Statistics Fundamentals

• Statistics deals with error measurement, acknowledging the impossibility of perfect census outside small universes.
• Probability (P) is calculated as successes divided by total outcomes (x/n).
• Q represents the probability of an event not occurring, equaling 1-P.
• Estimating a parameter (true universal value) involves subsampling, which is unknowable in large universes, requiring a perfect census.
• A distribution of possibilities has higher probabilities closer to the truth.
• Error arises from small samples and inaccurate data collection, influencing the distribution's shape.
• Models are used to predict the future and explain variation within data; useless models indicate equal chances of all or no subjects expressing a trait.
• Parameters include µ (location parameter/mean) and Pearson product moment correlation (r) indicating how things rise and fall together.
• B is the standardized coefficient of linear models (y=mx+b).
• The Law of Large Numbers states small samples deviate from the truth, while larger censuses approach mu, requiring random data.
• Conducting a good census involves either low n with numerous experiments or high n with one experiment.
• Data organization involves creating a tidy table where each variable forms a column, measurements require a column, observations form a row (sample size), and each observational unit forms a table.

Types of Data

• Numeric data includes continuous (any value on a number line) and count (integers between 0 and infinity).
• Nonnumeric data includes ordinal (categorical with rank), binomial (two outcomes), and categorical (unordered with 2+ outcomes).

Visualizing Data

• Both axes should have units, and correct plot geometry tells the story.
• There are various plots
• Scatter
• Box
• Bar

Central Tendency

• Median is the middle value, suited for data not from a bell curve.
• Mode is the greatest frequency in a dataset, measuring current observations without predictions.
• Mean is the average, predicting the future, requiring a normal distribution to avoid equal probability for each value and ensure the mean's likelihood.

Normal Distribution

• Characterized by symmetry around the mean, median, and mode, but has issues when making predictions because there's an equal chance of ending up on either side of the parameter and outliers
• It can be changed through transformations and estimates based on average distance from from the truth
• Central Limit Theorem states mean estimates are normally distributed, with the mean of the means being mu.
• Shape of Distribution includes width (estimate quality),variance, and SD.
• Variance indicates a point's average distance from the mean.
• SD, the squared variance, explains variation shape, with 0 indicating the shape parameter.
• 1 SD covers 68.3% of points, 2 SD covers 95%, and 3 SD covers 99.7%.
• Degrees of Freedom are the maximum amount of data that can vary
• SE is the mean of how far something is from the truth, and is influenced by n, and is defined as: SE = σ/√n.
• Reducing SE increases claim confidence, measuring precision of and variation from mu and the reliability of the mean.
• Confidence Interval indicates distribution extremes still accepted by the null, often 1.96 times the SD for a 95% confidence interval, with wider intervals indicating more uncertainty.
• Repeating sampling means 95% of confidence intervals would contain mu.

Hypothesis Steps

• An initial observation identifies the study area.
• A single statement research question adds to inductive models.
• A statistical hypothesis deals with the data's shape/pattern, using paired statements (H0 and HA) to examine the null's likelihood.
• A biological hypothesis answers the research question with a proposed mechanism.
• The Null Hypothesis states no pattern/difference in data (equality), with an alternate directional statement.

Normality Tests

• KS test involves measuring distance between sample and normal data, with D (distance equaling zero indicating normal data, though weak to repeating values and not specific to normal distribution.
• Shaprio Wilks test is weak to data w/ repeating values and extremely weak to small sample sizes
• D'agostino's K² test is specific to normal distribution and is powerful.
• The Eyeball test is efficient but subjective
• Skew (deviations from normality): how far the mean is from the middle
• P-value: probability of sample data drawn from distribution given D=0
• The goal for types of experiments it to eliminate bias, reduce error, use appropriate experimental units, and balance groups

Types of Experiments

• Natural experiments are observations without manipulation, either static/snapshot longitudinal/trajectory
• Manipulative experiments involve controlling/modifying a study factor, either static or longitudinal, with in vivo or in vitro settings.
• Error is defined as random chance plus bias, and eliminating bias increases the variance explained by the model.

Eliminating Bias

• Simultaneous control groups, randomly assigned to treatment and control.

Minimizing Random Error

• Replication which decreases SE
• Sample Size measures true random variables in the study
• Common Experimental Designs
• Block: Block Sampling sites
• Nested: Multiple treatments happen in each experimental unit
• Paired: each experimental unit contains one of each treatment
• t-scores compares means, indicating how many SEs out the null is
• P-value: the null will give the distribution
• Types of error
• Type 1
• Type 2