Allman E.S., Rhodes J.A. Mathematical Models in Biology: An Introduction

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356 Basic Analysis of Numerical Data

half (half being above the median and half being below the median), the ﬁrst

quartile point is the value that has one quarter of the data points below it and

three quarters above. The second quartile point is just the median, and the third

quartile point has three quarters of the data values below it, with one quarter

above. The interquartile range is the interval from the ﬁrst quartile point to

the third quartile point. It always contains the median and 50% of the data.

Finding all of these is easily done from an ordered list of data.



For the 20 bean heights of Section 2, ﬁnd the interquartile range.



From looking at a histogram or distribution, how can you judge where

the ﬁrst and third quartiles lie? Estimate them for all the distributions

in this appendix.

Of course there is nothing special about quartiles. Sometimes quintiles

(ﬁfths), or deciles (tenths), or even percentiles (hundredths), are used. Speci-

fying the full range of the data, by giving the smallest and largest values, also

gives the reader a better understanding of the variability of the data.

If the mean is chosen as the way of specifying the data’s central tendency,

then it is usual to also report the standard deviation as the measure of data

spread. To develop the idea of the standard deviation, consider an example.

Suppose the height (in centimeters) of ﬁve bean plants is our data:

12.1, 16.3, 13.2, 13.5, 14.9.

The mean of this data is µ = 14.0. Notice that two of the data values are

larger than the mean, and three are smaller, as is reasonable.

A ﬁrst approach to understanding the spread of the data would be to see

how far each data point is from the mean. Hence, we calculate

12.1 − 14.0 =−1.9

16.3 − 14.0 = 2.3

13.2 − 14.0 =−0.8

13.5 − 14.0 =−0.5

14.9 − 14.0 = 0.9.

We get negative values when the data point is smaller than the mean and

positive values when it is larger than the mean.

A seemingly good idea would be to average these differences from the

mean that we have just calculated. So, we should add them up and divide

A.4. The Spread of Data 357

by 5:

(−1.9 + 2.3 − 0.8 − 0.5 + 0.9)

= 0.

Unfortunately, zero is not a good measure of spread. In fact, this calculation

of average differences will always give zero, as a little algebra can show. The

crux of the matter is that some of the differences will be positive and some

will be negative, and adding them up always results in cancellation.

The next natural idea would be to just make all these differences from the

mean positive before averaging them (i.e., average their absolute values). If

we do this we get:

(1.9 + 2.3 + 0.8 + 0.5 + 0.9)

6.4

= 1.28.

This looks a bit better, and means that, on average, our data points differ from

the mean of 14.0 cm by 1.28 cm. This quantity 1.28 is referred to as the mean

deviation and is a reasonable measure of data spread. To summarize for a set

of n data points,

mean deviation =



i=1

− µ|.

Another way of handling the problem of cancellations is to square the

differences and then average the squares:



(−1.9)

+ (2.3)

+ (−0.8)

+ (−0.5)

+ (0.9)



10.6

= 2.12.

This quantity is called the variance or mean square deviation of the data.

Note that, since our data was in centimeters (cm), the calculation produced a

quantity whose units should be squared centimeters (cm

). To have the same

units as our original data, we take the square root and get

√

2.12 ≈ 1.46.

This last quantity is called the standard deviation of the data.

To summarize for a set of n data points, the standard deviation, denoted

usually by σ ,is

σ =









i=1

− µ)

The variance, σ

, is calculated by just leaving the square root out of this

formula.

358 Basic Analysis of Numerical Data

µ µ+σµ−σ

Figure A.6. Normal distribution, with mean µ and standard deviation σ .



Calculate the standard deviation for the 20 bean heights of Section 2.

The standard deviation is the most commonly used measure of data spread,

though the reasons for this require more statistical theory than can be explained

here. When the mean of data is reported, the standard deviation usually should

be as well.

For the theoretical model of data given by the normal distribution, there is

a good graphical interpretation of the standard deviation, as shown in Figure

A.6. It is the horizontal distance from the peak of the normal curve (the

mean) to the inﬂection points (the points where the curve changes from being

concave up to concave down or vice versa). The larger the standard deviation,

the wider the bell curve.

It can be shown that, for a normal distribution, approximately 34% of the

area under the graph is between the mean and the inﬂection point to the left of

the mean. Because the graph is symmetrical, this holds for the area between the

mean and the inﬂection point to the right as well. Because area corresponds to

the number of data points, that means that if your data is normally distributed,

about 68% of your data should be within one standard deviation of the mean.

Similarly, about 95% will be within two standard deviations from the mean,

and more than 99% within three standard deviations.

A more precise deﬁnition of the normal distribution can now be given. The

normal distribution with mean µ and standard deviation σ , where µ is any

number and σ>0, is

f (x) =

√

2π

−

(x −µ)

2σ

Although other curves appear to be bell-shaped, only the particular curves

given by this formula are called normal.

A.5. Populations and Samples 359

A.5. Populations and Samples

There is one last point to be made concerning these basic statistical concepts.

In the discussion so far, the focus has been on having some data (a list of

numbers) and ﬁnding ways of describing that data. But we might want to

make conclusions that go beyond the particular data we have collected.

Again, thinking of bean heights as our data set, we can adopt a slightly

more sophisticated viewpoint. While we only have the heights of 20 beans

recorded, we can certainly imagine performing our experiment on all beans in

the world. We will consider all beans as the population we are trying to study,

and the 20 beans we actually experimented with as being a sample from that

population. Although a histogram is a good way of graphically treating the

data from our sample, the distribution curve is what describes the population

as a whole. Of course, we cannot know exactly what the distribution curve

for the population really is without experimenting on every bean, but we can

make well-informed guesses based on histograms for data sets involving some

reasonably large number of beans.

With this viewpoint, there is a change in what we would like to get from

our bean data. Although we understand the mean and standard deviation of

the data, our real interest is the mean and standard deviation of the entire

population. But, without data on the entire population, we can’t ﬁnd these

exactly. We will, however, be able to estimate them. Let

µ = mean of population

σ = standard deviation of population

be the two quantities we would like to estimate.

Not surprisingly, the best estimate you can give for the mean µ of the entire

population is simply the mean of the sample. More formally, the mean of the

sample is

x =

(

+ x

+···+x

)



i=1

and

µ ≈

Although the standard deviation of the sample is not a bad estimate of

the standard deviation of the population, there is a better one. Statistics texts

prove that if

s =







n − 1



i=1

− x)

360 Basic Analysis of Numerical Data

then

σ ≈ s

and that s is the best estimator σ .



Calculate the estimate for the population standard deviation for all beans

based on the 20 bean heights of Section 2 and compare it to the standard

deviation of that data set that you calculated before.



Compare this formula to the formula for standard deviation of a data

set and ﬁnd all the differences. What effect will these differences have

on the value you would obtain?

There are, of course, two differences between the formulas for s and σ .

First, rather than use µ, which we do not know exactly, we use

x, which

estimates it. Then, after adding up the squared deviations from the data mean,

we divide by one less than the number of data points, rather than by the number

of data points. Since we divide by a smaller number, we end up with a bigger

number. Taking the square root afterward still leaves us with a bigger number.

Thus, the estimate for the population standard deviation will be inﬂated a bit

from the data’s standard deviation.

The informal reason why this inﬂation is desirable is subtle: In the formula

for s, we use not only our data points, but also the mean

x of the data as an

estimate of the unknown mean µ of the population. The data points are likely

to be clustered more closely around their own mean

x than they are around

the population’s mean µ. Thus if we do not modify the formula, we would get

a standard deviation that was smaller than σ . Inﬂating the standard deviation

slightly gives a better estimate. The full argument why replacing n by n − 1

is precisely the right thing to do can be found in statistics books.

If we have a large sample, so that n is big, then n and n − 1 are really

about the same size, so whichever one we use should not matter too much.

This is reasonable, since if the sample size is large, then the sample mean

is likely to be quite close to the population mean µ, so little adjustment is

necessary.

A.6. Practice

Returning to beans for one last time, suppose we grow 10 beans under a set

of conditions we will refer to as condition A, and 10 beans under a different

set of conditions which we will refer to as condition B. We count the number

A.6. Practice 361

of leaves on each plant and get the following data:

Condition A: 4, 6, 5, 6, 8, 4, 6, 5, 10, 5

Condition B: 7, 5, 9, 6, 10, 8, 9, 7, 8, 7

Analyze this data using all the concepts in this appendix (histograms and

distributions, mean, median, mode, interquartile range, standard deviation,

sample vs. population).

Appendix B

For Further Reading

For further study, there are many textbooks focusing on mathematical models

in biology. They generally assume a solid knowledge of calculus and some

differential equations and linear algebra, though sections may be read by those

with less mathematical background. Among the books covering a variety of

biological topics are:

Leah Edelstein-Keshet. Mathematical Models in Biology. McGraw-

Hill, New York, 1988.

Frank C. Hoppenstaedt and Charles S. Peskin. Modeling and Simulation

in Medicine and the Life Sciences. Springer, New York, second edition,

2002.

J. Mazumdar. An Introduction to Mathematical Physiology and Biology.

Cambridge University Press, Cambridge, second edition, 1999.

James D. Murray. Mathematical Biology I: An Introduction and Math-

ematical Biology II: Spatial Models and Biomedical Applications.

Springer, New York, third edition, 2002.

Clifford Taubes. Modeling Differential Equations in Biology. Prentice

Hall, Upper Saddle River, NJ, 2001.

S. I. Rubinow. Introduction to Mathematical Biology. John Wiley, New

York, 1975.

E. Yeargers, R. Shonkwiler, and J. Herod. An Introduction to the Mathe-

matics of Biology: With Computer Algebra Models. Birkhauser, Boston,

1996.

For linear models, including ones using differential equations, recommended

books are:

Hal Caswell. Matrix Population Models: Construction, Analysis, and

Interpretation. Sinauer Associates, Sunderland, MA, 1989.

Michael R. Cullen. Linear Models in Biology. Ellis Horwood, Chich-

ester, England, 1985.

362

Appendix B 363

In addition to sections of the books above, infectious disease models have

been the focus of a number of texts and survey papers:

L. J. S. Allen. Some discrete-time SI, SIR, and SIS epidemic models.

Math. Biosci., 124:83–105, 1994.

Roy M. Anderson and Robert M. May. Infectious Diseases of Humans:

Dynamics and Control. Oxford University Press, Oxford, England,

1992.

Fred Brauer and Carlos Castillo-Chavez. Mathematical Models in Pop-

ulation Biology and Epidemiology. Springer, New York, 2001.

Herbert W. Hethcote. The mathematics of infectious diseases. SIAM

Rev., 42(4):599–653, 2000 (electronic).

Material on molecular evolution and phylogenetic tree construction has not

yet appeared in other texts at this level. Several good surveys exist, directed

at researchers and advanced students, as does low-cost or free software:

W.-H. Li. Molecular Evolution. Sinauer Associates, Sunderland, MA,

1997.

J. Felsenstein. PHYLIP (Phylogeny Inference Package), Version 3.5c.

Department of Genetics, University of Washington, 1993.

D. L. Swofford. PAUP* (Phylogenetic Analysis Using Parsimony *and

Other Methods), Version 4. Sinauer Associates, Sunderland, MA, 2002.

David L. Swofford, Gary J. Olsen, Peter J. Waddell, and David M. Hillis.

Phylogenetic Inference,inMolecular Systematics. Sinauer Associates,

Sunderland, MA, second edition, 1996.

More on the classical genetics topics can be found in:

J. F. Crow and M. Kimura. An Introduction to Population Genetics

Theory. Harper and Row, New York, 1970.

Electronic Scholarly Publishing. Foundations of Classical Genetics,a

collection of important papers in the development of classical genetics

[http://www.esp.org].

Daniel L. Hartl and Andrew G. Clark. Principles of Population Genetics.

Sinauer Associates, Sunderland, MA, second edition, 1989.

Books providing a solid background on some of the mathematical and statis-

tical topics introduced here include:

David C. Lay. Linear Algebra and Its Applications. Addison-Welsey,

Boston, third edition, 2002.

364 For Further Reading

Marcello Pagano and Kimberlee Gauvreau. Principles of Biostatistics.

Duxbury, Paciﬁc Grove, CA, second edition, 2000.

Sheldon Ross. A First Course in Probability. Prentice Hall, Upper Sad-

dle River, NJ, ﬁfth edition, 1997.

Gilbert Strang. Introduction to Linear Algebra. Wellesley-Cambridge

Press, Wellesley, MA, second edition, 1993.

Dennis D. Wackerly, William Mendenhall III, and Richard L. Scheaffer.

Mathematical Statistics with Applications. Duxbury, Paciﬁc Grove, CA,

sixth edition, 2002.

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