Statistics Cheat Sheet - Probability and Descriptive/Inferential Statistics

Oct 13, 2026

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Statistics Cheat Sheet covering core probability axioms and key properties, linking descriptive methods to inferential reasoning. Defines population, sample, discrete vs continuous random variables, sample mean/median, trimmed mean, variability measures (range, sample variance, standard deviation), and box plot construction. Includes probability foundations: sample spaces and events, probability axioms, addition and inclusion–exclusion formulas, product rule, conditional probability, Bayes’ theorem, and independence. Also summarizes common distributions, pmf conditions, cdf interpretation, and expected value as a weighted average.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages6

Reading Time2 min

AccessFree to download

Axioms & Properties of Probability

Overview & Descriptive Stats

Probability and Conditional Probability

Discrete and Continuous Variables

Measures of Location

Measures of Variability

Box Plots

Probability

Sample Space and Events

Counting Techniques

Independence

Random Variables and Distributions

CDF and Expected Value

Statistics Cheat Sheet Axioms & Properties of Probability: 1. 8A 2 S ; 0 􀀔 P (A) 􀀔 1 Ch 1: Overview & Descriptive Stats Populations, Samples and Processes Population: well-de􀀌ned collection of objects Sample: a subset of the population Descriptive Stats: summarize & describe features of data Inferential Stats: generalizing from sample to population Probability: bridge btwn descriptive & inferential techniques. In probability, properties of the population are assumed known & questions regarding a sample taken from the population are posed and answered. 2. P (S) = 1 3. If A1 ; A2 ; : : : is an in􀀌nite collection of disjoint events, P (A1 [ A2 [ 􀀁 􀀁 􀀁 ) = P P (Ai) 4. P (;) = 0 5. 8A; P(A) + P(A0 ) = 1 from which P (A) = 1 􀀀 P (A0 ) 6. For any two events A; B 2 S ; P (A [ B) = P (A) + P (B) 􀀀 P (A \ B) 7. For any three events A; B; C 2 S; P(A [ B [ C) = P (A) + P (B) + P (C) 􀀀 P (A \ B) 􀀀 P (A \ C) 􀀀 P (B \ C) + P (A \ B \ C) Discrete and Continuous Variables: A numerical variable is discrete if its set of possible values is at most countable. A numerical value is continuous if its set of possible values isan uncountable set. Probability: pop ! sample Stats: sample ! pop Measures of Location For observations x 1; x 2 ; 􀀁 􀀁 􀀁 ; xn Sample Mean = Pn~~1~~~~1 ~~x~~i~~ Sample Median = ( ~~n+~~2~~1~~ )nth observation Trimmed Mean btwn and , compute by removing smallest and largest observations Measures of Variability Range Sample Variance, 􀀛 2 S xx Sample Standard Deviation, 􀀛 =lgst-smllst observation = P (xi 􀀀 ))2 = S xx n􀀀1 n􀀀1 = Px2i 􀀀 (Pxn~~i~~)2 = p􀀛2 Box Plots Order the n observations from small to large. Separate the smallest half from the largest (If n is odd then is in both halves) . The lower fourth is the median of the smallest half (upper fourth..largest..) . A measure of the spread that is resistant to outliers is the fourth spread fs given by fs = upper fourth-lower fourth. Box from lower to upper fourth with line at median. Whiskers from smallest to largest xi . Ch 2: Probability Sample Space and Events Experiment Sample Space (S) Event activity with uncertain outcome the set of all possible outcomes any collection of outcomes in S Axioms, Interpretations and Properties of Probability Given an experiment and a sample space S, the objective probability is to assign to each event A a number P (A), called the probability of event A, which will give a precise measure of the chance that A will occur. Behaves very much like norm. Equally Likely Outcomes : P (A) = NN(A) Counting Techniques Product Rule for Ordered k-Tuples: If the 􀀌rst element can be selected in n 1 ways, the second in n 2 ways and so on, then there are n 1n 2 􀀁 􀀁 􀀁 nk possible k-tuples. Permutations: An ordered subset. The number of permutations of size k that can be formed from a set of n elements is Pk;n Pk;n = (n)(n 􀀀 1) 􀀁 􀀁 􀀁 (n 􀀀 k + 1) = (n~~!~~k)! Combinations: An unordered subset. 􀀀nk􀀁 = Pkk~~ ~~;!n = k!(~~n~~n􀀀~~! ~~k)! Conditional Probability P (Aj B) is the conditional probability of A given that the event B has occurred. B is the conditioning event. P (A \ B) P (Aj B) = P (B) Multiplication Rule: P (A \ B) = P (Aj B) 􀀁 P(B) Baye's Theorem Let A1; A 2; : : : ; Ak be disjoint and exhaustive events (that partition the sample space) . Then for any other event BP (B) = P (Bj A1)P (A1) + 􀀁 􀀁 􀀁 + P(B jAk)P (Ak) = Pki=1 P (BjAi)P (Ai) Independence Two events A and B are independent if P (Aj B) = P (A) and are dependent otherwise. A and B are independent i􀀋 P (A \ B) = P (A) 􀀁 P(B) and this can be generalized to the case of n mutually independent events. Random Variables Random Variable: any function X : 􀀊 ! R Prob Dist .: describes how the probability of 􀀊 is distributed along the range of X Discrete rv: rv whose domain is at most countable Continuous rv: rv whose domain is uncountable and where 8c 2 R; P(X = c) = 0 Bernoulli r

FAQs

What is the difference between population and sample in this cheat sheet?

A population is a well-defined collection of objects, while a sample is a subset of that population used for analysis.

How are discrete and continuous random variables distinguished?

A numerical variable is discrete if its set of possible values is at most countable, and continuous if its set of possible values is uncountable.

What do Bayes’ theorem and conditional probability relate?

Conditional probability gives the probability of A given that B has occurred, and Bayes’ theorem rewrites probabilities using the partition of the sample space into disjoint exhaustive events.

Oct 13, 2026

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