Math Review for Physiology Students (LPC)
A) Understanding Range
If a set of numerical readings are taken by a scientist or clinician, it is likely that not all of
the readings will be identical. For example, a person’s cholesterol level will tend to vary
according to the time of the day or depending on what foods have been recently
consumed. One isolated measurement may not be enough to figure out the individual’s
typical, day-to-day serum cholesterol level. Some measurements will be higher than
others. Thus, often, several measurements of the same thing are more revealing than a
single one. However, in the end you will want to give a single measurement that is
representative of all the others. This can be the range, and the average or mean.
If all the values are arranged in numerical order, from lowest to highest or vice versa, the
range of values is the difference between the highest and lowest values in the series. For
example, suppose that cholesterol readings over several days are 120 mg/100mL, 205
mg/100mL, 176 mg/100mL, 147 mg/100mL, and 234 mg/100mL. The lowest value is
120, the highest is 234. The difference between these two measurements is 114, meaning
that the person’s serum cholesterol varied by 114 mg/100mL over the days when the
measurements were taken. A second, more informational way of expressing range is to
state the extremes. In the example from above, we can say the patient’s cholesterol
ranged between 120 - 234 mg/100mL.
1) Suppose that over a particularly rainy winter week the rainfall amount was measured
at 2.5 cm on Sunday, 3.53 cm on Monday, 1.51 cm on Wednesday, 0.75 cm on
Thursday, and 2.12 cm on Saturday. State the range of rainfall, which fell during this
week in two different ways.
1) The range of rainfall varied by ________________
2) The rainfall ranged between ___________________
2) Suppose that a person interested in his daily urine production measured his urine over
a five-day period at 1,940 mL, 796 mL, 2,121 mL, 2,051 mL and 1,996 mL; all on
different days. State the range of urine volume over the course of the study, in two
different ways.
1) The range of urine production varied by ________________
2) The urine production ranged between ___________________
B) Understanding Average and Mean
Another way to express data is to calculate the average. The average, or mean, is found
by adding the values together and dividing by the total number of values. In the serum
cholesterol example given above the sum of the measurements equals 882 mg/100mL.
The total number of measurements taken was 5. Therefore, the average is 882 ÷ 5, or
176.4 mg/100mL. (By the way, the unit of measurement “mg/100 ml” is frequently used
in laboratory reports, including blood [glucose].)
1) Calculate the average, or mean, of the rainfall per day during the week described
above.
_________________
2) Apparently it did not rain on Tuesday and Friday, so the rainfall for those days was 0
cm. If these days were included in the average, would it increase or decrease the result?
If you did include them, your result shows the average daily rain. If you didn’t include
them, your result shows the average rain when it rains. The two averages have slightly
different meanings. The lesson here is that it is important to be clear on what exactly an
average is means.
3) What is the average or mean amount of urine produced per day in the example on
page 1?
___________________mL
4) Suppose you need to calculate how much urine an outpatient’s kidneys produce each
day. You tell the patient to measure his urine production over a week’s time. He took
measurements every day, but was traveling on one of the days and therefore unable to
measure his urine that day. Would it be accurate to add 0 mL into the total and divide by
7? Why or why not?
5) Go back and look at the urine volume values given on the previous page. Notice that
one day’s measurement of urine is considerably lower than that of the other days, and
therefore it lowered the average considerably. You might be justified in leaving that
measurement out of your calculation if you knew it was due to extremely abnormal
circumstances (i.e. the patient spilled the urine sample or didn’t drink any liquids that
day) but otherwise you should include it.
C) Understanding Percent
Calculating Percent from Data
Percent (designated by the sign %) refers to the number of instances out of every
hundred. Thus, if 25% of the students in an early morning physiology class have not
eaten any breakfast, it means that 25 out of 100 students have not eaten breakfast.
However, calculating percent is usually not that easy, because the total number of
instances being studied does not add up to a convenient 100. For example, assume that
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the above class consists of 20 students. If 25% have not eaten breakfast, exactly how
many students is that? _________
What if the class was made up of 16 students and four of them have not eaten breakfast?
This means __________% have not eaten breakfast.
This was easy to calculate mentally. But what if it is not that easy? Not to worry: there is
an easy way to calculate percent, regardless of the total number in the sample. You only
need to identify which is the part under consideration, and which is the whole. Then you
divide the part by the whole, multiply by 100, and add the % sign. In the above
example the whole is the 16 students in the class, and the part being studied is the four
who have not eaten breakfast. Doing the calculation, 4  16 = 0.25, and 0.25 x 100 = 25.
Add the % sign and you have the result: Four students are 25% of the class.
Notice how easy it is to multiply mentally by 100. You just have to move the decimal
point two places to the right. For example: 0.25 x 100 = 25, and 4 x 100 = 400. If no
decimal point is visible, it is assumed to be at the end of the number, and zeros are added
to fill any empty spaces, as in the above example.
1) Suppose that 5% of the students in the class have not yet bought their textbook. If
there are 100 students in the class, how many have not yet bought their book?
__________
2) Suppose that the class is made up of 40 students, and two of them have not yet bought
their textbook. What % of them has not bought their book? ______________%
Be careful not to make the common mistake of always dividing the small number into the
larger! In most percent problems, the part is smaller that the whole. However, it is
possible to encounter a situation in which the part is greater than the whole. This is
possible when 100% is considered the normal expected amount, and the number of
instances being studied is greater than that.
For example, the normal annual rainfall for a particular year is 28 cm, but in a certain
year the region gets 34.4 cm. Once again the rule to follow is to divide the part by the
whole and multiply by 100, then add the % sign. In the above example the whole is the
normal 28 cm and the part being studied is the 34.4 cm in that particular year. Doing the
calculation 34.4  28 = 1.22, and 1.22 x 100 = 122. The rainfall in that year, then was
122% of normal.
3) A patient’s blood cell count shows 15,000 white blood cells (WBCs) per µL of blood.
A normal WBC count is about 9,000 cells/µL (the normal range is actually 4,000 to
11,000 WBCs/µL. What percent of normal is this particular patent’s WBC count?
_____________
4) Obesity is defined medically as 20% above a person’s ideal weight. If a person’s
ideal weight is 72 kg, when (above what weight) would that person be considered
obese? ________________kg
5) If a man’s ideal weight is 165 lbs. and he weighs 185 lbs., how many pounds above
his ideal weight is he? What percent above his ideal weight is he?
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______________lbs. above ideal weight
______________% above ideal weight
Calculating Data from Percent:
It is even easier to calculate the number of measurements or occurrences when you are
given the percent. Remember that “percent” means “the number of instances out of one
hundred”. A helpful way to understand percent is to translate “per” as “divided by”, and
“-cent” as 100. Thus, 50% is the same as 50  100, or 0.5.
It is easy to determine percent if the number of instances is exactly 100. Suppose that 100
victims of smoke inhalation are brought into a hospital. 92% have been sent home after
basic treatment. It is easy to calculate mentally that 92 victims have been sent home and 8
have been retained for additional treatment.
It is really not much more difficult if the number of instances is not exactly 100. Suppose
the number of smoke inhalation victims equals 133, and 92% needed only basic
treatment. 92% = 0.92, and 0.92 x 133 = 122.36 victims (or realistically 122 victims).
6) In one year, 120 individuals were diagnosed with AIDS in a particular city. Three
years later, 68% of those individuals had shown some improvement, while 3% had died.
How many patients had shown improvement, and how many had died.
_________ showed improvement
__________died
D) Scientific or Exponential Notation
Scientists often need to work with numbers that range from the astronomical to the
subatomic. Such numbers are cumbersome to work with. For example an important
concept in chemistry is the “mole”, which is 602,000,000,000,000,000,000,000 atoms,
molecules or particles of any sort. How much more convenient it is to write that number
as 6.02 x1023 instead!
In biology extremely large or small numbers are frequently encountered. Bacteria range
in size between 0.0002 – 0.005 mm (millimeters). The average number of red blood cells
in an adult male is about 5,500,000 cells/µL. Ultramicroscopic features in cells are often
measured in Angstrom (Å) units, an Å being 0.0000000001 m or 0.0000001 mm. These
numbers can be written in a shorter, more manageable form using standard exponential
(or scientific) notation.
For example, 100 can be written as “10 squared” or 102; 1,000 can be written as 103;
10,000 as 104 etc. Likewise, a number like 186,000 can be written as 1.86 x 105.
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In the scientific notation system any number can be expressed as the product of a number
between 1 and 10 (including 1 but not 10), and a power of 10. Thus, the last example
would be mathematically correct but scientifically unconventional to write as 18.6 x 104.
Similarly, very small numbers can be expressed as the product of a number between 1
and 10 (but not including 10), and a negative power of 10. Thus, 0.1 can be expressed
10-1, 0.01 as 10-2 etc. One Å can be expressed as 10-10 m. Again the power is equal to
the number of spaces you need to move the decimal point, but this time to the left, in
order to restore the number to its original form. The power is given a negative sign.
If bacteria range in size between 0.0002 – 0.005 mm, we can also express this range as
2 x 10-4 to 5 x 10-3 mm.
1) Suppose that the national budget surplus is 3.3 trillion dollars ($ 3,300,000,000,000).
Express this number in standard scientific notation. _____________
2) Express the following numbers in scientific notation:
The diameter of a human red blood cell is about 0.0075 mm. _______________
The smallest viruses are about 0.000018 mm wide. _______________
3) Express the following numbers in conventional form:
The length of an Argentine ant is 2.4 x 10-1 cm. _______________
An electron weighs 9.11 x 10-28 g. ___________________________________
E) UNITS OF MEASUREMENT: THE METRIC SYSTEM
For a rich resource of metric information go to: http://lamar.colostate.edu/~hillger/
Introduction:
The Metric system (from the Greek word metrikos, meaning “measure”), which was first
developed in late eighteenth century France, is worldwide employed as the standard in
scientific literature and in the field of medicine. The modern definitions of the units
used in the metric system are those adopted by the General Conference on Weights and
Measures, which in 1960 established the International System of Units (also called the
modern metric system) abbreviated SI in all languages.
Technically, the United States does use the metric system: It’s been the nation’s official
system since 1893. Unfortunately, most Americans prefer to ignore the official system in
favor of weights and measures that are relics of when North America was a British
colony. In terms of the family of nations, America’s only kin on the metric issue are
Liberia and Burma!!
There has been some progress in the move to metric. Think of 35 millimeter film or soft
drinks sold in 1- and 2-liter bottles. Furthermore, anyone who reads nutrition labels
encounters milligrams. Cars and other products seeking to compete overseas generally
are made to metric specifications. In the hospital, drugs are prescribed in mg or g per kg
body weight. Clinical laboratories will report routine blood profiles with listings such as
HDL Cholesterol: 53 mg/dL
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Glucose: 104 mg/dL
Albumin: 4.9 g/dL
This means, that you will now need to get used to the metric terminology. In our class we
will exclusively use the metric system. You will need to recall the definitions for the
metric units of length, mass, volume, and temperature. You probably should have a good
understanding of converting from one system to the other, but for the purpose of this
course you will only need to convert from one unit to another within the metric system.
meter (m) – unit of length equal to 1,650,763.73 wavelengths in a vacuum of the
orange-red line of the spectrum of krypton-86. (1 m = 1.09 yards)
gram (g) – unit of mass based on the mass of 1 cubic centimeter (cm3) of water at the
temperature of its maximum density. (1 kg = 2.2 pounds)
liter (L) – unit of volume equal to 1 cubic decimeter (dm3) or 0.001 cubic meter (m3).
(1L = 1.05 quarts)
Celsius (C) – temperature scale in which 0C is the freezing point of water and 100C is
the boiling point of water; this is equivalent to the centigrade scale. (0C = 32F)
Conversions between different orders of magnitude in the metric system are based on
powers of ten. You can convert from one order of magnitude to another simply by
moving the decimal point either to the right (for multiplying by whole numbers) or to the
left (for multiplying by decimal fractions). Sample conversions are illustrated in table 1.
Table 1: Sample metric conversions
To Convert From
To
Factor
Move Decimal Point
Meter (liter, gram)
Meter (liter, gram)
MilliMicroMilliMicro-
MilliMicroMeter (liter, gram)
Meter (liter, gram)
MicroMilli-
x 1,000 (103)
x 1,000,000 (106)
 1,000 (10-3)
 1,000,000 (10-6)
x 1,000 (103)
 1,000 (10-3)
3 places to right
6 places to right
3 places to left
6 places to left
3 places to right
3 places to left
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A few Practice Examples:
2.5 m = __________ mm = ____________________µm
950 µm = ____________ mm
6,500 mg = _____________ g = ____________________µg
350 µl = ____________ mL
4,700 mL = _____________ L =_____________________µL
Estimating Uncertainties
Scientists do not rely exclusively on the markings on measuring instruments to make
measurements. They also estimate smaller quantities between the markings. For
example, on the ruler A below, we might be content with stating the length of the bar as
about 9, even though the bar is a little shorter that that. A scientist, however, would
imagine that the space between 8 and 9 is divided into ten equal parts, and might use his
or her best judgment to state the length of the bar as 8.8. Because of the uncertainty,
another worker might decide that the measurement is closer to 8.7. Only one estimated
digit is allowed, therefore: 8.75 would not be an acceptable estimate!
It is important to remember that the last digit is uncertain, and therefore two workers
might disagree about its exact value.
Likewise, ruler B is divided into smaller units, but the scientists will imagine that these
very small units also are divided into equal parts, and do their best to estimate between
them. Reasonable measurements for this bar might be 7.15 or 7.16.
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My estimate:__________
My collegues estimates:
My estimate:__________
My colleagues estimates:
My estimate:__________
My colleagues estimates:
My estimate:__________
My colleagues estimates:
My estimate for G:
My estimate for H:
My estimate for I:
Practice estimating the measurements in rulers C and D, then in tubes E and F. Liquids
in tubes or cylinders have a curved surface, called a meniscus, with a top and a bottom.
Scientists always read the measurement by viewing the bottom of the meniscus, as in the
illustration. Be careful to not be content with “about” this or that, but to imagine that each
interval is divided up into ten smaller, equal intervals, and estimate to the best of you
ability between the intervals. Then ask three other classmates for their estimate, and
record them next to yours (with listing their name). Remember that the last digit is a
personal estimate and that not everyone might agree, but all estimates must be
reasonable. Finally, estimate the measurements in thermometers G and H, and in gauge I.
Remember always to estimate between the calibration marks on the instrument.
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Length
1) Examine a meter stick which has a yard printed on the back. The metric side will be
calibrated in centimeters and millimeters (the numbered intervals are centimeters). First,
what is the very approximate comparison in length between a yard (36 inches) and a
meter? Which is longer? _____________________________
Measure the height of your lab bench, from the floor to the top. Is it higher or shorter than
a yard? _____________
Express the height in centimeters: _______cm.
Now express the height in meters: __________m.
Have your partner measure your height in the metric system: ___________ cm or
___________m.
2) Examine a six-inch ruler having both a metric scale and a scale of inches. The metric
scale is calibrated in cm and mm. If you were using this ruler to measure something in
cm, and if you estimated between calibration marks, as you are supposed to do, how
many decimal places would appear in your measurement? __________
Compare the size of an inch and a cm. Line up a piece of paper with the one-inch mark
and use it as a guide to help you estimate (to the allowable number of decimal points of
course) the size of an inch in cm. ____________
To help you remember the approximate size of a cm, find one of your fingernails that is
as close as possible to being 1cm wide. Keep this in your memory for future reference!
3) Measure the diameter of a penny, then a nickel and a quarter in cm and remember to
estimate to the correct number of decimal places if necessary.
Penny:___________
Nickel:__________
Quarter__________
Suppose a child were to ask you how large a cm is. Using the information you just
gathered, write a statement, which expresses to the child the approximate length of a cm:
________________________________________________________________________
__________________________________________________.
Express the following diameters in millimeters:
Penny:___________
Nickel:__________
Quarter__________
4) Obtain a piece of uncooked spaghetti. Measure the thickness of the spaghetti. Be sure
to measure to as many decimal places as you are allowed.
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Thickness of a spaghetti noodle in mm: ___________mm
Thickness of a spaghetti noodle in cm: ___________cm
How would you use your newfound information to give a child an idea of how large a
millimeter is? Write a clear statement expressing that information:
________________________________________________________________________
________________________________________________________________________
__________.
A micrometer (µm) is also called a micron, but the term micrometer is preferred!
Because of its very small size, we have no easy references to help us remember the size
of a µm.
Express the thickness of an uncooked spaghetti in micrometers: ___________µm
5) A typical human hair is about 50µm wide. How many human hairs lined up side by
side would it take to fill in the space of a mm on your ruler? __________
Volume
Volume is space – it is a measurement of the amount of space something takes up. In the
Metric system, volume is measured in liter. Examine a measuring cup which has a
capacity of one quart and is also calibrated in liters. Approximately how many cups are
there in a liter? ___________
Actually, a liter equals 1.06 quarts, so a quart and a liter are very close to the same. Does
this mean that a quart is a little larger or a little smaller than a liter? _____________
One liter is subdivided into 10 parts or 10 deciliters (dl). A dl is subdivided into 10 parts
or 10 centiliters (cL). A cL is subdivided into 10 parts or 10 milliliters (mL).
Since 10 x 10 x 10 equals 1,000: 1 L = _________mL
Examine a measuring cup that has a capacity of one cup, or use the one quart measuring
cup you used above. Find the mL graduations. If a muffin recipe calls for a cup of milk,
approximately how many mL would this be? ___________
You now know that a mL is equal to 1 / 1000th of a liter. You can get another idea of the
volume of a mL by filling a 1-mL calibrated plastic transfer pipette with water to the 1
mL mark and counting the number of drops as you slowly empty the pipette. Look for the
calibration mark just below the bulb of the pipette, and then do this experiment.
Number of water drops in 1 mL: ___________________
Not all liquids have the same degree of surface tension, intermolecular attraction and
density. Therefore, not all liquids will form droplets which are exactly the same size.
Repeat the above experiment with ethanol (can you remember the chemical formula?
_______________). Catch the droplets in a small beaker and wash the alcohol down the
sink when you are done. How many drops of this alcohol are there in a mL? _________
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Calibrating a Pipette
Finally, calibrate a plastic transfer pipette of you own, both for the learning experience
and for use in a future experiment.
To do this, fill a 10 mL graduated cylinder with demineralized water exactly to the 10 mL
mark. Remember that the bottom of the meniscus must rest precisely on the 10 mL
mark!!
Have a permanent felt marker handy. Obtain a clean 2 mL plastic transfer pipette. (They
are called 2 mL pipettes, but they actually take up more than 3 mL.) Use it to draw up
exactly 3 mL of H2O from the cylinder. Remember that the bottom of the meniscus now
must rest precisely on the 7 mL mark!! After making sure that there is no air space in the
tip of the pipette, use the marker to mark the 3 mL level on the side of the pipette. Use as
thin a line as you can. Then carefully release H2O into the cylinder until the meniscus is
resting exactly on the 8 mL mark, and use the marker to mark the 2 mL level on the
pipette. Do the same to mark the 1 mL level.
If you accidentally lose some H2O while calibrating your pipette, or if you are aware of
any airspace in the tip, you must do it over. When you are finished, put your name on the
pipette to identify it as your own in a future experiment. Turn it in to your instructor.
Milliliters and Cubic Centimeters:
Try to imagine H2O in the shape of a cube, which measures exactly 1 cm on each side.
This figure would be one cubic centimeter of H2O, or 1 cm3.
Now imagine this H2O being transferred to an unmarked glass cylinder, similar to the one
you used while calibrating your transfer pipette. The bottom of the meniscus could be
marked to indicate the 1 cm3 mark. This volume is identical to the volume of 1 mL. The
unit cm3 is also sometimes abbreviated as cc, but it means the same thing: cubic
centimeter.
Therefore, remember that:
1 mL = 1 cm3 = 1cc
Since the development of biotechnology in recent decades, many volume measurements
in biology are made in microliters (µL). A µL is 1/1,000th of a mL, or /1,000,000th of a
liter. In a later experiment, you will practice measuring liquids in µl, but for now make
sure you prepare for that experiment by remembering what a µl is.
Mass
Mass is a measure of the amount of matter a substance is made up of. We often use the
term “weight” interchangeably, but technically there is a difference between weight and
mass. When Neil Armstrong set foot on the moon in 1969, he was composed of
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approximately the same amount of matter as when he walked on the earth; that is, his
mass remained the same. However, the gravity of the moon is less than the gravity of the
earth, and so Armstrong's weight was less on the moon.
The unit of mass in the Metric system is the kilogram, which is equal to 1,000 grams.
Inspect one of the cans or boxes used for household goods on display in the lab. Do not
weigh the container. Simply write down the name of the former contents of the container,
then write down its weight in grams (from the label, not the scale), then write down its
weight in ounces (from the label, not the scale). To learn the approximate relationship
between a gram and an ounce, divide the weight in grams by the weight in ounces. The
correct answer will tell you the number of grams in an ounce, or grams per ounce.
Inspect two additional containers and perform the same calculations. If you have done
this correctly, your three answers should be approximately the same.
Now use a centigram balance to weigh a penny (preferably a relatively new one, as the
mass of pennies changed in 1983), a nickel and finally a quarter.
Weight of a penny: ________
Weight of a nickel: ________
Weight of a quarter: ________
Lastly, measure the mass of a 1/4 teaspoon of common table salt. Put a weighing paper
on the centigram balance and zero the balance. Measure out one level 1/4-tsp. of table
salt and pour it onto the weighing paper. Read the mass of the salt and round it to one
decimal place: _______________
Use these measurements to help you remember the relative size of a gram (g).
In medicine and nutrition, quantities are very often expressed in units smaller than grams:
A mg is __________________ of a gram.
A µg is _______________ __ of a mg or __________________________ or a g.
Remember that the standard unit of mass is the kilogram. How many µg are there in a
kg? ___________. Express this number in scientific notation: ____________
Las Positas College
Livermore, CA
Adams, Zingg FS 01
12
Physiology 1, LPC
Math Review Exercise
1 cc = ________mL = _________µL
3050 ml = ________L
2.05 L = ________ mL
520 cc = ________L
Add: 5 L, 45 cc, 175 mL, 3.6 L, and 0.3 L = ________ cc
1 kg = ________mg = ________ µg (use scientific notation !)
300mg = ________ µg
250 mg = ________ g
0.8 mg = ________ µg
40,000 µg ________ mg
A medicine is available as 0.5 g / 2 cc and you must give 125 mg. You will give
________cc.
A medicine is available as 0.005 g per tablet and you must administer 10 mg. You will
give ________ tablet(s).
Write in scientific notation:
7951 = ___________
0.0000087 =___________
1 = ___________
The human body is made up of trillions of cells. For the sake of our calculation, let’s say
the number is 45 trillion cells. In scientific notation: ___________________
A RBC measures approximately 7.5 x 103 nm in diameter.
This is___________ µm or ___________ mm.
A person’s body weight is 72 kg. Using first the regular number then scientific notation,
this is the same as
_____________________ or____________ mg
______________________or_____________µg
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