Description
DIRECTIONS: Use a pencil and ruler for constructing graphs. Be sure to include appropriate
title, labels, and scales on all graphs. Include units on answers and be sure to answer to the
listed decimal place when listed.
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ENVR 1401 – EXERCISE
Lab 1 – Math & Environmental Science
DIRECTIONS: Use a pencil and ruler for constructing graphs. Be sure to include appropriate
title, labels, and scales on all graphs. Include units on answers and be sure to answer to the
listed decimal place when listed.
1. Construct a bar graph for the following pH data for water samples. (HINT: expand the y-axis)
Test Tube # pH
A
7.5
B
6.8
C
8.1
D
7.0
E
7.3
2. Use the following data and given the assumptions listed below, construct a line graph and a paired bar
graph:
Fall Semester
Month
Science Grade
September ‘18
96
October ‘18
92
November ‘18
86
December ‘18
76
LINE GRAPH – assume these grades represent
performance of one student over two semesters
in one academic year
Spring Semester
Month
Science Grade
February ‘19
81
March ‘19
85
April ‘19
79
May ‘19
95
PAIRED BAR GRAPH – compare average
grades assuming two groups of students taking
the same course in sequential semesters
1
3. Complete the table and construct a pie chart for the following data for the Quick-Mart Service Station.
(Hint: A circle contains 360 degrees). Include a title, remediation method labels, and cost amounts
on the pie chart below.
Remediation Method for
2018 the Quick-Mart
Service Station
Cost
Analytical testing
$ 538,900
Decontamination
$ 177,665
Groundwater treatment
$ 254,347
Health & Safety
$ 49,475
Soil excavation & disposal
$ 188,950
Percentage of Budget
(to two decimal
places)
Degrees of Circle
(to three decimal
places)
TOTALS
2
Scientific Notation (Use commas appropriately for numbers exceeding 1,000) – If the value given is a
number, then write it in scientific notation. If the value given is expressed in scientific notation, then write
it as a number.
4. 28,736,000,000
=
5. 1,000,000,000,000
=
6. 7,538,000
=
7. 4,742
=
8. 0.0000049328517
=
9. 6.242 x 10-7
=
10. 4.37 E + 12
=
11. 5.39 x 1011
=
_________________________
From the Background, here are the rules for significant figures again:
1. Every nonzero digit in a recorded measurement is significant. 24.7 m, 0.743 m and 714 m all
have three significant figures.
2. Zeroes appearing between nonzero digits are significant. The measurements 7003 m, 40.79 m,
and 1.503 m all have four significant figures.
3. Zeroes in front of (before) all nonzero digits are merely placeholders; they are not significant.
0.0000099 only has two significant figures.
4. Zeroes at the end of the number if a decimal point is present and also zeroes to the right of the
decimal are significant. The measurements 1241.20 m, 210.100 m and 5600.00 all have six
significant digits.
5. Zeroes at the end of a measurement and to the left of an omitted decimal point are ambiguous.
They are not significant if they are only place holders: 6,000,000 live in New York—the zeroes
are just to represent the magnitude of how many people are in N.Y. But the zeroes can be
significant if they are the result of precise measurements.
Rounding & Significant Figures (retain the following as whole or decimal numbers):
12. 47,830,000,000 to 2 significant figures
=
13. 35.6971 to 3 significant figures
=
14. 7.689 to nearest whole number
=
3
Calculating Percentage Changes (calculate to two decimal places, rounding as appropriate)
SHOW EQUATION TO BE SOLVED
Answer
% change
15.
342.91 to 477.38 =
16.
477.38 to 342.91 =
% change
The Dallas County budget includes several departmental line-items that are related to environmental
concerns. The following table lists amounts expended in 2017 and 2018, along with the projected
budgetary allowance for 2019.
Line Item
Fire Prevention & Public Education
Bioterrorism & Environmental Health
Clearing trash & debris from roadways
On-site Sewage Facility Inspections
Building Inspections
Permitting
17.
$ in 2017
$ in 2018
$ in 2019
158,000
181,000
134,000
220,000
213,000
124,000
271,000
246,000
145,000
233,000
216,000
126,000
308,000
250,000
154,000
228,000
219,000
129,000
Calculate Annual Budget Totals:
What percentage (round to 2 decimal places) of the overall spending for various environmental services
was allocated in 2019 to each of the following line items?
SHOW EQUATION TO BE SOLVED
18. Fire Prevention &
Public Education
=
19. Clearing trash &
=
Answer
debris from roadways
20. On-site Sewage
=
Facility Inspections
21. Permitting
=
4
Calculate the following for these environmental issues. (round your answer to nearest whole number).
SHOW EQUATION TO BE SOLVED
22. 2017 Line item mean
=
23. 2018 Line item mean
=
24. 2019 Line item mean
=
25. Budget total mean
=
Answer
Based on discussion in the background document Math & Environmental Science, list three (3)
advantages of understanding and using scientific notation and/or percentages.
26)
27)
28)
5
ENVR 1401 – BACKGROUND
Lab 1 – Introduction to Mathematics & Environmental Science
Environmental science assesses phenomena using both qualitative and quantitative techniques.
Particularly, the measurement and accumulation of scientific data necessitates the use of mathematical
analysis, expression of the data, and results to facilitate effective scientific communication. This
background document is a review of basic mathematical methods for data analysis and presentation.
PERCENTAGES
Percentage is a convenient method of representing the part of a whole expressed in hundredths. Per cent
(%) means per hundred, that is, it means ‘divided by 100′ which results in the apparent “SHORTCUT”
that the decimal point must be moved two places to the left:
25% = 25/100 = 0.25
To determine the ratio of a part to the whole, expressed as a percent:
Process: find the ratio of the part to the whole; then calculate as a decimal fraction
and multiply by 100 to convert the decimal fraction to a percentage
To compare two different values or to determine percent change to analyze trend:
Process:
New value – original value * 100 = %
Original value
apply the formula
If the answer is a “negative value” then there is a “decreasing
trend” If the answer is a “positive value” then there is an
“increasing trend”
184 ➔ 327 … (327-184) * 100 /184 = 77.717 % = 77.7 %
327 ➔ 184 … (184-327) * 100 / 327 = – 43.7308 % = – 43.7 %
Scientific Notation & Calculations with Exponential Numbers
In science, numbers large and small are commonplace and a shorthand notation call scientific notation
was developed to simplify their specification and utilization. It is based on place value and base ten, as
shown in the following table.
10000 = 1 x 104
89,472 = 8.9472 x 104
1000 = 1 x 103
8947 = 8.947 x 103
100 = 1 x 102
894 = 8.94 x 102
10 = 1 x 101
89 = 8.9 x 101
1 = 100
8 = 8.0 x 100
1/10 = 0.1 = 1 x 10-1
0.8 = 8.0 x 10-1
1/100 = 0.01 = 1 x 10-2
0.089 = 8.9 x 10-2
1/1000 = 0.001 = 1 x 10-3
0.00894 = 8.94 x 10-3
1/10000 = 0.0001 = 1 x 10-4
0.0008947 = 8.947 x 10-4
Revised: August 2019
1
Consider the value expressed as: 8.94 x 10-3 The first number (8.94) is called the coefficient and it
must be greater than or equal to 1 and less than 10. The second number is called the base and it
must always be 10 in scientific notation. The base number 10 is always written with exponent form.
In the number 102 the number 2 is referred to as the exponent or power of ten.
With the ubiquitous use of computers, methods of typing exponential numbers have been
developed. Therefore, exponential notation may be expressed using “E” for exponential:
8.94 x 10-3 ➔ 8.94 E-3
Significant Figures and Rounding
•
The rules for determining which digits in a measurement are significant are:
1. Every nonzero digit in a recorded measurement is significant. 24.7 m, 0.743 m and
714 m all have three significant figures.
2. Zeroes appearing between nonzero digits are significant. The measurements 7003 m,
40.79 m, and 1.503 m all have four significant figures.
3. Zeroes in front of (before) all nonzero digits are merely placeholders; they are not
significant. 0.0000099 only has two significant figures.
4. Zeroes at the end of the number if a decimal point is present and also zeroes to the
right of the decimal are significant. The measurements 1241.20 m, 210.100 m and
5600.00 all have six significant digits.
5. Zeroes at the end of a measurement and to the left of an omitted decimal point are
ambiguous. They are not significant if they are only place holders: 6,000,000 live in
New York—the zeroes are just to represent the magnitude of how many people are in
N.Y. But the zeroes can be significant if they are the result of precise measurements.
•
The significant figures in a number in scientific notation are the number of digits in the
coefficient. The number 4×105 has only one digit in the coefficient, so it has one significant
figure. 9.344×105 has 4 significant figures. The number 1200 is unclear as to how many
significant figures it has; if it is more clearly expressed as 1.200×10 3 then it has 4 significant
figures or as 1.2×103 then it has 2 significant figures.
•
When calculating with significant figures, an answer cannot be more precise than the least
precise measurement. Values should be rounded, not truncated.
•
When rounding numbers to a certain number of significant figures, do so to the nearest
value. If the value beyond the significant figure is 5 or greater, then round to the next higher
number.
o
Round to 3 significant figures: 2.3467 x 104 ➔ 2.35 x 104
o
Round to 2 significant figures: 1.612 x 103 ➔ 1.6 x 103
o
Round to 2 significant figures: 4,451 ➔ 4,500
Importance of Units
The metric system (Système International or SI) is in use the world over and is the preferred
system for science since it uses the base 10. The fundamental units in SI include the meter and
kilogram. For derived values, the most commonly used prefixes are listed in the following table.
Revised: August 2019
2
Prefix
Symbol
Factor
Value
Descriptor
Giga-
G
109
1,000,000,000
billion
Mega-
M
106
1,000,000
million
kilo-
K
103
1,000
thousand
centi-
c
10-2
0.01
hundredth
milli-
m
10-3
0.001
thousandth
micro-
10-6
0.000 001
millionth
nano-
n
10-9
0.000 000 001
billionth
Pico-
p
10-12
0.000 000 000 001
trillionth
The National Institute of Standards and Technology (former National Bureau of Standards) is the official source of
standard weights and measures in the United States. Though in the United States, imperial (often referred to as
common measures: e.g., feet, miles, ounces, pounds, tons, etc.) are also used.
•
Units of time: seconds. There are 60 seconds per minute (angle or time), 60 minutes per hour (or
degree), 24 hours per day, 7 days per week, and approximately 365 days per year. Time should be
expressed using military format:
o
1:00 a.m. ➔ 0100 hours
o
Noon ➔ 1200 hours
o
2:26 p.m. ➔ 1426 hours
•
Units of distance: meters. Imperial units include inches, feet, yards, and both nautical and statute
miles. Feet are often abbreviated as single quotes and inches as double quotes (height of 5 feet, 6
inches = 5′ 6″). These same quote symbols are used for angle measurement in degrees, minutes, and
seconds (a right angle = 90° 0’ 0″) such as describing an angle of slope or global position system
(GPS) using latitude and longitude.
•
Units of area: hectares. Imperial units include acres, square miles, and land sections (36 square
miles).
•
Units of “weight” or mass: gram or kilogram. Imperial units include ounces, pounds (lb), and tons.
Note: “ton” is imperial and “tonnes” are metric (1 ton = 1.016 tonnes).
•
Units of temperature: degrees Celsius. Commonly measured as degrees Fahrenheit.
•
Units of volume: liter. Imperial units include ounces, cups, quarts and gallons.
Unit Conversions
Converting within the metric system. Metric conversions are simplified because the metric system uses the base
10 and requires manipulation of the decimal by the proper number of places in the correct direction. For example:
27,000 grams equals 27 kilograms and 45 millimeters equals 0.045 meters. The most important relationship is: 1
milliliter = 1 cubic centimeter ~ 1 gram.
Converting within the imperial system. When using imperial units, it is necessary to document specific units of
measurement and use the necessary conversion factor to ensure appropriate measure and calculation. For example,
to determine how many inches are in 186,000 statute miles, a specific conversion factor must be known and used
to calculate the correct answer: 11,785,000,000 inches.
Converting from one type of unit to another is necessity in science. The conversion factor is typically written as an
identity:
Revised: August 2019
3
•
Number of feet in a mile ➔ 5280 feet / 1 mile
TABLES & GRAPHS
The first step in graphing data is to organize your data into a logical TABLE that reflects what you are
analyzing, and the results obtained. The table may also help you to see what interim steps are needed (such as
calculating totals or percentages) before the data may be logically displayed on an appropriate graph.
For example, analysis of dissolved oxygen content at several locations along a river indicated the following
values: 3.3, 4.8, 5.2, 5.3, 7.2 mg/L. The following is a logical table that presents these results — NOTE: the
locations must be linked to a figure depicting where the samples were collected along the river to facilitate
interpretation!! Note that the values do NOT decrease as one moves downstream!!
Measured Dissolved Oxygen
Concentrations in the
Trinity River
Location
Dissolved Oxygen
(figure 1)
(mg/L)
A
5.3
B
5.2
C
3.3
D
4.8
E
7.2
A variety of graphing methods are used to summarize data so the results may be easily understood. The type of
graph is typically selected based on the variables analyzed and the intent of the data analysis and presentation.
The adjacent figure presents one possible decision tree that may be used in selecting an appropriate graphing
method to analyze and depict data and results of investigations in environmental science.
Commonly used graphs include:
•
Line graphs, Scatter graphs, Bar charts (including Histograms), Pie charts
NOTE:
Every graph should have an appropriate title and the axes and data
depicted should be clearly labeled (including units) to facilitate interpretation.
Line Graphs
In laboratory experiments, usually one variable is controlled, and the investigation is designed to examine how it
affects another variable. Line graphs can clearly show these relationships. For example, in an experiment in which
the growth of a plant is measured over time to determine the rate of plant growth, the time intervals are controlled.
Therefore, time is called the independent variable (controlled variable). The height of the plant is the dependent
variable (uncontrolled or measured variable). The following table and graph illustrate some sample data for an
experiment to measure plant growth.
Revised: August 2019
4
Plant Height (mm)
Plant Growth Over a Month
50
45
40
35
30
25
20
15
10
5
0
Time (days) Plant height (mm)
0
11.73
7
24.64
14
29.97
21
33.15
28
43.04
0
5
10
15
20
25
30
Time (Days)
Scatter Plots
Some experiments or groups of data are best represented on a scatter plot to find trends in the data. Scatter plots
can also be used when there are two or more trends within one group of data or when there is no distinct trend at
all. As in a line graph, the data points are plotted on the graph by using values on an x-axis and a y-axis. Instead of
connecting the data points with a line, a trend can be represented by a best-fit line that represents all the data
points without necessarily going through any or all of them. To estimate a best-fit line, pick a line that is
equidistant from as many data points as possible as shown in the following figure. Note that the best-fit line may
not be the ‘best fit’ if only considering a small portion of the data. The actual best-fit line may be calculated using
the mathematical process of linear regression to derive the equation for the line (that is, y = mx + b where m is the
slope of the line and b is the value at which the line intercepts the y axis).
SLOPE
The slope of the line indicates the rate of change. Slope of the best-fit line (which averages the data) can be
calculated either by linear regression to obtain the value of “m” as noted above, or it may be determined from
the graph (see figures on previous page):
Revised: August 2019
5
Slope =
change in the value of “y”
change in the value of “x”
In the example: slope = 40 – 18 water analyses = 22 analyses/20 years = 1.1 analyses/year
1940 – 1920 years
When addressing environmental conditions, the term gradient is equivalent to slope. As shown in the
following figure, the irregular sloping surface of the mountainside contains segments that are both gentle and
steep. The slope of the mountainside from point “A” to point “B” is the average slope over the distance
between the two points.
The calculations for the given slope values are as follows:
Bar Graphs
Bar graphs (graphical display of magnitude or frequency) make it easy to compare data quickly; bar graphs can
also be used to identify trends, especially trends among differing quantities. Compare the following two figures.
Both figures indicate Jupiter has the largest radius and Mercury has the smallest radius. However, in the figure
on the left where the planets are listed in order of increasing distance from the sun, the relative position of the
largest and smallest planets is obvious. The figure on the right reflects only decreasing planet radius.
Revised: August 2019
6
To compare the shapes of two distributions with different sample sizes on the same axis it is necessary to use
percentage as the vertical axis. For example: A survey of the opinions of 50 men and 80 women on an issue
with “Yes,” “No” or “None” as possible responses is tabulated and graphed below for comparison.
Response of Women & Men
Response Men Women Men Women
#
#
%
%
Yes
15
40
30
50
No
25
35
50
43.75
None
10
5
20
6.25
Percent is used for comparison since
the Sample size (total number) of
Men is 50 and Women is 80.
The shape of the graphed distribution may be indicative of certain phenomena and aid in interpretation, as
shown in the following histograms (bar chart showing graphical display of tabulated frequencies):
Normal Distribution
Bell-shaped curve (Symmetrical shape)
Most random events or processes, such as:
Average Height
Standardized Test scores
Revised: August 2019
7
Skewed (Non-symmetrical curve)
Positively Skewed curve
Longer right tail (few high values) with greater Frequency to left
(more low values)
Examples:
Income or Salary Levels
Environmental Science tests
Negatively Skewed (skewed to the left)
Longer left tail (few low values)
Frequency bunched to right (more high values)
Examples:
Fuel Consumption at various speeds
Easy tests
J-shaped curves (Exponential)
Extremely skewed curves (exponential curves)
Highest point at extremes
Fast growth or decay rates
Examples:
Automobile depreciation
Unrestricted Population Growth
Bimodal Distribution
M-shaped distribution
Two separated high points reflecting two different homogeneous
characteristics mixed together Or two characteristic frequencies
Examples:
Average Heights of women and men mixed together
Precipitation regimes
U-shaped distribution
Extreme bimodal distribution where two modes are at the
extremes (more high and low values than intermediate)
Examples:
Distribution of precipitation
Failure rates of automobiles
Rectangular (Uniform) Distribution
Equal Frequencies at each point or category
Each value randomly determined
Examples:
Frequency of values on a fair die
Temperature regimes
Pie Charts
Pie charts are an easy way to visualize how parts make up a whole. Pie charts are made from
percentage data such as the data on the elemental composition of the crust of the Earth. Note that
percentage data may have to be calculated from raw data in order to plot as a pie chart.
Revised: August 2019
8
Element
Percent of Crust
Oxygen
Silicon
Aluminum
Iron
Calcium
Sodium
Magnesium
Potassium
Titanium
All other
46
28
8
6
4
2
2
2
1
1
Nomograms
A nomogram is a graphical calculating device presented as a two-dimensional diagram designed to allow the
approximate computation of a function. These scales may be linear, logarithmic or have some more complex
relationship. Since the accuracy of the nomogram is limited by the scale of the drawing, it is best utilized when an
approximate answer will suffice. A simple nomogram typically has three scales: two scales represent known values
and one scale is the result. Therefore, if any two values are known, the third value may be determined graphically.
Values may be quantitative or qualitative. For example, to determine the level of environmental risk for a teenage
exposed to Chemical 10-A-c-X for several weeks, use a straight line to connect the two known values and read the
level of risk from the crossing point. As shown on the following nomogram, the red line connects teenage (10 – 20
years age) with duration of exposure (weeks) and the resulting interpretation is that this age group would have
MEDIUM level of risk.
Revised: August 2019
9
Triangular Graphs
On a triangular graph, the data for each of the three components represents a percentage value and the three
component percentage values must add up to 100 percent. Values on each axis, therefore, range from 0 % to 100
%. The position of the plots indicates the relative dominance of each of the three components and the value of the
graph arises in giving a quick visual comparison of contrasting component dominance for different areas. It is also
useful in identifying changes over time, since a position on the graph will change as the relative dominance of the
components change.
For example, in the following Soil Texture Triangle, soil “A” is a “clay” and soil “B” is a “silt loam. Soil “A” has
significantly greater clay content than soil “B.” Soil “A” and soil “B” have the same sand content. Soil “A”
consists of approximately 26 % sand, 12 % silt and 62 % clay. Soil “B” consists of approximately 26 % sand, 62 %
silt and 12 % clay.
Common Equivalents
1 inch = 2.54 cm
1 mile = 5,280 feet = 1609 meters
1 square mile = 640 acres
4 quarts = 1 gallon = 3.875 liters
Temperature Conversion
Celsius
Fahrenheit
Fahrenheit
(9 x C) + 32 = F
5
Celsius
5 * (F – 32) = C
9
Revised: August 2019
10
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