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1.4: An Overview of Essential Mathematics Used in Science

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    Math, Statistics, and Science

    Mathematics is the abstract science of numbers, quantity and space. Mathematics may be studied in its own right (pure mathematics), or as it is applied to scientific disciplines such as (applied mathematics).

    Statistics is a mathematical body of science that pertains to the collection, analysis, interpretation or explanation, and presentation of data, or as a branch of mathematics.

    Mathematics and statistics are vital for scientific inquiry. It is an especially imperative tool for disciplines that are inherently variable and require an extensive collection of data. Ideally, scientists would like to collect data from every individual in the population of interest, but this is rarely possible. As a result, scientists often must use data collected from a representative sample of individuals to draw inferences about basic biological phenomena for a population. This is where mathematics and statistics step into the party.

    Math and Ecology

    Quantitative ecology is the application of advanced mathematical and statistical tools to any number of problems in the field of ecology. It is a small but growing subfield in ecology, reflecting the demand among practicing ecologists to interpret ever larger and more complex data sets using quantitative reasoning. Quantitative ecologists might apply some combination of deterministic or stochastic mathematical models to theoretical questions or they might use sophisticated methods in applied statistics for experimental design and hypothesis testing. Typical problems in quantitative ecology include estimating the dynamics and status of wild populations, modeling the impacts of anthropogenic or climatic change on ecological communities, and predicting the spread of invasive species or disease outbreaks.

    Quantitative ecology, which mainly focuses on statistical and computational methods for addressing applied problems, is distinct from theoretical ecology which tends to explore focus on understanding the dynamics of simple mechanistic models and their implications for a general set of biological systems using mathematical arguments.

    Why Statistics?

    Some of the most important functions of statistics include:

    • Description and summary of basic findings (Descriptive statistics).
    • Testing hypotheses regarding relationships between variables so the cause and effect can begin to be understood (inferential statistics).
    • Presenting findings in an easily understood manner – misunderstandings keep science from moving progressing!

    Essential Mathematical Terms and Concepts

    General terms:

    • Data – Systematically recorded information.
    • Value – Each measurement or observation
    • Variable – The object being controlled, manipulated, measured or observed. There are two main types:
      • Independent (explanatory) – The variable that you think will affect what is being measured/observed.
      • Dependent (response) – The variable that is being measured.
    • Population – Entire set of objects to be studied.
      • Parameter – Numerical characteristic of population.
    • Sample – Sub-collection of objects from population.
      • Statistic – Numerical characteristic of sample from population.
    Example \(\PageIndex{1}\)

    You are a biologist who studies how monarch butterfly (Danaus plexippus) populations are affected by habitat destruction. You set up a long-term study to monitor populations in degraded, intact, and restored habitats where monarchs historically have been recorded/observed.

    • Population: All monarch butterflies.
    • Parameter: Not possible to collect a population worth of data for monarchs. Thus, no parameters can be calculated.
    • Sample: Total monarchs observed at each field site during each year of the study.
    • Statistic: Any calculations/manipulations from the field site data.

    Types of statistics:

    • Descriptive statistics – Are calculations to summarize trends in the data. Minimally, measures of center (averages) and spread (standard deviations) from data recorded.
    • Inferential statistics – The point of inferential statistics is to take data from the sample to make inferences about the population. Calculations here test hypotheses and try to find/infer cause and effect relationships and/or correlations.

    It is important to note that statistics can only be helpful if the data from the sample is representative of the population and the interpretation of the data is unbiased!

    Types of data:

    • Qualitative (categorical) data – Data expressed not in terms of numbers, but rather by means of a natural language description. There are two main types of qualitative data:
      • Ordinal – When categories are in a particular order (ex: large, medium, small)
      • Nominal – When categories have no natural ordering (ex: dog breed, color)
      • Graph types used: Pie, bar
    • Quantitative (numerical) data – Data expressed not by means of a natural language description, but rather in terms of numbers. There are two main types of quantitative data:
      • Continuous – Numbers where any integer or fraction can be observed (ex: time, height, or weight)
      • Discrete – A fixed number of outcomes is possible such that there are only whole integers possible (ex: counts)
      • Graph types used: Histograms, line-graphs, scatterplots

     

    Attribution

    Rachel Schleiger (CC-BY-NC) and Quantitative ecology From Wikipedia, the free encyclopedia

    This page titled 1.4: An Overview of Essential Mathematics Used in Science is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Melissa Ha and Rachel Schleiger (ASCCC Open Educational Resources Initiative) .