Metric Measurements and Biological Scaling
INTRODUCTION
In many biology investigations, precise measurements must be made before observations
can be interpreted. For everyday measuring, we still use English units such as the inch,
the gallon, and the pound. But for scientific work, and for everyday measuring in most
countries, the SI System is used. The SI System (Systeme International d'Unites or
International System of Units) is commonly referred to as the metric system. Like our
monetary system, SI is a metric system. All units are based on the number 10. In the SI
system it is easy to change one unit to another because all units are related to one another
by a power of 10. The basic measures (known as units) are listed in the following table:
Type of measure Standard Unit Symbol
length
meter
m
mass (weight)
kilogram
kg
temperature
degree Kelvin
K
time
second
s
TABLE 2-1
Although the kilogram is commonly used as a measure of weight, it is actually defined as
a measure of mass. Weight is a measure of the pull of gravity on an object which is
proportional to that object’s mass; whereas mass is a measure of the amount of matter in
an object. To illustrate this idea, a 120 pound woman would weigh only 20 pounds on the
moon (due to the moon having lower gravity than the earth), whereas a 50kg woman is
50kg on both the earth and the moon. In other words, her weight (how heavy she is) is
less on the moon but her mass (how much matter she has) is unchanged. For everyday
use, the difference between mass and weight is irrelevant, as everyone is together on the
earth. However, for scientific calculations (e.g. objects being launched into space), the
difference is important.
Also, temperature is officially measured in degrees Kelvin, but in day-to-day use the
Celsius system is used instead. The original metric measure of temperature was the
Celsius system, where 0°C was set to the freezing point of water. Officially, this has now
been replaced by the Kelvin system, where 0°K is set to absolute zero. However, it is
much more natural to say 'what a hot day, it must be 40°C' than to say 'what a hot day, it
must be 313°K'. Consequently, the system Celsius lives on in common usage. (In the
United States we often measure temperature in degrees Fahrenheit – where water freezes
at 32° and boils at 212° -- but this scale is not used in science.)
The key to the simplicity of the metric system is its use of multiples of 10, and its
reliance on only seven different base units. These units can then be combined to make
other measures. For example, speed can be expressed as meters per second or volume in
terms of cubic meter (a liter is the volume equal to a cube with each side 0.1 meters).
In the metric system, the standard units of measurement are multiplied by 10 or divided
by 10 in order to produce larger or smaller units. For example, a decimeter is a tenth of a
meter and a dekameter is 10 meters. For even larger or smaller units, one continues to
divide or multiply by 10. For example, one can either say '1000 meters' or one can say 'a
kilometer', as kilo means '1000' (which is 10 * 10 *10).
This use of base 10 greatly simplifies calculations and is a major reason that the metric
system has been so widely adopted. The following table is a complete list of the prefixes,
but many of them are not in common usage and consequently may not be understood by
everyone. For example, most people will say '10 meters' rather than '1 dekameter'. Both
are correct, but the former is much more frequently used and understood.
Multiplication factor Prefix Symbol
(1024)
yotta
Y
(1021)
zetta
Z
(1018)
exa
E
(1015)
peta
P
12
(10 )
tera
T
1 000 000 000 (109)
giga
G
1 000 000 (106)
mega
M
1000 (103)
kilo
k
100 (102)
hecto
h
1
10 (10 )
deka
da
0.1 (10-1)
deci
d
0.01 (10-2)
centi
c
0.001 (10-3)
milli
m
0.000 001 (10-6)
micro
µ
0.000 000 001 (10-9)
nano
n
-12
(10 )
pico
p
(10-15)
femto
f
(10-18)
atto
a
(10-21)
zepto
z
(10-24)
yocto
y
TABLE 2-2
Biologists use a wide variety of laboratory tools to make scientific measurements. Some
basic tools of measurement are the metric ruler, the electronic balance, the graduated
cylinder, and the Celsius thermometer. More precise measurement tools allow biologists
to measure increasingly smaller quantities with more accuracy. As an example,
micropipettes can be used to measure volumes that are expressed in microliters or
millionths of a liter.
The accuracy of a scientific measurement depends on three things: the smallest unit on
the measuring scale, the ability of the observer to read the scale properly, and the degree
of precision of the measuring instrument and scale. Since accurate measurement is
essential in many fields, and since all measurements are necessarily approximations, a
great deal of effort must be taken to make measurements as accurate as possible.
Nevertheless it is important that you realize the limitations of each measurement you take
and that you appreciate the precision that is possible with a particular measuring tool.
The numerical value of a laboratory measurement should always be recorded with the
proper number of significant figures. The number of significant figures depends on the
instrument or measuring device used and is equal to the digits definitely known from the
scale divisions marked on the instrument plus one estimated or doubtful digit. The last
estimated digit represents the uncertainty in the measurement and indicates the precision
of the instrument.
Measurements made with rulers and graduated cylinders should always be estimated to
one place beyond the smallest scale division that is marked. If the smallest scale division
on a ruler is centimeters, measurements of length should be estimated to the nearest 0.1
cm. If a ruler is marked in millimeters, readings are usually estimated to the nearest 0.2 or
0.5 mm, depending on the observer. The same reasoning applies to volume measurements
made using a graduated cylinder. A 10 ml graduated cylinder has major scale divisions
every one ml and minor scale divisions every 0.1 ml. It is therefore possible to estimate
the volume of liquid in a 10 ml graduated cylinder to the nearest 0.02 or 0.05 ml. Three
observers might estimate a particular volume of liquid in a 10 ml graduated cylinder as
8.32, 8.30, and 8.33 ml respectively. These are all valid readings. But it would not be
correct to record the volume of liquid simply as 8.3 ml. Likewise, a reading of 8.325
would be too precise.
Some instruments, such as electronic balances, give a direct reading – there are no
obvious marked scale divisions. This does not mean that there is no uncertainty in an
electronic balance measurement. It means that the estimation has been carried out
internally (inside of the electronic balance) and the result of that estimation is reported
digitally. There is still uncertainty in the last digit. On an electronic balance, three
researchers might measure the mass of the same object as 5.65, 5.67, and 5.68 grams
respectively. The uncertainty of an electronic balance measurement is one unit in the
smallest scale division that is reported. On this particular balance, it would be 0.01 grams.
Accuracy and precision are two different ways to describe the error associated with
measurement. Accuracy describes how ‘correct’ a measured or calculated value is – how
close the measured value is to the actual value. The only way to determine the accuracy
of an experimental measurement is to compare it to a ‘true value’, but that is typically not
known. Precision describes the closeness with which several measurements of the same
quantity agree. The precision of a measurement is limited by the uncertainty of the
measuring device. Uncertainty is often represented by the symbol ± (“plus or minus”)
followed by an amount. If the measured length of an object is 24.72 cm and the estimated
uncertainty is 0.05 cm, the length can be reported as 24.72 ± 0.05 cm to take into account
precision.
Variations among measured results that do not result from carelessness, mistakes, or
incorrect procedure are called experimental error. Experimental error is unavoidable. The
magnitude and sources of experimental error should always be taken into account when
evaluating the results of an experiment.
Once you have made measurements, it is important to be able to analyze those data such
that you can look for patterns and address whether or not the particular hypothesis at
hand is rejected or supported by the data. Such analyses often require a visual
representation which often takes the form of a graph. A graph serves as a pictorial
representation used to show the relationship between two or more different variables.
This lab is set up as a series of stations that will give you the opportunity to practice
measuring a variety of objects using different measuring instruments. There are questions
throughout that you should answer as you go. Those answers should guide you as you
write your final report for Lab 1. Following the measurement activities, you will
complete a series of exercises that enable you to apply measurement in a meaningful way
to answer some specific questions about the size and shape of biological structures. Size,
shape, and the relationships of each to function are important topics in biology. Whether
you are interested in understanding the significance of a particular form or interpreting
the factors which may underlie a particular pattern, it is important that you understand the
physical consequences and interdependence of organismal form and size. You will take a
sample of related biological specimens of different sizes, devise hypotheses about
whether the shapes of these specimens vary with size and how that might relate to some
biologically relevant function, and then test your hypotheses by making a series of
measurements and graphing those data.
MATERIALS
a variety of objects to measure
metric ruler
metric measuring tape
graduated cylinders with colored water
Eppindorf tubes with colored water
micropipette with tips
electronic balance
a series of bivalve shells of increasing size
PROCEDURE
Measuring Length
Use the metric ruler and/or the metric tape measure to determine the length, width, and
height of the objects provided. Record your measurements to the nearest one tenth
centimeter (millimeter) in the following table. Be sure to use the appropriate number of
significant figures. Calculate the surface area (SA) and the volume (V) of each object and
enter those values in the table. Be sure to use the appropriate number of significant
figures. The surface area of a cube can be calculated by adding together the surface area
(length x width) of all six sides. The volume of a cube is equal to (length x width x
height).
OBJECT #
Length (cm)
Width (cm)
Height (cm)
SA (cm2)
V (cm3)
TABLE 2-3
Measuring Mass
The electronic balance has many advantages over other types of balance. The most
obvious is the ease with which a measurement is obtained. All that is needed is to place
an object on the balance pan and the measurement can be read on the display to
hundredths of a gram. A second advantage is being able to use the Zero button on the
front of the balance. Because one must never place a chemical directly on the balance
pan, some container must be used. Place the container on the balance and the mass of the
container will be displayed. By pressing the Zero button at this point, the balance will
reset to zero and ignore the mass of the container. You may now place the substance to be
weighed into the container and the balance will show only the mass of the substance. This
saves calculation time and effort. However, when the container is removed from the
balance, the display will go into negative numbers until the Zero button is pressed again.
Measures the mass of the objects provided to the
nearest one hundredth gram and enter it in the
table below.
Calculating Density
The density of a material is defined as its mass
per unit volume. The density of a certain material
can be calculated easily if you know the mass and
volume of that object. Just divide the mass by the
volume. Be sure to use the appropriate number of
significant figures.
FIGURE 2-1
The density of pure water is 1.0 gm/cm3 and the density of rubbing alcohol is roughly
0.79 g/cm3 at 15°C. For each of the three objects for which you measured mass and
volume, calculate the object’s density and predict whether or not the object will float in
water and whether or not it will float in rubbing alcohol. Record your predictions and
observations in the table.
PREDICTION
OBJECT #
Mass (g)
Volume
(cm3)
Density
(g/cm3)
Floats in
Water
Floats in
Alcohol
ACTUAL TEST
Floats in
Water
Floats in
Alcohol
TABLE 2-4
Estimating Error
Compare the measurement data from your group for three objects with the measurement
data from a neighboring group for the same three objects. Enter that data in the tables
here.
Data for Object # 1
GROUP
Length
(cm)
Width
(cm)
Height
(cm)
SA (cm2)
V (cm3)
Mass (g)
Density
Your Group
Neighbor Group
AVERAGE
TABLE 2-5
Data for Object # 2
GROUP
Length
(cm)
Width
(cm)
Height
(cm)
SA (cm2)
V (cm3)
Mass (g)
Density
Your Group
Neighbor Group
AVERAGE
TABLE 2-6
Data for Object # 3
GROUP
Length
(cm)
Width
(cm)
Height
(cm)
SA (cm2)
V (cm3)
Mass (g)
Density
Your Group
Neighbor Group
AVERAGE
TABLE 2-7
Study these data tables. Do each group’s measurements have the exact same value for
each object?
Using the tools and techniques available, were some measurements more nearly alike
than others? (For example, are all of the values for length of a particular object more
similar than values for the mass of a particular object?) Why were some measurements
more consistent between groups than others? Be specific – there are likely several
reasons.
Calculate the average (mean) of each measurement for both groups. Why might an
average of the two measurements be more precise (closer to the actual value) than a
single measurement?
Measuring Volume with a Graduated Cylinder
The surface of a liquid confined in a cylinder curves to form what is known as a
meniscus. The meniscus of most liquids curves up the sides of the container, making the
center of the curve appear lower than the edges. Since reading the meniscus at the top or
at the bottom of the curve will make a difference in the volume measured, it is generally
agreed to always read the bottom of the curve. The smaller the container, the greater the
curve of the meniscus. To measure the volume accurately, your eye must be at the same
level as the bottom of the meniscus. This is the mark on the graduated cylinder that you
must read. Use the graduated cylinder to measure the volume of colored liquid provided
in each test tube. Be sure to use the appropriate number of significant figures.
Volume (ml)
Volume (ml)
Large Cylinders
Small Cylinders
TABLE 2-8
FIGURE 2-2
Micro-volumes
In subsequent labs such as the DNA Fingerprinting lab and the ELISA lab you will
measure very small volumes using a tool called a micropipette. A micropipette measures
volumes in microliters (μl) or 10-6 liters (millionths of a liter). We have a setup of small
tubes with colored water showing a variety of micro-volumes ranging from 10 μl to 1000
μl. You should examine the tubes and take some notes here. See if you can describe the
range of volumes in a way that would help you remember about how much each volume
represents – maybe in “drops”?
Line Graphing
The type of graph that best shows the relationship between two variables is the line
graph. The line graph will usually be the most useful type of graph in this course. A line
graph has one or more lines connecting a series of points.
When plotting data on your graph, you must decide which variable to place along the xaxis and which variable to place along the y-axis. Usually, along the horizontal axis, or xaxis, you will find the most consistent variable in the experiment (often called the
independent variable). Along the vertical axis, or y axis, you will find the other variable
(often called the dependent variable). You must then decide on the scale of each axis; that
is, how much each unit along the axis represents. Scales should be chosen to make the
graph as large as possible within the limits of the paper and still include the largest item
of data. If the scale unit is too large, your graph will be cramped into a small area and
will be hard to read and interpret. If the scale unit is too small, the graph will run off the
paper. Scale units should also be selected for ease of locating points on the graph.
Multiples of 1, 2, 5, and 10, are the easiest to work with.
Practice reading the information in this line graph by answering the questions that follow.
FIGURE 2-3
1) What might you consider to be the independent variable in this experiment? Why?
2) In what units is the independent variable expressed?
3) What might you consider to be the dependent variable in this experiment? Why?
4) In what units is the dependent variable expressed?
5) Which plant grew the tallest?
6) How many plants grew to be at least 6 cm tall?
7) Which plant grew the fastest in the first five days?
8) After 10 days, how much had plant 3 grown?
9) How long did it take plant 1 to grow 6 cm?
Many of the lab reports this semester will require you to produce a graph. You could
draw out a line graph by hand using graph paper, but it’s even easier to create
professional looking line graphs using a program life Microsoft Office Excel. Here’s how
you would produce a line graph for the data presented in the following table.
Breathing Rate of the Freshwater Sunfish
Temperature
Breathing Rate
(breaths per minute)
(°C)
10
15
15
25
18
30
20
38
23
60
25
57
27
25
All you need to do is enter the data as two columns:
10
15
18
20
23
25
27
15
25
30
38
60
57
25
TABLE 2-10
TABLE 2-9
Then go to the menu tab “Insert” and select “Chart”. Under ‘chart type’ select “XY
(Scatter)”. There are lots of additional tabs that allow to you make your graph look
exactly the way that you want it to look and you can even add text for the axes, but in the
simplest form you can obtain a graph with these simple steps like the one show here:
70
60
50
40
Series1
30
20
10
0
0
10
20
30
FIGURE 2-4
You should practice making graphs using Excel. This is an excellent way to prepare
graphs to include in your reports.
Scaling Exercises
This part of the exercises allows you to use measurement in order to address important
biological questions about scaling and geometric scaling. Two objects are considered
geometrically similar if they have the same shape. For geometrically similar objects, one
linear measure is proportional to another linear measure. For example if one object has a
length that is twice that of another, you would expect its width to also be twice that of the
second object if the two are geometrically similar.
Objects that exhibit geometric similarity (where different sizes share the same shape) are
said have Isometric Scaling. Objects that differ in shape as well as size are not
geometrically similar and are said to have Allometric Scaling.
This concept of geometric similarity can be extended to measurements such as area and
volume too. To test whether objects scale isometrically or allometrically, two sets of
measurements (x and y) are log transformed and then plotted against each other in a line
graph. There is only one slope for that line that indicates isometric scaling – no difference
in shape. Any departure from that one slope of isometry indicates a non-isometric or an
allometric scaling relationship. If the measurements are both linear, log transformation is
unnecessary because they are both expressed in the same units. But for area and volume
measurements that are in cm2 and cm3 respectively, log transformation will be important
– and easy to do with Excel or a calculator.
NOTE: You should be familiar with logarithms as they will be incorporated in several
labs this semester. In mathematics, the logarithm of a number to a given base is the power
or exponent to which the base must be raised in order to produce that number. For
example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must
be raised to produce 1000: 103 = 1000, so log101000 = 3. The study of any natural
relationship that scales exponentially, involves working with numbers that are very small
or very large. By using logarithmic values (exponents) those large and small numbers
become easier to work with. An example might be changes in the number of bacteria in a
colony that doubles every 20 minutes. The relationship between population size and
number of generations is exponential and would typically be better studied by converting
raw values into logs.
As another example, picture a box that is 1 cm on each side – the length equals 1 cm and
the volume equals 1 cm3. But what happens if that box gets larger (isometrically –
without changing shape) such that it is now 10 cm on each side. The length increased by
10….but what happened to the volume. Notice that the volume increased to 1000 cm3! So
if you plotted a simple graph of length vs. volume for a series of boxes of increasing size
you would not end up with a line of slope = 3. You would not even get a straight line.
You would have an exponential curve. However, if you take the log of each measurement
and then make your graph, you will end up with a linear relationship, in this case with a
slope of 3 – the expectation under isometric scaling when examining length vs. volume.
Examine these data sets and corresponding graphs. Notice that the raw data does not
produce a linear plot (because it is exponential), but the log transformed data does
produce a linear plot. Keep in mind that BOTH graphs show the same data set, but
because the numbers cover a large range and scale exponentially; a clear scaling
relationship is only apparent when the data set is transformed.
Raw Data
1
10
100
1000
10000
100000
1000000
Log Transformed Data
0
1
2
3
4
5
6
1200000
7
1000000
6
5
800000
4
600000
Series1
Series1
TABLE 2-11
3
400000
2
200000
1
0
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
FIGURE 2-5
The isometric expectation for log transformed data sets is:
Length (cm) vs. Length (cm)
Length (cm) vs. Area (cm2)
Length (cm) vs. Volume (cm3)
SLOPE = 1
SLOPE = 2
SLOPE = 3
By taking measurements, log transforming the data sets, and plotting that log-transformed
data, you will be able to test the shells for isometric scaling.
As these bivalves grow in size, does the overall shape of the shell change? Choose two
linear dimensions of the shell (length and width) that you can determine consistently on
all specimens and use a metric ruler to measure these. Use the landmarks indicated on the
shell here to consistently measure length and width the same way on each shell.
length
width
FIGURE 2-6
Before you actually take your measurements and test to see whether the length and width
of the shells scale isometrically, make some predictions. What do you think are some of
the functional implications of shell shape? Are there reasons why you might expect the
shell’s shape to change as the bivalve grows? Think about the purpose of the bivalve’s
shell. What are the advantages of having a shell for the bivalve? Are there any
disadvantages to having a shell? In light of those benefits and potential limitations, might
there be a good argument as to why the shell should change in shape as the animal
grows?
Measure the length and width of your shells to the nearest tenth of a centimeter and enter
those values in the data table. Start with the smallest shell and work up to the largest
shell. Because you are using two linear measurements, it is not necessary to log transform
the data. Enter the data into Excel and produce a line graph for your report.
Shell #
1
2
3
4
5
6
7
Length (cm)
Width (cm)
TABLE 2-12
What do you expect the slope of the graph to be if the shells scale isometrically (do not
change in shape as the animal grows)?
What is the slope of your graph? (You can get a y = mx + b equation by right clicking on
one of your points, selecting “Add Trendline”, and clicking the box that adds the
“Equation”. m = the slope.)
What conclusions can you draw from this?
Now let’s test to see if the thickness of the shell increases as the animal grows in length.
You will use the mass of the shell as a proxy for thickness. This in essence is a volume
measurement so we are thinking of the units as cm3 rather than grams – but you are
measuring in grams. So you are comparing a linear measurement (length) to volume
(mass). If the shape does not change (i.e. length scales isometrically with volume), what
should the slope of the line graph be after the data is log-transformed?
Again, before testing for isometry, make predictions. Do you think the shells should
increase in thickness proportionally as they increase in length? Based on the biology of
these bivalves, why do you think the thickness should scale the way you predict? What
are the advantages and disadvantages of having a shell that thickens proportionally to its
length?
Measure the mass of your shells with the electronic balance and enter those values in the
data table along with the length values from the previous table. Enter the data into Excel
and log-transform the data before producing a line graph.
Shell #
1
2
3
4
5
6
7
Length (cm)
Mass (cm3)
TABLE 2-13
What number would you expect the slope of the graph to be if the shells scale
isometrically (1, 2, or 3)?
What is the slope of your graph?
What conclusions can you draw from this?
DISCUSSION QUESTIONS
1) Since you know that the density of pure water is equal to 1 g/ml, what is the mass of
40 ml of pure water?
Would you expect the mass of 40 ml of vegetable oil to be more or less than the mass
of the water? Explain your answer.
2) How is a graph similar to a data table?
3) Does a steep curve on a line graph indicate a rapid or a slow rate of change? Why?
4) Of the following graduated cylinders, 100 ml, 25 ml, or 10 ml, which would you use
to most accurately measure 8 ml of a liquid? Explain your answer.
5) You are conducting an experiment to measure the gain in mass of a young mouse over
a ten-week period. In constructing a line graph to represent your data, which variable
should you place along the x-axis and which variable should you place along the yaxis. Explain your answer.
6) When might you graph multiple lines on the same line graph? What would be the
advantage of doing so?
7) Why is it important to have all parts of a graph clearly labeled?