BIO 211: Statistics and Data Analysis

lectures 1-4 so far...

Science

A way of understanding the natural world by testing explanations against the natural world.

Fact (Scientific)

Confirmed observations

Hypothesis

A testable statement

Theory (Scientific)

A logical explanation of a natural phenomenon built from facts, tested hypotheses, and laws.

Law (Scientific)

A descriptive generalization

Scientific Hierarchy of Importance

Theories at the top, followed by laws, hypotheses, and lastly facts.

Layperson Perception of Theory

The common public misconception that a scientific theory is merely a guess, hunch, or something not to be taken seriously.

Steps of Scientific Method

Ask a question or address a problem → Research → Hypothesis → Experiment → Analysis → Conclusion

What are the critical elements of the realistic scientific method?

self-correcting, iterative, built on reasoned evidence, aware of limitations, built on feedback between models and data

Model

A simplified representation of the world that helps us describe, understand, predict, and test mechanisms in complex systems

Why are models useful?

they organize knowledge, explore the logical consequences of our assumptions, generate testable predictions, guide data collection and experimental design, and integrate across scales and systems

What are the 4 types of models in ecology?

Conceptual, analytics, simulation-based, statistical

Conceptual models

food webs, flow diagrams

Analytical models

Lotka-volterra, logistic growth

Simulation-based

Spatial models, agent-based models

Statistical

Regression, Bayesian hierarchical models

Statistics

Provides the theoretical framework for reasoning under uncertainty

Data Analysis

applies statistics principles to real datasets to extract insights and support inference

Descriptive statistics

Organizing and summarizing data

Inferential statistics

Using probability to determine how confident we can be that our conclusions are correct

Methods of showing frequency distributions

bar graphs and histograms

methods of showing associations between variables

grouped bar graphs, mosaic plots, box plots, scatterplots, dot plot

Area of a histogram

frequency distribution of a numeric variable

height of a histogram

frequency

Bin width

smaller to be specific, too big can show nothing

Outliers in scatterplots help..

identify distribution skews, data collection and entry errors, and provide insight

science is uncertain because

It is built on limited reasoning and limited data, explanations rely on indirect information, and evidence is not always neat, alternative explanations can exist

numerical variables

take numerical values and it is sensible to add, subtract, or take averages with those values

categorical variables

The responses themselves are categories; the possible values are called the variable’s “levels”

Nominal scale

categories with no natural order

Example of nominal scale values

land types, alleles, species

Ordinal scale

categories ranked in some natural order

Examples of ordinal scale values

letter grades, sequences, clothing sizes

Discrete variable

can only take on a numerical value with jumps

Examples of discrete variables

whole numbers, years

Continuous variable

can take on any number, even beyond the decimal point

examples of continuous variables

wind speed, body temperature, beak length

Interval scale

numbers with a meaningful interval but no quantitatively meaningful zero

examples of interval scale

temperature in Celsius, years

Ratio scale

numbers with a meaningful interval and a meaningful zero

examples of ratio scale

length in cm, age, temperature in kelvin

univariate

one set of data that must be described

example of univariate data

snout vent length of salamander

bivariate data

two sets of data that must be described and correlated to each other

example of bivariate data

snout vent length vs elevation

multivariate data

2 or more sets of data that can be a mix of categorical and numerical data

example of multivariate data

snout vent length vs elevation and diet and species

Histogram

• Shows numerical variables

• has a zero baseline

• must choose number of bins

• no spaces between bins

• fixed order

Bar Plot

• Shows categorical data

• no order

• zero baseline

• choose number of bins

• spaces between bins

• height = frequency of 2 or more categories

Pie chart

1 nominal variable shown, not used much

mosaic plot

lose some information

Scatter plot x-axis

independent/explanatory variable

Scatter plot y-axis

dependent/response variable

Mean

the point at which a distribution would balance

median

the value that minimizes the sum of absolute deviations; splits the data in half when ordered in ascending order

What is special about the mean?

it is sensitive to extreme values

Unimodal distribution

has one peak

bimodal distribution

has two peaks

multimodal distribution

has multiple peaks

uniform distribution

has no peaks, plateau

Positive/right skew

long tail on right side

Negative/left skew

long tail on left side

Severe skew

All measures are different, so no measures of central tendency will be good to use, but median is more useful

What measure to use for symmetrical distributions?

the mean because its more precise

In a bell curve what is special?

Mean = Mode = Median

What is not a measure of central tendency?

interquartile range

Variability

how spread out a group of measurements is

what are the measures of variability?

range, interquartile range, variance, and standard deviation

Crude range

highest value minus lowest value

Variance

average squared difference of each score from the mean

Actual range

Crude range - 1

Variance in a population of size N

Sigma² = Sum (Mean-mew)² / N

Variance estimated from sample size n

s² = sum (Mean-X|)² / (n-1)

Why is s² better that sigma² ?

it is unbiased and accounts for samples being smaller than the population

standard deviation

the square root of the variance

population standard deviation

sigma

sample standard deviation

s

What happens to sample variance when you increase sample size n?

it converges toward population variance

percentage of the population within 1 standard deviation

68.3%

Percentage of the population within 2 standard deviations?

95.5%

Percentage of the population within 3 standard deviations?

99.7%