Statistics and Probability Formulae AI SL & AI HL - Top Study Guide

Statistical and Probability Formulas

Interquartile range

IQR = Q₃ - Q₁

Mean, x̄ , of a set of data

x̄ = ( ∑i=1k fi xi ) / n, where n = ∑i=1k fi

Probability of an event A

P(A) = n(A) / n(u)

Complementary events

P(A) + P(A') = 1

Combined events

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Mutually exclusive events

P(A ∪ B) = P(A) + P(B)

Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Independent events

P(A ∩ B) = P(A) P(B)

Expected value of a Discrete random variable X

E(X) = ∑ x P(X = x)

Binomial distribution Mean; Variance

X ~ B(n, p);
E(X) = np;
Var(X) = np(1 − p)

*Note: Some complex mathematical notations are approximated using standard characters and HTML subscripts/superscripts due to the constraint of not using external math rendering libraries.

Frequently Asked Questions about Probability and Statistics

What is Probability in Statistics? +

Probability is a mathematical framework used in statistics to quantify the likelihood or chance of an event occurring. It's a number between 0 (impossible event) and 1 (certain event), often expressed as a percentage (0% to 100%). It provides the foundation for statistical inference, allowing us to make statements and predictions about uncertain situations based on data.

What are Statistics and Probability, and how are they related? +

Probability and Statistics are closely related fields of mathematics that both deal with randomness and uncertainty, but they approach it from different directions.

**Probability:** Starts with a known model or process and predicts the likelihood of specific outcomes (e.g., if you know a coin is fair, what's the probability of getting heads?). It goes from theory to data.
**Statistics:** Starts with observed data and uses it to infer information about the underlying process or population from which the data came (e.g., observing a sequence of coin flips to determine if the coin is fair). It goes from data to theory.

Probability is a fundamental tool *within* statistics, particularly in inferential statistics, to measure the reliability of conclusions drawn from data and to quantify uncertainty.

What is the difference between Probability and Statistics? +

The core difference lies in their focus:

**Probability:** Works with populations to predict sample outcomes. It's deductive reasoning.
**Statistics:** Works with sample outcomes to make inferences about populations. It's inductive reasoning.

Probability answers questions like "What is the chance this will happen?", while Statistics answers questions like "Based on what happened, what can we conclude about the system?"

How is Probability Calculated or Found in Statistics? +

Calculating probability depends heavily on the context and the nature of the events. For simple cases with equally likely outcomes:

Probability of an event = (Number of ways the event can occur) / (Total number of possible outcomes)

More complex calculations involve:

Using rules of probability (addition rule, multiplication rule).
Applying permutations and combinations for counting outcomes.
Using conditional probability (Bayes' Theorem) for dependent events.
Working with probability distributions (like binomial, normal, Poisson) that describe the probabilities for all possible outcomes of a random variable.
Using statistical software or calculators for complex distributions or large datasets.

What is a Probability Value (P-value) in Statistics? +

A P-value (Probability value) is specifically used in hypothesis testing. It's the probability of observing a test statistic (or something more extreme) given that the null hypothesis is true. A small P-value suggests that your observed data is unlikely under the null hypothesis, leading you to potentially reject the null hypothesis in favor of the alternative hypothesis.

What is a Probability Distribution in Statistics? +

A probability distribution is a function or table that shows the probabilities of all possible outcomes of a random variable. It essentially maps every possible outcome to its probability of occurrence. Distributions can be discrete (for countable outcomes, like the number of heads in flips) or continuous (for outcomes on a continuous scale, like height or weight).

How difficult is Probability and Statistics as a course? +

The difficulty is subjective and depends on your mathematical background and the level of the course. Basic introductory courses focus on concepts, simple calculations, and interpreting results, which many find manageable. More advanced courses delve into rigorous mathematical proofs, measure theory, and complex models, requiring a strong foundation in calculus and abstract thinking. Generally, it requires logical thinking, attention to detail, and understanding conceptual frameworks rather than just memorizing formulas.

Why is Probability Important in Statistics? +

Probability is crucial because it allows statisticians to make inferences about a larger population based on a smaller sample. Without probability, statistics would primarily be descriptive (summarizing data). Probability provides the tools to quantify the uncertainty involved when we extrapolate from a sample to a population. Concepts like confidence intervals, hypothesis testing, and statistical significance all rely on probability theory to determine how likely our observed results are due to random chance.