# Statistics

Statistics is the mathematical science of collecting, organizing, summarizing, and interpreting information in a numerical form. There are two main branches of statistics. Descriptive statistics summarizes particular data about a given situation through the collection, organization, and presentation of those data. Inferential statistics is used to test hypotheses, to make predictions, and to draw conclusions, often about larger groups than the one from which the data have been collected.

Statistics enables the discernment of patterns and trends, and causes and effects in a world that may otherwise seem made up of random events and phenomena. Much of the current debate over issues like global warming, the effects of industrial pollution , and conservation of endangered species rests on statistics. Arguments for and against the human impact on the environment can be made or broken based on the quality of statistics supporting the argument, the interpretation of those data, or both. Providing accurate statistics to help researchers and policy-makers come to the most sound environmental conclusions is the domain of statistical ecology in particular, and environmental science in general.

## Basic concepts

Statistical analysis begins with the collection of datathe values and measurements describing an event or phenomenon. Researchers collect two types of data: qualitative and quantitative. Qualitative data refers to information that cannot be ascribed a numerical value but that can be counted.

For example, suppose a city wants to gauge the success of its curbside recycling program. Researchers might survey residents to better understand their attitudes toward recycling. The survey might ask whether or not people actually recycle, or whether they feel the program is adequate. The results of that surveythe number of people in the city who recycle, the number who think the program could be improved, and so onwould provide quantitative data on the city's program. Quantitative data, then, refers to information that can be ascribed numeric valuesfor example, a study of how many tons each of glass and paper get recycled over a certain period of time. A more detailed breakdown might include data on the recycling habits of individual residents by measuring the amount of recycled materials people place in curbside bins.

Sorting through and weighing the contents of every recycling bin in a town would be not only time consuming but expensive. When researchers are unable to collect data on every member of a population or group under consideration, they collect data from a sample, or subset, of that population. In this case, the population would be everyone who recycles in the city.

To bring the scope of the study to a manageable size, the researchers might study the recycling habits of a particular neighborhood as a sample of all recyclers. A certain neighborhood's habits may not reflect the behavior of an entire city, however. To avoid this type of potential bias, researchers try to take random samples. That is, researchers collect data in such a way that each member of the population stands an equal chance of being selected for the sample. In this case, investigators might collect data on the contents of one bin from every block in the city.

Researchers must also define the particular characteristics of a population they intend to study. A measurement on a population that characterizes one of its features is called a parameter. A statistic is a characteristic of or fact about a sample. Researchers often use statistics from a sample to estimate the values of an entire population's parameters when it is impossible or impractical to collect data on every member of the population. The aim of random sampling is to help make that estimate as accurate as possible.

## Descriptive statistics

Once the data have been collected, they must be organized into a readily understandable form. Organizing data according to groups in a table is called a frequency distribution. Graphic representations of data include bar charts, histograms, frequency polygons (line graphs), and pie charts.

Measures of central tendency, or the average, provide an idea of what the "typical" value is for a group of data. There are three measures of central tendency: the mean, the median, and the mode. The mean, which is what most people refer to when they use the term "average," is derived by adding all the values in a data set and then dividing the resulting sum by the number of values. The median is the middle value in a data set when all the numbers are arranged in ascending or descending order. When there is an even number of values, the median is derived by calculating the mean of the two middle values. The mode is the value that most often occurs in a set of data.

Although averages describe a "typical" member of a group or set of data, it can also be helpful to know about the exceptions. Statisticians have therefore devised several measures of variabilitythe extent to which data fluctuate between different measures. The range of data set is the difference between the highest and lowest values. Deviation is the difference between any one measure and the mean.

Range and deviation provide information on the variability of the individual members of a group, but there are also ways to describe the variability of the group as a whole if, for example, a statistician wants to compare the variability of two sets of data. Variance is derived from squaring the deviations of a set of measures and then calculating the mean of those squares. Standard deviation is the most common statistic used to describe a data sets variability because it can be expressed in the same units as the original data. Standard deviation is derived by calculating the square root of the variance.

## Inferential statistics

Inferential statistics is largely concerned with predicting the probabilitythe likelihood (or not)of certain outcomes, and establishing relationships or links between different variables. Variables are the changing factors or measurements that can affect the outcome of a study or experiment.

Inferential statistics is particularly important in fields such as ecological and environmental studies. For example, there are chemicals contained in cigarette smoke that are considered to be carcinogenic. Researchers rely in no small part on the methods of inferential statistics to justify such a conclusion.

The process begins by establishing a statistical link. To use a common example, health experts begin noticing an elevated incidence of lung cancer among cigarette smokers. The experts may suspect that the cause of the cancer is a particular chemical (if there are 40 suspected chemical carcinogens in the smoke , each chemical must be evaluated separately) in the cigarette smoke. Thus, there is a suspected association, or possible relationship, between a chemical in the smoke and lung cancer.

The next step is to examine the type of correlation that exists, if any. Correlation is the statistical measure of association, that is, the extent or degree to which two or more variables (a potential chemical carcinogen in cigarette smoke and lung cancer, in this case) are related. If statistical evidence shows that lung cancer rates consistently rise among a group of smokers who smoke cigarettes containing the suspected chemical compared with a group of nonsmokers (who are similar in other ways, such as age and general health), then researchers may say that a correlation exists.

Correlation does not prove a cause and effect relationship, however. The reason, in part, is the possible presence of confoundersother variables that might cause or contribute to the observed effect. Therefore, before proposing a cause-effect relationship between a chemical in cigarette smoke and lung cancer, researchers would have to consider whether other contributing factors (confounders)such as diet, exposure to environmental toxins , stress, or genetics may have contributed to onset of lung cancer in the study population. For example, do some smokers also live in homes containing asbestos ? Are there high levels of naturally occurring carcinogens such as radon in the work or home environment?

Teasing out the many possible confounders in the real world can be extremely difficult, so although statistics based on such observations are useful in establishing correlation, researchers must find a way to limit confounders to better determine whether a cause-effect relationship exists. Much of the available information on environmentally related causes and effects is verified with data from lab experiments; in a lab setting, variables can be better controlled than in the field.

Statistically and scientifically, cause and effect can never be proved or disproved 100%. Researchers test hypotheses, or explanations for observed phenomena, with an approach that may at first appear backwards. They begin by positing a null hypothesis, which states that the effects of the experiment will be opposite of what is expected. For example, researchers testing a chemical (called, for example, chemical x) in cigarette smoke might start with a null hypothesis such as: "exposure to chemical x does not produce cancer in lab mice." If the results of the experiment disprove the null hypothesis, then researchers are justified in advancing an alternative hypothesis. To establish that there is an effect, an experiment of this nature would rely on comparing an experimental group (mice exposed to chemical x, in this case) with a control groupan unexposed group used as a standard for comparison.

The next step is to determine whether the results are statistically significant. Researchers establish a test or experiments P value, the likelihood that the observed results are due to chance. Frequently, the results of an experiment or test are deemed statistically significant if the P value is equal to or less than 0.05. A P value of 0.05 means there are five or fewer chances in 100 that the observed results were due to random processes or statistical variability. In other words, researchers are 95% sure they have documented a real cause and effect.

Other important considerations include whether the results can be confirmed and are reliable. Findings are considered confirmed if another person running the same test or experiment can produce the same results. Reliability means that the same results can be reproduced in similar studies.

[Darrin Gunkel ]

## RESOURCES

### BOOKS

Cohn, Victor. News and Numbers. Ames, IA: Iowa State University Press, 1989.

Graham, Alan. Statistics. Lincolnwood, IL: NTC/Contemporary Publishing, 1999.

Jaisingh, Lloyd R. Statistics for the Utterly Confused. New York: McGraw-Hill, 2000.

Slavin, Stephen. Chances Are: The Only Statistics Book You'll Ever Need. Lanham, MD: Madison Books, 1998.

### OTHER

Elementary Concepts in Statistics. 19842002 [cited July 6, 2002]. <http://www.statsoft.com/textbook/stathome.htm>.

Introduction to Statistics. 19972002 [cited July 6, 2002]. <http://writing.colostate.edu/references/research/stats/pop2a.cfm>.

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