In almost every specialty area within biology it is necessary to express
some things in **quantitative** ways: in numbers, fractions, percentages,
averages, ranges, tables, graphs, equations, formulae, and so on. And often
the points of interest involve comparisons: one thing is larger than another,
one group has fewer diseases than another or lives longer, something is
faster than something else or more dense or narrower or cooler and so on.
The ability to deal with quantitative aspects of biology is as important
as mastery of a working vocabulary and understanding of the scientific
method, among other things.

**1**. Quantitative expressions of comparison may appear in many
forms. For example, suppose that a person weighed 120 pounds in 1995 and
240 pounds in 1999. How might that change or difference be expressed?

a. Obviously, he gained 120 pounds.

b. You could say his weight doubled during the period of time or that he weighed twice as much in 1999 as in 1995. Both statements are correct and probably obvious to you.

c. It is also correct to say that his weight **increased** by 100%
(not 200%). If we take his 1995 weight as the point of reference and ask
about the percentage of **change**, then we have **(**(240 lb. -
120 lb.) ÷ 120 lb.**) **X 100 = 100%.

Note well that his weight did not **increase** by 200%. However,
it *is* correct to say that his 1999 weight is 200% of his 1995 weight.
That is, (240 lb. ÷ 120 lb.) X 100 = 200%. It is important to realize
that in both of these equations the lb. units cancel when you divide; so,
the percent answer has no units. Here the point of interest is his total
weight in 1999 relative to his total weight in 1995. In the previous comparison
the focus was on only the part of his weight that was gained during the
4 years. His 1995 weight (120 lb.) was 100% of his weight at that time.
So, if he **added** that much again, he added 100% of his starting weight;
that is, he **doubled** it. Looking at these two equations, you see
that the difference lies in whether you subtract the starting value from
the end value before you divide.

d. It is also correct to say that during the 4 year period his weight
increased **1-fold** (one-fold). The common "fold" expression means
"100%", and it is very often misused in expressing comparisons. If something
increases by 100% (that means it **doubles**), it increases one-fold.
An increase of 180% is an increase of 1.8-fold. An increase from 200 to
1000 is a 400% **increase**, which means a 4-fold increase. Here, again,
the focus is on the amount of change that has occurred relative to the
starting value. If you compare the end value (1000) to the starting value
(200), then it is correct to say that the end value is 5-fold of the starting
value. Similarly, if Bill's salary were $50,000/year and Sam's salary were
$12,500/year, then Bill's salary is 4-fold Sam's. And if Sam's salary rose
to $50,000, we would say that:

(i) his salary increased by 300%, (ii) his salary increased 3-fold, (iii) his salary quadrupled, (iv) his current salary is 4-fold his former salary, (v) his current salary is 400% of his former salary, (vi) his current salary is 4 times his former salary.

**2**. The expression "orders of magnitude" sometimes appears in
comparisons. One **order of magnitude** means one power of ten. So,
the numbers 100 and 10,000 differ by two orders of magnitude; or we can
say that 10,000 is two orders of magnitude greater than 100. Similarly,
a meter and a micrometer differ by six orders of magnitude:

**4**. Suppose that in a study of the squirrel population in a large
city park we found that the number of squirrels was 165 in 1987. If the
number fell by 40% in the next 10 years, how many squirrels were there
in 1997?

The solution can also be written this way:

**(**(171 sqrl. - 248 sqrl) ÷ 248 sqrl.**)** X 100 = (-77
sqrl. ÷ 248 sqrl.) X 100 = -31%.

Here the answer is a negative number, which means, of course, that the change is a negative one, i.e. a decrease. The minus sign has replaced the word "decrease" in the first solution.

Suppose that Marsha's blood cholesterol level is 180 mg/dL (that's milligrams per deciliter of blood) and that this is a 20% reduction from what it used to be before she altered her diet and exercise habits. What was the cholesterol value previously?

One approach: If 180 mg/dL is a 20% reduction from what it used to be (call the former value "x"), then 180 mg/dL is 80% of "x", that is:

225 mg/dL - (0.2 X 225 mg/dL) = 225 mg/dL - 45 mg/dL = 180 mg/dL. So, our solution must be right. Got that? Do the units come out right here?

**6**. In using percentage comparisons some people become confused
when they lose sight of what the reference point is. Here are two examples.

a. Bob managed to reduce his weight from 200 lb. to 150 lb. That's a 25% decrease. Later, though, he lost his self control and his weight went back up by 25%, which put him at 187.5 lb.

Some people would say that's incorrect. If he lost 25% and then gained
25%, his weight must be back at 200 lb, where he started. **Not so!**
The 25% loss was relative to the 200 lb. weight. However, the 25% gain
was relative to the 150 lb. weight.

b. Mary and Sarah both entered a physics competition that involved taking a difficult exam. Neither of them won an award. The next year both women entered the competition again, and each improved her score by 20% over the previous year. But only Sarah won an award, even though the minimum score for an award (75) was the same as the year before. How can that be? Some people will assume that the women's scores in the second year were the same because the degree of improvement was the same. Not necessarily! If the first scores were 60 (Mary) and 70 (Sarah), the equal percentage improvement would still leave Mary with a score (72), below the required minimum, while Sarah's score (84) was above that minimum value.

**7**. Here's one last example of expressing magnitude of change;
and in this sort of instance the meaning is not just wrong but silly. Suppose
you hear someone talking about his pulse (heartbeat rate) before, during,
and after strenuous exercise. He says that his pulse increases 100% during
maximum physical exertion and that after resting for a while it decreases
100%. A 100% decrease means that it has dropped to zero; a few minutes
of that and he's dead! Whatever the pulse was, 100% is all there is.