1.5.1: Essential mathematical and statistical terms and concepts
- Page ID
- 75928
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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
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.
- 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)
Graphs 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)
Graphs types used: Histograms, line-graphs, scatterplots
