Even in the valley of the shadow of death, two and two do not make six.
Leo Tolstoy
Through and through the world is infested with quantity: to talk sense is to talk quantities. It is no use saying the nation is large-how large? It is no use saying that radium is scarce-how scarce? You cannot evade quantity.
Alfred North Whitehead
Mathematics is both the door and the key to the sciences.
Roger Bacon
Numerical precision is the very soul of science.
Sir D'Arcy Thompson
All science as it grows toward perfection becomes mathematical in its ideas.
Alfred North Whitehead
Botanists try to remove the mystery from the world by using experiments and effective writing to describe and understand plants. Precise language is critical to these descriptions. Since the most precise language is math, it is easy to understand why mathematics is becoming increasingly important in all sciences. For example, consider Gregor Mendel, a botanist and monk who made the most lasting impression on what would become the science of genetics. Mendel's crosses with garden peas were not original; other botanists had already made and studied those crosses for many years. Mendel's genius involved using mathematics to analyze his data. Counting and comparing the different types of offspring enabled Mendel to make a monumental discovery.
Botanists must know how to write effectively about numbers. This requires that botanists understand scientific systems of measurement (i.e., the metric system) as well as how to use statistics to understand populations.
Metric units commonly used in botany include:
| meter (m) | The basic unit of length | |
| liter (L) | The basic unit of volume | |
| gram (g) | The basic unit of mass | |
| degree Celsius (C) | The basic unit of temperature |
| Prefix | Metric Symbol | Multiple of Metric Unit | ||
|---|---|---|---|---|
| deci | d | 10-1 = 0.1 | ||
| centi | c | 10-2 = 0.01 | ||
| milli | m | 10-3 = 0.001 | ||
| micro | µ | 10-6 = 0.000001 | ||
| nano | n | 10-9 = 0.000000001 | ||
| pico | p | 10-12 = 0.000000000001 | ||
| deka | da | 101 = 10 | ||
| hecto | h | 102 = 100 | ||
| kilo | k | 103 = 1000 | ||
| mega | M | 106 = 1000000 | ||
| giga | G | 109 = 1000000000 | ||
| tera | T | 1012 = 1000000000000 |
Thus, multiply by:
| 0.01 | to convert cg to g | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.1 | to convert dm to m | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1000 | to convert kg to g | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.000001 | to convert to 1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0.1 | to convert mm to cm |
| 1 m = 39.4 inches (in) = 1.1 yards (yd) | 1 in = 2.54 cm |
| 1 km = 1000 m = 103 m = 0.62 miles | 1 ft = 30.5 cm |
| 1 cm = 0.01 m = 10-2 m = 0.39 in = 10 mm | 1 yd = 0.91 m |
| 1 nm = 10-9 m = 10-6 mm | 1 mi = 1.61 km |
| 470 m = 0.470 km |
| 1 L = 1000 cm3 = 1000 mL | 1 tsp = 15 mL | |
| 1 L = 2.1 pints = 1.06 qt = 0.26 gal = 1 dm3 | 1 cup = 0.24 L | |
| 1 mL = 0.035 fl oz |
For example,
The following is not meant to be a comprehensive description of statistical techniques. Rather, I present only the basic measures you'll need to include in most lab reports. More comprehensive descriptions of statistical techniques are presented elsewhere (Glantz, 1992).
Let's start our discussion with the mean and median.
To determine the median, first arrange the measurements in numerical order. Our sample would look like this: 9 cm, 40 cm, 45 cm, 51 cm, 61 cm, 63 cm, 64 cm, 64 cm, 65 cm, 67 cm, 69 cm, 69 cm, 73 cm, 80 cm. The median is between the seventh and eighth measurement; that is, the median is 64 cm. Note that in this example, the mean differs from the median. What causes this difference? How would the mean change if the shortest plant were not in the sample?
The mean of two samples could be the same, yet the samples could differ significantly. For example, consider these samples:
| Group 1: 25 cm, 35 cm, 28 cm, 32 cm | Mean = 30 cm | |
| Group 2: 10 cm, 15 cm, 20 cm, 75 cm | Mean = 30 cm |
Thus, reporting only the mean does not give readers a good description of the sample. To better describe the sample, you must provide some measure of the spread and variability of the data. You can do that by reporting the range and standard deviation.
The range is the difference between the largest and smallest values of the sample. For example, the range of Group 1 is 10 (35-25), whereas that of Group 2 is 65 (75-10). Although these measures are informative, they ignore all other data between the extremes.
The standard deviation measures how much each value differs from the mean. It's easy to calculate: to start, calculate the mean, measure the deviation of each sample from the mean, square each deviation, and then sum the deviations. For example, consider a group of trees that are 22 years, 19 years, 21 years, and 18 years old. The mean age is 20 years.
| Individual | Mean | Deviation | (Deviation)2 | |||
|---|---|---|---|---|---|---|
| 22 | 20 | 2 | 4 | |||
| 19 | 20 | -1 | 1 | |||
| 21 | 20 | 1 | 1 | |||
| 18 | 20 | -2 | 4 |
The sum of the squared deviations is 4 + 1 + 1 + 4 = 10. Divide this number by the number of observations minus one (i.e., 4 - 1 = 3). This produces a value of 10/3 = 3.3 years2, which is the variance (note that the units are years squared). The standard deviation is the square root of the variance: 1.8 years. The standard deviation is usually reported with the mean; for example, "The mean age of the trees was 20 ± 1.8 years."
The standard deviation (SD) is important for understanding the spread of a sample. For many distributions of values for a variable, the mean ± 1 SD encompasses 68% of the observations, whereas the mean ± 2 SD encompasses 95% of the observations. These distributions are the basis for determining what values of some variable are considered normal; those that are not considered normal (in terms of these distributions) may be biologically significant and worthy of more study.
Your future studies in botany will require that you learn more about using statistics to understand plants. For example, you'll need to understand how to do a t test, a chi-square analysis, and an analysis of variance (ANOVA). The specific test that you'll use will depend on the amount and type of your data and the nature of your hypothesis.
| Year | Population (millions) |
|
|---|---|---|
| 8000 B.C. | 5 | |
| 400 B.C. | 86 | |
| 1 A.D. | 133 | |
| 1650 | 545 | |
| 1750 | 728 | |
| 1800 | 906 | |
| 1850 | 1130 | |
| 1900 | 1610 | |
| 1950 | 2400 | |
| 1960 | 2998 | |
| 1970 | 3659 | |
| 1980 | 4551 | |
| 1990 | 5300 | |
| 2000 | 6500+ (projected) |
What would be the best way to present these data to an audience of your peers? Write a short essay describing the importance of these data. Do not repeat the data; rather, discuss what they mean.
4. Re-examine the essay you produced at the end of Chapter 4. If you've not done so already, add some quantitative data to support your ideas. In the space below, describe the source of the data, the importance of the data, and how you would best present those data.
Source:
Importance:
Presentation: