BIO 211: Statistics and Data Analysis

lectures 1-7 (quiz 1), Lecture 9-11

Science

A way of understanding the natural world by testing explanations

marginal probability

probability based on a single variable

joint probability

probability of outcomes for two or more variables or processes

Disjoint / mutually exclusive

cannot happen at the same time

non-mutually exclusive events

can happen at the same time

sum rule for disjoint outcomes

if outcome A and B are disjoint, then Pr(A or B) = Pr(A) + Pr(B)

Probability of one event or another occurring when they are non-mutually exclusive

Pr(A or B) = Pr(A) + Pr(B) - Pr(A and B)

Do the sum of probabilities of two disjoint events always add up to 1?

Not necessarily, there may be more than 2 events in the sample space

independence

knowing the outcome of one, provides no useful information about the outcome of another

gambler’s fallacy

thinking that random processes compensate for what happened in the past

product rule for independent outcomes

Pr(A and B) = Pr(A) x Pr(B)

probability distribution

lists all possible events and the probabilities with which they occur, must total 1

complementary events

two mutually exclusive events who’s probabilities add up to 1

sample space of one coin flip vs two

• one : (H,T)

• two : (HH, HT, TH, TT)

Pr [event] =

number of times the event happened / total number of samples

The probability of an outcome is the proportion of times an outcome would occur if…

we observed the random process an infinite amount of times

sample space

the collection of all possible outcomes of a trial

random / stochastic process

a process that can produce different possible outcomes; we can use a probability model to describe how likely each outcome is

examples of random processes

• coin toss

• die rolls

probability

a number assigned to an event that represents its long-run relative frequency or likelihood, with 0 meaning impossible and 1 meaning certain

if you sample too many times…

the observed frequency converges to the probability

central limit theorem

Given a population of any distribution with mean u and variance o² , the sampling distribution of sample size n approaches a normal distribution with mean u and variance o² / n

central limit theorem conditions

• Sample observations must be independent

• if the population distribution is skewed, a larger sample size must be used

why use means?

to average out outliers

The standard error of the mean is synonymous with…

the standard deviation of the sampling distribution

Is the variability of the sampling distribution smaller, equal, or larger than the population variability, and why?

it is smaller because sample means vary less than individual observations

as sample size increases…

the distribution of means has lower standard deviation

variance of sampling distribution of mean =

variance of population/sample size

the sampling distribution is…

symmetric and centered at the population true mean

The mean of the sampling distribution of the mean is…

equal to the mean of the population

What does it mean that the distribution of the sample mean is well approximated by a normal model?

95% of the sample means are within two standard deviations of the sampling distribution

What are the important properties of the sampling distribution of the mean?

• The distribution of the sample mean is well approximated by a normal model

• The sampling distribution is symmetric and centered at the population true mean

• The standard deviation of the sampling distribution of the mean is the standard error of the mean

• Standard error decreases as sample size increases

larger sample sizes have…

lower sampling variability and higher precision

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, laws, hypotheses, and facts.

usual definition of Theory

merely a guess or hunch

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

Why are models useful?

they organize knowledge, explore 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 models

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 (5)

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

it is sensible to add, subtract, or take averages with these 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, numerical 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