What happened?
Summarize variation before explaining it.
Center, spread, shape, outliers and relationships turn raw observations into a visible empirical object.
Side 06
Statistics &
Probability
A study of uncertainty from both directions: probability starts with a model and asks what data could occur; statistics starts with data and asks what the underlying world may be. The point is not calculation alone. It is learning what evidence can and cannot justify.
variation→uncertainty→evidence→decision
05learning stages
06distribution families
06failure modes
06Side
From description to decision.
The dependency chain matters. Inference is fragile when variation, probability and sampling have not been understood first.
What could happen?
Assign structure to uncertain outcomes.
Events, conditional probability, independence and random variables create the grammar for uncertainty.
What did we get to observe?
Separate population from sample.
Sampling variation explains why repeated studies differ even when nothing fundamental has changed.
What does the sample support?
Estimate before declaring.
Intervals, likelihoods, tests and posterior distributions quantify uncertainty around claims.
What action follows?
Evidence does not choose the loss function.
Decisions require costs, benefits, thresholds and consequences in addition to statistical evidence.
data↔model↔uncertainty↔decision rule
Probability before statistics.
Probability is the forward problem: assume a model of the world, then reason about possible observations.
P(A)
A number from 0 to 1 representing uncertainty about an event under a specified model.
P(A | B)
The probability of A after restricting attention to cases where B is known to have occurred.
P(A | B) = P(A)
B supplies no information about A under the model. Independence is an assumption to examine, not a default.
E[X]
The probability-weighted long-run average of a random variable. It need not be a value that can actually occur.
Var(X)
A measure of spread around the expectation. Two processes can share a mean while carrying very different risk.
Prior → posterior
New evidence updates an existing probability through the likelihood of seeing that evidence under competing possibilities.
Base-rate machineSuppose 1% of 10,000 people have a condition. A test catches 90% of true cases and falsely flags 5% of others.
90 true positives + 495 false positives → about 15.4% of positive tests are true cases.
90 true positives + 495 false positives → about 15.4% of positive tests are true cases.
The neglected variable is often the base rate.
A highly sensitive test can still produce mostly false positives when the underlying condition is rare.
Recurring shapes of uncertainty.
Distributions are not decorative curves. Each one encodes assumptions about what can happen and how often.
| Family | Question it models | Structural clue | Typical use |
|---|---|---|---|
| Bernoulli | Did one binary event occur? | Two outcomes; one probability p. | Success/failure, yes/no events. |
| Binomial | How many successes in n trials? | Fixed number of comparable Bernoulli trials. | Counts of successes. |
| Normal | How does continuous variation cluster around a center? | Symmetric bell shape; fully described by mean and variance. | Measurement error and many aggregate phenomena. |
| Poisson | How many events arrive in a fixed interval? | Count data generated by a rate. | Arrivals, incidents, defects. |
| Exponential | How long until the next event? | Waiting time associated with a constant event rate. | Reliability and inter-arrival times. |
| Power law | What if rare large events matter disproportionately? | Heavy tail; scale lacks a single “typical” size. | Some network, wealth and event-size phenomena. |
A named distribution is useful only when its assumptions are plausible enough for the question being asked.
What the sample can support.
Inference is the reverse problem: data are observed; the generative process is partly unknown.
Frequentist frame
Parameters are fixed but unknown. Repeated hypothetical samples define the long-run behavior of estimators, confidence procedures and tests.
Estimate
Point + interval
Report a plausible range and its procedure, not merely a single best estimate.
Test
Data under a null model
A p-value describes how surprising the data or more extreme results would be if the null model were true. It is not the probability that the null is true.
Bayesian frame
Uncertainty about parameters is represented directly with probability distributions. Prior information is updated by the likelihood to obtain a posterior.
Prior
What was plausible before?
Make assumptions visible rather than pretending the analysis began without them.
Posterior
What is plausible now?
The result depends jointly on prior information, the model and observed data.
Statistical significance, practical importance and decision value are different questions.
An effect can be precisely estimated and trivial, or large enough to matter while still uncertain.
Association is the beginning.
Causal claims require a model of what would have happened under an alternative exposure or intervention, not merely a correlation.
A third variable moves both.
The observed association can partly or wholly reflect a common cause.
The sample is not neutral.
Who enters, remains in or responds to a study can create systematic distortion.
Conditioning can create association.
Restricting or controlling for a common effect can manufacture a relationship that was absent.
Extreme values drift inward.
When noisy measurements are selected for extremity, later measurements tend to be less extreme even without intervention.
More tests create more accidents.
Searching many outcomes, subgroups or models raises the chance of apparently notable results.
The variable may be a proxy.
Error, categorization and construct validity can limit what a measured number actually represents.
randomizationhelpsbalance confoundersbut notbad measurement or poor generalization
Claim clinic.
Open a claim and interrogate it before deciding whether the result deserves belief, action or another study.
“Users who adopted feature X retained 23% better.”
Ask: 23% relative or absolute? How were adopters selected? What was baseline retention? Was adoption randomized? Could more engaged users simply be more likely to adopt? What uncertainty surrounds the estimate?
“The treatment produced p = 0.03.”
Ask: What effect size was estimated? What is the interval? How many tests were run? Was the analysis pre-specified? Does the result survive plausible modeling choices? What decision would change if the p-value were 0.06?
“The model is 94% accurate.”
Ask: Compared with what baseline? On which population and time period? Is the class distribution imbalanced? Which errors matter most? Was the evaluation set genuinely held out?
“Average income increased after the policy.”
Ask: Compared with what counterfactual? Did composition change? Was there a broader economic trend? Which groups moved? Is the mean hiding distributional changes?
OpenIntro StatisticsDiez, Barr & Çetinkaya-Rundel · statistical foundations
Statistical RethinkingRichard McElreath · model-based Bayesian reasoning
The Book of WhyJudea Pearl & Dana Mackenzie · causal reasoning
Causal Inference: The MixtapeScott Cunningham · applied causal methods