the effects of pesticide pollution on the
hatching success of fish eggs. He has noticed that fish
eggs in streams near agricultural fields fare poorly, while
those in undisturbed areas hatch successfully. The
researcher sets up a lab experiment in which ten groups
of fish eggs are allowed to develop in unmanipulated
stream water, and ten groups of eggs develop in stream
water with the addition of pesticide. The proportion of
eggs hatching in each group is tallied (Table 3), and
descriptive statistics for the two groups compared.
Descriptive Statistics:
Provide descriptive statistics for the different experimental groups
Table 1:
Mean
Median
Which measure of central tendency is the most appropriate for thi
Data presentation:
Insert a graph depicting the data. Don’t forget to include error bars
rent experimental groups in the problem in the table, and write a title for the table in the space provided (after Table 1:).
Mode
Std. Dev.
most appropriate for this data set? Explain your answer.
orget to include error bars, axes titles, and a legend.
d (after Table 1:).
Problem #2: Comparing amphibian populations
The Data:
Pond
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Irrigated
Natural
Natural
Natural
Natural
Natural
Natural
Natural
Natural
Natural
Natural
Descriptive Statistics:
No. of
masses
0
2
1
2
1
0
0
0
86
7
10
10
35
105
62
8
5
99
115
257
Provide descriptive statistics for the different experimental groups in the problem in the
Table 1:
Mean
Median
Mode
Std. Dev.
Which measure of central tendency is the most appropriate for this data set? Explain yo
Data presentation:
Insert a graph depicting the data. Don’t forget to include error bars, axes titles, and a leg
Problem #2: Comparing amphibian populations
Abstract: To determine if the spraying of wastewater effluent was adversely affecting Jefferson salamanders (Amb
jeffersonianum ) in central Pennsylvania, researchers estimated population sizes in areas with and without wastewa
species deposits their eggs in temporary ponds that fill in the late winter and usually dry in the mid to late summer.
deposited in “masses”, with each mass being roughly the size of a baseball and containing 10 – 35 embryos. As cou
numbers of adults is quite difficult, breeding populations in each pond were estimated by counting the number of eg
during the breeding season. The number of Jefferson salamander egg masses were counted in 10 wastewater-irrig
natural temporary ponds in the spring of 1997 after breeding had ceased for the year.
Reference: Laposata, M. M. and W. A. Dunson (2000). Effects of spray-irrigated wastewater effluent on temporary
amphibians. Ecotoxicology and Environmental Safety 46:192-201.
Variable Names:
Pond type: Ponds irrigated with wastewater effluent (Irrigated) or natural ponds with no effluent inputs (Natural)
No. of masses: Number of Jefferson salamander egg masses counted
oups in the problem in the table, and write a title for the table in the space provided (after Table 1:).
or this data set? Explain your answer.
bars, axes titles, and a legend.
rson salamanders (Ambystoma
ith and without wastewater spraying. This
he mid to late summer. Eggs are
10 – 35 embryos. As counting the
unting the number of egg mass deposited
ed in 10 wastewater-irrigated and 10
er effluent on temporary pond-breeding
uent inputs (Natural)
Problem #3: Effects of copper concentration on Lemna gibba
The Data:
Copper No. of live
(ppm)
fronds
0
167
0
162
0
195
0
184
0
172
0,001
158
0,001
160
0,001
163
0,001
177
0,001
184
Descriptive Statistics:
Provide descriptive statistics for the different experimental groups in the problem in the
Table 1:
Mean
Median
Mode
Std. Dev.
Which measure of central tendency is the most appropriate for this data set? Explain yo
Data presentation:
Insert a graph depicting the data. Don’t forget to include error bars, axes titles, and a leg
Problem #3: Effects of copper concentration on Lemna gibba
Abstract: A researcher investigated the effects of potentially toxic copper concentrations in water on the growth of
aquatic plant, Lemna gibba . Five large glass bowls were filled with artificial pond water with no copper (0 parts per
or ppm), and five with artificial pond water with 0.001 ppm, a naturally-occurring copper concentration in many lakes
southeastern United States. Thirty-eight Lemna fronds placed in each bowl, and after seven days the number of fro
each replicate was tallied.
Reference: Experiment and data based on Sutton, H.D. (1996). Contaminant-induced peroxidase response in subm
and wetland plants . Ph.D. Dissertation, Clemson University.
Variable Names:
Copper: Copper concentration in water in parts per million
No. of live fronds: Number of living fronds after 7 day exposure period.
oups in the problem in the table, and write a title for the table in the space provided (after Table 1:).
r this data set? Explain your answer.
bars, axes titles, and a legend.
water on the growth of the
no copper (0 parts per million,
centration in many lakes in the
n days the number of fronds in
xidase response in submerged
Problem #9: Effects of nitrate on embryonic development in wood frogs
The Data:
Nitrate
Deform
(ppm)
0 0,07
0 0,07
0 0,07
0 0,20
0 0,14
25 0,27
25 0,60
25 0,50
25 0,27
25 0,27
Descriptive Statistics:
Provide descriptive statistics for the different experimental groups in the problem in the tab
Table 1:
Mean
Median
Mode
Std. Dev.
Which measure of central tendency is the most appropriate for this data set? Explain your a
Data presentation:
Insert a graph depicting the data. Don’t forget to include error bars, axes titles, and a legend
Problem #9: Effects of nitrate on embryonic development in wood frogs
Abstract: To determine if elevated levels of nitrate (a nitrogen compound found in wastewater) led to higher inciden
of deformities in amphibians, researchers reared eggs of the wood frog (Rana sylvatica ) in experimental solutions w
varying nitrate levels. Five sets of 15 eggs were exposed to solutions with zero parts per million (ppm) of nitrate, an
additional five sets were exposed to solutions with 25 ppm of nitrate. The tadpoles were examined under a microsc
after hatching, and the proportion of tadpoles with deformities was determined.
Reference: Laposata, M. M. and W. A. Dunson (1998). Effects of boron and nitrate on hatching success of amphib
eggs. Archives of Environmental Contamination and Toxicology 35:615-619.
Variable Names:
Nitrate: Nitrate level of 0 or 25 ppm.
Deform: Proportion of tadpoles exhibiting developmental deformities.
ups in the problem in the table, and write a title for the table in the space provided (after Table 1:).
r this data set? Explain your answer.
bars, axes titles, and a legend.
water) led to higher incidence
in experimental solutions with
million (ppm) of nitrate, and an
examined under a microscope
atching success of amphibian
Statistics and Graphing
Table of Contents:
(1.) Statistics: An Introduction
(2.) Descriptive Statistics: Measures of Central Tendency
(3.) Descriptive Statistics: Comparing Measures of Central Tendency
(4.) Descriptive Statistics: Measures of Dispersion
(5.) Descriptive Statistics: Central Tendency and Dispersion
(6.) Presenting Data: Tables and Figures
(7.) Presenting Data: Types of Graphs
(8.) Presenting Data: Choosing the Values to Graph
(9.) Practice Problem Answers
(10.) Sample Problems
Statistics: An Introduction
Statistics – they’re all around us, and we’re exposed to them every day. They allow us to
summarize large volumes of mathematical information, and so are found wherever there’s data
to be presented. Let’s say you’re watching your favorite college football team, and the game has
come down to a last second thirty-five yard field goal attempt for the win. When describing the
kicker’s past performance in this range, the play-by-play announcer doesn’t list all of the kicker’s
attempts one by one and their outcome, he simply states the percentage of kicks made in this
yard range. If the kicker’s success rate between 30-40 yards has been 90% this season, then
you’re feeling pretty confident of victory. If it’s 40%, you’re gripping the chair a little tighter when
the ball is snapped.
The most recognizable use of statistics for the typical college student is when exam grades are
posted. You want to know two things: your score and the class average. The class average
matters because it is an easy way for you to compare your performance with the “average”
student in the class, and to get an idea for the potential for the “curving” of exam scores.
If you’re a sports fan, statistics can be important when comparing players’ performance,
determining success or failure in fantasy sports leagues, and in the all-too-frequent debates on
the greatest players of all time. They allow you to quickly summarize an entire player’s career or
performance into a few numbers, and to make comparisons between players for the same
statistic. Statistics also play a prominent role in politics, where supporters or detractors of
legislation often use statistics to back their position and discredit their opponents’ point of view.
When politicians are proposing tax cuts, opponents of the cut will state the vast majority of the
benefit goes to the top 1% of taxpayers, while proponents will state that the average taxpayer
stands to have their tax burden reduced by a sizable amount. In many of these cases, both
sides are using the same data source and are technically telling the truth, but the impression
one gets from the two points of view is radically different. Advertising works in much the same
way, where commercials will present statistics from industry ratings (automobile ratings,
consumer research studies) that show their products favorably and their competitor’s products
unfavorably. Curiously enough, their competitor will run a commercial giving the exact opposite
impression while citing the same source.
Statistics and Graphing 1
The gathering and analysis of data forms the backbone of science, and so statistics obviously
play an important role in this discipline. Statistics are used to describe numerous parameters
(temperature, growth rates, age, etc.) and to make comparisons between experimental groups.
In this week’s exercise, you will be given an introduction to statistics, with emphasis on the
statistical measures you will be using throughout the semester to analyze and present data. We
will begin by examining descriptive statistics, which are used to summarize groups of data. We
will then cover data presentation, in which the proper methods for displaying data in tables and
figures will be described. You will frequently be using the skills you learn here throughout the
semester, so you are strongly encouraged to review the material in this exercise diligently.
Descriptive Statistics: Measures of Central Tendency
When looking at a data set, we often are interested in knowing
the “center” of the group of values we’re examining. If you are
evaluating your exam score, you want to know the “average”
score for the class. If you are predicting the winner of a
basketball game, you want to know the “average” heights of the
two teams. There are three such measures of central tendency:
the mode, the median, and the mean. To evaluate these
measures, let’s examine a set of exam scores. To protect the
innocent, we will refer to the different students by a coded letter.
To examine measures of central tendency, it is helpful to
arrange the values in descending order.
97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
Student
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Score
pts.
93
88
77
85
74
69
90
66
82
88
59
83
75
97
71
Mode
The mode is the value that appears most frequently in your data set. The term “mode” comes
from the French word for “fashion”, as the most fashionable clothes are the ones seen most
commonly on the street. The mode of a data set does not need to be near the center of the data
set, it simply has to be the most common. It is important to note that some data sets can have
more than one mode if two or more values appear the same number of times. Modes are useful
in that they tell you the most common value in the data set, and can shed some light on the data
set’s tendencies. Modes give you an idea of the tendencies of the data set by showing you
which value is the most common, but the mode is not always representative of the value that
forms the “center” of the data set. To determine this, we need to examine the median or mean
of the values.
PRACTICE QUESTION #1:
Determine the mode of the data set below. Answer provided at the end of the exercise.
Data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
Statistics and Graphing 2
Median
The median is the value that occurs in the middle of the data set. To determine the median,
examine the values in descending order and find the value in the exact middle of the data set. If
the data set contains an odd number of values, the median will be the middle value. For
example, in an ordered data set of 19 values, value #10 would be the median as there would be
nine values below it and nine values above it. In a data set of 20 values, the median would be
the value halfway between the tenth and eleventh values.
PRACTICE QUESTION #2:
Determine the median of the data set below. Answer provided at the end of the exercise.
Data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
Mean
You are already familiar with the mean of a data set, commonly called the “average”. To
calculate the mean you sum all of the values and divide by the number of values. While the
terms “mean” and “average” are synonymous in common language, we will not refer to the
mean as the average, as the median also gives a good representation of the average, and yet is
a different statistic.
PRACTICE QUESTION #3:
Determine the mean of the data set below. Answer provided at the end of the exercise.
Data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
Descriptive Statistics: Comparing Measures of Central Tendency
So which measure should I use? If I have to describe the
central tendency of a data set, which statistic should I use?
Well, it depends. Let’s compare the three statistics for the exam
score data set.
Mode
88
Median
82
Mean
79.8
In most cases, the mode is usually not representative of the central value in a data set, so it is
rarely used in this capacity. Values can be the most common but rather unrepresentative of the
middle of the data set, as in this case. When deciding whether to use the median or mean, it
often falls to a value judgment on your part. Which one do you think is most representative of
the central tendency of the data set? In this example, the two values were quite similar, but this
is not always the case. Let’s take the original data set, make a change, and see how this affects
the two statistics. Let’s change the lowest exam score from a 59 to a 17, and see what happens.
Old data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
New data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 17
If we calculate the statistics for the new data set we get:
Mode
88
Median
82
Mean
77.0
Note that the difference between the median and the mean went from 2.1 points to 5 points with
a change in only one grade. This shows the influence of “outliers” on means. Outliers are values
that fall well outside the range of the other values in the data set. Because means are obtained
Statistics and Graphing 3
by adding all of the values together and dividing by the number of values, means will be heavily
influenced by outliers. Medians, which fall in the middle of the data set regardless of the
magnitude of the largest and smallest values, will be less affected. So medians are the way to
go then, right? Not necessarily…
Let’s look at a totally different group of exam scores, already ordered for you:
All new data set: 97, 96, 94, 94, 93, 92, 91, 90, 71, 70, 70, 69, 69, 67, 59
If we calculate the statistics for this data set we get:
Mode
Median
94, 70, 69
Mean
81.5
90
As this example illustrates, the median can also be unrepresentative of the central tendencies of
a data set. While many students scored in the 90-100% range, there were many students with
much lower scores, and the median doesn’t give you any indication of that. At the same time,
the mean value for this data set is 81.5, and yet not one of the 15 students scored even
remotely close to this value on the exam. So what have we decided is the best way to show the
central tendency of a given data set? We’ve seen that modes, medians, and means all have
their shortcomings, and the choice of the most appropriate statistic often depends upon the data
set. What we really need is a statistic that gives us an indication of the data’s “center”, and
another that describes the “spread” of the data around that value. Well guess what, it’s your
lucky day…
Descriptive Statistics: Measures of Dispersion
We saw in the previous sections that having an idea of the spread (dispersion) of data points is
necessary to accurately interpret a data set. If you are traveling to a faraway city and are given
only the mean daily temperature, you will have no idea of the temperature extremes (high and
low temperature for the day), and so would have little idea of the types of clothes to pack. An
understanding of data dispersion can therefore be quite important. In this section, we will look at
three measures of dispersion: the range, variance, and standard deviation.
Range
The range describes the highest and lowest values in a data set. The range is commonly used
in weather reports, where the daily high and low temperatures are reported. The range tells you
that all of the values in the set fall within two values, and so gives an indication of the spread of
the data.
PRACTICE QUESTION #4:
Determine the range of the data set below. Answer provided at the end of the exercise.
Data set: 97, 93, 90, 88, 88, 85, 83, 82, 77, 75, 74, 71, 69, 66, 59
Variance and Standard Deviation
While the range is useful for describing the “borders” of a data set, it tells us almost nothing
about the points that fall between the two extremes. In most cases, we are interested in knowing
how far each of the data points is from the mean or median of the data set, as this would enable
us to see the spread in all of the data points. Since the median doesn’t take the magnitude of
each data point into account and the mean does, the mean is the better option for a central
Statistics and Graphing 4
point. So let’s determine the individual point deviations for the original exam score data set by
subtracting each score from the mean.
Value
Deviation
97
93
90
88
-17.2 -13.2 -10.2 -8.2
88
85
83
82
77
75
74
71
-8.2
-5.2
-3.2
-2.2
2.8
4.8
5.8
8.8
69
66
59
10.8 13.8 20.8
If we sum all of the deviations from the mean, we end up with zero (which shouldn’t be
surprising given how a mean is calculated), and this tells us nothing about the data’s dispersion.
To get around this, we need to look at the absolute values of the deviations, and one way to do
this is to square each of them and add them up. If we divide this sum by one less than the
number of values, this will give us the variance of each data point. The variance describes how
far each value is from the mean. So our problem is solved, right? Wrong. While the variance
gives you an indication of the deviation of each value from the mean, the variance is in
“squared” units and it is difficult to relate to the “unsquared” mean. We can undo the squaring
by taking the square root of the variance, which will provide us with a measure of data
dispersion in the same units as the mean. This value is known as the standard deviation (SD).
Coupled with the mean, the standard deviation is useful because it allows us to examine the
dispersion of a data set quickly and easily. Data that have a bell-shaped distribution when
plotted, such as those shown in Figure 1, are said to have a “normal” distribution. A normal
distribution is symmetrical, with the majority of the points near the center (mean) and fewer as
you progress away from the center. If the data set resembles a normal distribution, we can
conclude that about 68% of the data points will fall within one SD of the mean (area indicated in
red), 95% will fall within two SD’s of the mean (red area + green area), and 99% will fall within
three SD’s of the mean (red area + green area + blue area). By knowing the mean and the
spread of points around it, we can get a good indication of the characteristics of a data set.
For the data set above, for example, the SD = 10.8 points, so we can conclude that 68% of the
scores fell between 69.0 and 90.6 points.
Figure 1. Normal distribution with standard deviations
NOTE: When listing the mean and standard deviation of a data set, convention dictates that you
list the mean first, followed by the standard deviation in parentheses with a plus/minus sign.
Statistics and Graphing 5
The mean and SD for the exam scores example would therefore be written as: mean (SD) =
79.8 (±10.8)
Descriptive Statistics: Central Tendency and Dispersion
By this point, you’ve seen that there are several measures of central tendency and dispersion
for data, and that each one can be inappropriate for certain data sets. The point of data analysis
is to present the data in a concise but clear manner, so that you have sufficient information to
make conclusions. While presenting all of the statistical measures we’ve discussed for a data
set might be informative, the sheer volume of information hinders interpretation. For every data
set, you must strive to describe the data in the most effective manner possible. We’ve seen that
the mean, coupled with the standard deviation, is a good way to summarize a data set as it
provides the reader with a measure of both the central tendency and the dispersion of the data.
You should not, however, always limit yourself to these two statistics. If you determine that the
median, range, mode, or other statistic is useful for a particular data set, you should provide it.
Statistical Tools
Calculating descriptive statistics by hand can be a daunting task, particularly when large
numbers of data points are involved. To facilitate data analysis in the lab exercises this
semester, we suggest you use a statistical analysis tool. Microsoft Excel can calculate the
descriptive statistics in this exercise, as do some online tools. Some examples are listed below.
VassarStats
Online statistical package
Vassar College
QuickCalcs
Online statistical package
GraphPad Software
http://faculty.vassar.edu/lowry/VassarStats.html
choose “Miscellanea” then “Basic Sample Stats”
http://www.graphpad.com/quickcalcs/CImean1.cfm
Presenting Data: Tables
You have seen the value in using descriptive statistics to summarize data sets, and we will now
examine the various methods available to visually present data. We will describe how to choose
the proper format for presenting data, and then detail the procedures for constructing tables and
graphs.
A data table, if properly designed, can be a highly effective way to organize data. For example,
if we are conducting an experiment in which we will be monitoring oxygen concentrations in
samples of pond water over 45 minutes, we would design a table for our data like the one
below. Like all tables, the data would be entered in the appropriate cell, referenced by column
and row headings.
Statistics and Graphing 6
Time (min.)
Darkness
Ambient Light
0
.
.
5
.
.
10
.
.
15
.
.
20
.
.
25
.
.
30
.
.
35
.
.
40
.
.
45
.
.
Table 1. Oxygen concentrations (mg per L) over 45 minutes in pond
water samples exposed to darkness or ambient light levels.
Note that there are certain formatting characteristics for an effective table:
(a.) Each table is numbered sequentially, beginning with Table 1.
(b.) The title of the table should be descriptive, allowing readers to interpret the table based on
the title alone. It should include both the dependent and independent variables when applicable.
(c.) The cells in the table should be roughly equally spaced for an organized look, and the
column and row headings should stand out against the data.
(d.) The units for the data should always be identified, either within the table or in the title. We
would then enter data into the table during the experiment.
Time
(min.)
0
Darkness
Ambient Light
2.1
2.1
5
2.1
2.1
10
2.1
2.2
15
2.0
2.2
20
2.0
2.2
25
2.0
2.3
30
1.9
2.3
35
1.9
2.3
40
1.9
2.4
45
1.9
2.4
Tables can be an important tool for organizing and recording data during an experiment, but
also can serve role in presenting data. For example, let’s say that we wish to present the mode,
Statistics and Graphing 7
median, and mean exam scores for two groups of students taught with different educational
approaches. One group of students examined smog formation with a hands-on laboratory
activity, while the other used a computer simulation. Both groups were given the same exam
afterwards and the scores compared in the two groups. Compare the two ways to present this
information – first as a table, and then in paragraph form.
Laboratory activity
.
Hands-on
Simulation
Mode
76
79
Median
83.3
84.0
Mean
84.1
82.6
Table 2. Exam scores (percent correct) for students
using hands-on lab activity and computer simulation.
“The mode score for the hands-on group was 76%, and 79% for the simulation group. The
median score for the hands-on group was 83.3%, and 84.0% for the hands-on group. The mean
score for the hands-on group was 84.1%, and 82.6% for the simulation group”.
When comparing the two approaches, notice that although they both present the same exact
information, the data is more easily interpreted in the table. Because of this ability to organize
information and present values, tables are widely used in scientific reports and articles to
present data. You will similarly find many instances throughout the semester in which this
method of presentation will prove useful. While tables can be an effective way to record data
and present certain types of information, there are times when a figure (graph) is more
appropriate. Which takes us to our next section…
Presenting Data: Figures
Figures include illustrations, diagrams, maps, and graphs, and can be extremely useful in
scientific writing. Humans are a visual species (hence, “a picture is worth a thousand words”),
and figures are therefore beneficial in that they depict information visually. Let’s look at the
different types of figures often used in scientific writing.
Presenting Data: Illustrations, Diagrams, and Maps
The ultimate goal of science writing is to present information to the reader concisely, but
completely. The use of illustrations, diagrams, and maps can be a great way to efficiently
present large amounts of information, and should be used whenever appropriate. Illustrations
can be used to identify structures in an organism, or to show a new species or experimental
apparatus. Diagrams are often used to show the relationship between variables in complex
systems, or to visually depict a chemical reaction or other sequence of events. Maps are most
commonly used to show the locations of sampling sites when experiments are conducted in the
field. While using these types of figures can be helpful, their overuse can be detrimental. You
would not, for example, include an illustration of a rainbow trout in a report on the study of the
effects of low pH water on this species. Everyone reading your work should know what a
rainbow trout looks like, and the inclusion of the illustration is therefore simply wasted space. If
an illustration clarifies a concept or provides valuable information, include it in your report. If it
Statistics and Graphing 8
simply adds “visual appeal”, omit it. Don’t be afraid to use illustrations, diagrams, and maps, just
use them wisely.
Formatting characteristics for illustrations, diagrams, and maps:
(a.) Each illustration/diagram/map is numbered sequentially, beginning with Figure 1.
(b.) The title of the illustration/diagram/map should be descriptive, allowing readers to interpret
the table based on the title alone.
(c.) The illustration/diagram/map should be of high quality and appropriate size.
Graphs
The type of figure you will use most
frequently is a graph. Like tables, graphs
present data to the reader, but do so in a
visual manner that it often superior to tables.
Graphs can be an effective way to present
information, but it is important that they are
properly constructed. Let’s look at a sample
graph to demonstrate the basic
characteristics of a graph.
Note that the independent variable is
graphed on the x-axis and the dependent
variable on the y-axis, the x-axis (horizontal)
and y-axis (vertical) are labeled, units are
listed where appropriate, and the scale of
Figure 2. Sample graph
the y-axis is tailored to the data set. If “time” is a factor in your graph, it should be graphed on
the x-axis, as shown in Figure 3). Recall that graphs, illustrations, diagrams, and maps are all
called “Figures”, and should be numbered in the order in which they appear in the report.
Graphs should also be of high quality and appropriate size.
The scaling of the axes is an area that tends to give students trouble, and is deserving of further
discussion. The scale on an axis should cover the range of values in your data set, and does
not have to start at zero. If you are graphing a set of values that range from 70 to 85, the scale
on the axis should run from about 65 to 90 (start slightly below lowest value and slightly above
greatest value). The scale should also cover the entire height or width of the axis, so you should
begin the scale at one end of the graph, end it at the other end, and divide the increments
between equally. While this scaling is often automatically done by software applications with
graphing capability, you should nonetheless be diligent in checking them to ensure they are
correct. As shown in Figure 3, scaling can make a big difference in data interpretation. The
graph on the left seems to suggest no difference in pH level in the two treatments, but with
proper scaling the differences in the groups become apparent.
Statistics and Graphing 9
Figure 3. Effects of axis scale on data presentation
There are exceptions to this rule, though. If you are comparing percentages (e.g., exam scores),
it is conventional that you make the axis run from 0% to 100% even if the values graphed are
both near the top of the scale. When in doubt, use your best judgment.
Presenting Data: Types of Graphs
Although a variety of graphs exist, we will focus here on the two types that you will use most
often in this course: bar graphs and line graphs. Examples of each type of graph are shown
below.
Figure 4. Sample bar graph
Figure 5. Sample line graph
Choosing the appropriate type of graph is often difficult for students, even if provided with only
two choices. Let’s therefore outline a few criteria regarding graph choice.
When Bar Graphs are Useful
If you can break your data into discrete groups like those shown here, then a bar graph is
appropriate. Bar graphs are useful for graphing non-continuous data, such as data from different
experimental groups. Note that in each case, the dependent variable is on the y-axis and the
independent variable on the x-axis, and all of the formatting issues listed above are addressed.
Statistics and Graphing 10
Figure 6. Sample bar graph
Figure 7. Sample bar graph
When Line Graphs are Useful
If your data are continuous (each point is directly related to the next and can be connected by
an infinite number of intermediate points), then a line graph is the way to go. Line graphs are
commonly used in scientific studies to present data when time (a continuous variable) is one of
the variables involved. As before, note the dependent variable is on the y-axis, “time” is on the
x-axis, and all of the formatting issues are addressed.
Figure 8. Sample line graph
Figure 9. Sample line graph
Scatter Plots
While bar and line graphs are used commonly, there is another type of graph worth mentioning the scatter plot. In some cases you may need to graph two variables against one another to
determine their relationship. You graph one variable on the X-axis and the other on the Y-axis,
and then graph your points accordingly. By doing this, you can see if one variable increases as
the other increases (positive relationship), one increases as the other decreases (negative
relationship), or if the two variables show no relationship to one another. For example, assume
that you are given the data below and you want to find the relationship between the two
variables.
Statistics and Graphing 11
Variable 1 5.5 8.1 2.7 3.0 7.0 3.5 2.0 1.5 7.4 4.2 2.4 2.9 5.6 6.4 1.9 6.1
Variable 2 11.0 17.6 5.1 5.0 14.0 7.5 3.8 4.1 14.0 8.4 6.8 5.1 12.9 10.6 12.8 3.8
If you place variable 1 on the X-axis and variable 2 on the Y-axis and graph the points, you get
a graph that looks like this.
Figure 10. Sample scatter plot
Note that the points form an upward-sloping line, which indicates the two variables have a
positive relationship to one another. Had the line sloped downward, this would suggest a
negative relationship. If the points were scattered about and no clear relationship was visible,
this would mean the two variables are unrelated to one another. The relationship can be
indicated by drawing a “best-fit line” through the points. The line should go through all of the
points in such a way that it minimizes the distance between each point and the line.
Statistics and Graphing 12
Figure 11. Scatter plot with best-fit line
Presenting Data: Choosing the Values to Graph
Students can also have difficulty with graphing because they are unsure exactly what data to
graph. Let’s examine an example to show you what I mean.
Let’s assume that a researcher is attempting to
determine the effects of pesticide pollution on the
hatching success of fish eggs. He has noticed
that fish eggs in streams near agricultural fields
fare poorly, while those in undisturbed areas
hatch successfully. The researcher sets up a lab
experiment in which ten groups of fish eggs are
allowed to develop in unmanipulated stream
water, and ten groups of eggs develop in stream
water with the addition of pesticide. The
proportion of eggs hatching in each group is
tallied (Table 3), and descriptive statistics for the
two groups compared.
Statistic
Mean proportion eggs
hatching
Standard deviation
No
Pesticide
Pesticide
0.87
0.59
0.07
0.09
Proportion eggs
hatching
Group
No Pesticide Pesticide
1
0.80
0.50
2
0.76
0.45
3
0.81
0.68
4
0.90
0.77
5
0.95
0.64
6
0.84
0.60
7
0.88
0.54
8
0.99
0.57
9
0.86
0.62
10
0.93
0.57
Table 3. Effects of pesticide pollution
on hatching success of fish eggs.
Now that we have the data, how do we graph it? First, we determine that a bar graph is most
appropriate, as the independent variable is the two groups, and this is categorical, not
continuous. Second, we must determine what to graph. If we graph all of the values, we get a
graph like the one below.
Statistics and Graphing 13
Figure 12. Effects of pesticide pollution on
hatching success of fish eggs.
But what does this graph tell you? There are 20 bars, 2 categories, and lots of different group
numbers of mix things up. If we take a step back and look at the study, what we’re really
interested in doing is comparing the hatching success in groups exposed to pesticide and the
groups not exposed to pesticide. So what we really want to do is to graph the mean of the two
groups and compare them to one another. If we do that, we get a graph that looks like:
Figure 13. Effects of pesticide pollution on
hatching success of fish eggs.
Now while this graph is useful in that it readily allows us to compare the means of the two
groups, it doesn’t provide us with any indication of the spread of the data in each group.
Knowing the spread of the data can be important in making conclusions, so you need to provide
the reader with this information. This is done by adding “error bars” to data points or bars on a
graph that indicate the standard deviation of the data. In this example, we would add the
standard deviation error bar to the bar for each group. In bar and line charts, error bars are
drawn both above and below the mean to a distance equal to their value.
Statistics and Graphing 14
Figure 14. Effects of pesticide pollution on
hatching success of fish eggs.
Realize that error bars are only appropriate when graphing a group’s mean. If you are graphing
individual data points, you cannot add error bars as there is only one data point per point on the
graph (hence, no mean or standard deviation). Error bars are useful in that they allow you to
see the overlap in the data points from two groups. Remembering that all of the data points in a
distribution are within two standard deviations of the mean, you can visually estimate the data’s
distribution by doubling the height of the error bar (because it’s only drawn to one standard
deviation) for each group and seeing the degree to which their distributions overlap. As shown
in the examples below, including error bars can make a big difference in your interpretation of a
graph. Although the means of the two groups are identical in each of the graphs, the two groups
in the first figure appear to be more different from one another than the two in the second figure
due to the smaller error bars and lesser degree of overlap.
Figure 15. The role of error bars in estimating differences in treatment groups
Creating Graphs
You may choose to make your graphs by hand or with the assistance of a computer software
Statistics and Graphing 15
application. In either case, you need to make sure your choice of graphs, the data to graph, and
the formatting are all appropriate. There are several software applications that enable you to
create graphs (e.g., Microsoft Excel) and they are often available in campus computer labs.
There are also some web sites that allow you to create graphs online, and they are listed below.
Providing directions for graph preparation in each would be prohibitively long, so you will need
to become familiar with their operation on your own if you opt to use them.
Create a Graph
Online graphing application
National Center for Education Statistics
http://www.nces.ed.gov/nceskids/graphing/
Practice Question Answers
PRACTICE QUESTION #1: Mode = 88
PRACTICE QUESTION #2: Median = 82
PRACTICE QUESTION #3: Mean = 79.8
PRACTICE QUESTION #4: Range = 97 – 59
Sample Problems
The link below will take you to a PDF file with sample problems. Complete the problems
assigned by your instructor. If none were assigned, complete problems #1-4.
Sample Statistics Problems
http://esa21.kennesaw.edu/activities/stats/problems.pdf
Statistics and Graphing 16
ESA 21: Environmental Science Activities
Activity Sheet
Statistics and Graphing
Name:
Instructor:
Question #:
[you will need one activity sheet per question]
Descriptive Statistics:
Provide descriptive statistics for the different experimental groups in the problem in the table,
and write a title for the table in the space provided.
Table 1:
Mode
Median
Mean
Std. Dev.
Which measure of central tendency is the most appropriate for this data set? Explain your
answer.
Data presentation:
Figure 1:
Statistics and Graphing
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