Expressing Magnitude of Change and Degree of Difference

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:

1 meter = 100 meter = 106 micrometers. Zero and 6 are six "powers" apart, i.e. 6 orders of magnitude apart. 3. Suppose that a car's speed decreased from 100 mph to 75 mph. That is a decrease of 25%. We could also say that the speed decreased by 1/4 or that the speed decreased to 3/4 of the original speed.

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?

165 sqrl. - (0.4 X 165 sqrl.) = 165 sqrl. - 66 sqrl. = 99 squirrels If the number fell by 2/3 in that period, how many were there in 1997? 165 sqrl. - ((2/3) X 165 sqrl.) = 165 sqrl. - 110 sqrl. = 55 squirrels Suppose that in this squirrel study the number was 248 in 1990 and 171 in 2000. What is the percentage change? Obviously the change is a decrease; so, we write: ((248 sqrl. - 171 sqrl.) ÷ 248 sqrl.) X 100 = 31% decrease. Recall that % expressions don't have units. That means that the units in this problem, squirrels, canceled. Do you see that in the equation?

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.

5. You may see expressions of change or difference in other formats, e.g.:

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:

180 mg/dL = 0.8 x. Solving for x, x = 225 mg/dL. Another approach: Start with the former value (call it "x") and subtract from that 20% of itself to get the new value of 180 mg/dL. x - 0.2 x = 180 mg/dL; solving for x, x = 225 mg/dL. HINT: (a small point but a useful one sometimes). If this is the correct answer, you should be able to check it by working backwards to the given value. That is, if Marsha had a 225 mg/dL value and reduced that by 20%, what's her new value?

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.