In almost every specialty area within biology it is necessary to expresssome things in quantitative ways: in numbers, fractions, percentages,averages, ranges, tables, graphs, equations, formulae, and so on. And oftenthe points of interest involve comparisons: one thing is larger than another,one group has fewer diseases than another or lives longer, something isfaster than something else or more dense or narrower or cooler and so on.The ability to deal with quantitative aspects of biology is as importantas mastery of a working vocabulary and understanding of the scientificmethod, among other things.
1. Quantitative expressions of comparison may appear in manyforms. For example, suppose that a person weighed 120 pounds in 1995 and240 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 thathe weighed twice as much in 1999 as in 1995. Both statements are correctand 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 askabout 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 realizethat 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 totalweight in 1999 relative to his total weight in 1995. In the previous comparisonthe focus was on only the part of his weight that was gained during the4 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 seethat the difference lies in whether you subtract the starting value fromthe end value before you divide.
d. It is also correct to say that during the 4 year period his weightincreased 1-fold (one-fold). The common "fold" expression means"100%", and it is very often misused in expressing comparisons. If somethingincreases 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 to1000 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 thestarting 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 startingvalue. 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 roseto $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 formersalary, (v) his current salary is 400% of his former salary, (vi) his currentsalary is 4 times his former salary.
2. The expression "orders of magnitude" sometimes appears incomparisons. One order of magnitude means one power of ten. So,the numbers 100 and 10,000 differ by two orders of magnitude; or we cansay 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 largecity park we found that the number of squirrels was 165 in 1987. If thenumber fell by 40% in the next 10 years, how many squirrels were therein 1997?
The solution can also be written this way:
((171 sqrl. - 248 sqrl) ÷ 248 sqrl.) X 100 = (-77sqrl. ÷ 248 sqrl.) X 100 = -31%.
Here the answer is a negative number, which means, of course, that thechange is a negative one, i.e. a decrease. The minus sign has replacedthe word "decrease" in the first solution.
Suppose that Marsha's blood cholesterol level is 180 mg/dL (that's milligramsper deciliter of blood) and that this is a 20% reduction from what it usedto be before she altered her diet and exercise habits. What was the cholesterolvalue 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 confusedwhen 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 a25% decrease. Later, though, he lost his self control and his weight wentback up by 25%, which put him at 187.5 lb.
Some people would say that's incorrect. If he lost 25% and then gained25%, 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% gainwas relative to the 150 lb. weight.
b. Mary and Sarah both entered a physics competition that involved takinga difficult exam. Neither of them won an award. The next year both womenentered the competition again, and each improved her score by 20% overthe previous year. But only Sarah won an award, even though the minimumscore 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 werethe same because the degree of improvement was the same. Not necessarily!If the first scores were 60 (Mary) and 70 (Sarah), the equal percentageimprovement would still leave Mary with a score (72), below the requiredminimum, 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. Supposeyou hear someone talking about his pulse (heartbeat rate) before, during,and after strenuous exercise. He says that his pulse increases 100% duringmaximum physical exertion and that after resting for a while it decreases100%. A 100% decrease means that it has dropped to zero; a few minutesof that and he's dead! Whatever the pulse was, 100% is all there is.
