Math and Stats: Numbers & Algebra

• Mathematics and Statistics are interrelated disciplines.
• Mathematics provides the tools and language to express statistical concepts.
• Statistics applies mathematical principles to collect, analyze, and interpret data.

Numbers

• Real numbers consist of rational and irrational numbers.
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers cannot be expressed as a simple fraction (e.g., √2, π).
• Integers include positive whole numbers, negative whole numbers, and zero.
• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).

Algebra

• Variables represent unknown quantities and are denoted by letters (e.g., x, y, z).
• Expressions are combinations of variables, constants, and operations (+, -, ×, ÷).
• Equations state the equality between two expressions.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations have the form ax + b = 0, where a and b are constants and x is the variable.
• Quadratic equations have the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Systems of equations involve two or more equations with the same variables.
• Systems of equations can be solved by substitution, elimination, or matrix methods.

Geometry

• Euclidean geometry deals with points, lines, angles, surfaces, and solids.
• Key concepts in geometry include: points, lines, planes, angles, triangles, circles, and polygons.
• The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²).
• Trigonometry deals with the relationships between the sides and angles of triangles.
• Trigonometric functions include sine (sin), cosine (cos), and tangent (tan).

Calculus

• Differential calculus deals with rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and the areas under and between curves.
• Limits describe the behavior of a function as the input approaches a certain value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals calculate the area under a curve or the accumulation of a function.
• The fundamental theorem of calculus relates differentiation and integration, showing they are inverse operations.

Descriptive Statistics

• Descriptive statistics summarize and describe the main features of a dataset.
• Measures of central tendency include mean (average), median (middle value), and mode (most frequent value).
• Measures of dispersion (variability) include range (difference between max and min values), variance (average of squared differences from the mean), and standard deviation (square root of the variance).
• Frequency distributions show the number of occurrences of each value or range of values in a dataset.
• Histograms visually represent frequency distributions.

Probability

• Probability is the measure of the likelihood that an event will occur.
• Probability values range from 0 (impossible event) to 1 (certain event).
• Basic probability rules include the addition rule (for mutually exclusive events) and the multiplication rule (for independent events).
• Conditional probability is the probability of an event occurring given that another event has already occurred.
• Bayes' theorem relates conditional probabilities and is used to update beliefs given new evidence.

Probability Distributions

• A probability distribution describes the likelihood of different outcomes in a random experiment.
• Discrete probability distributions include the binomial, Poisson, and Bernoulli distributions.
• Continuous probability distributions include the normal, exponential, and uniform distributions.
• The normal distribution is a bell-shaped distribution characterized by its mean (μ) and standard deviation (σ).
• Many natural phenomena can be approximated by the normal distribution, due to the central limit theorem.

Inferential Statistics

• Inferential statistics involves making inferences or generalizations about a population based on a sample.
• Hypothesis testing is a method for testing claims or hypotheses about a population.
• Confidence intervals provide a range of values within which the population parameter is likely to fall.
• Common hypothesis tests include t-tests (for comparing means), chi-square tests (for categorical data), and ANOVA (for comparing means of multiple groups).
• Regression analysis is a method for modeling the relationship between a dependent variable and one or more independent variables.

Sampling

• Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population.
• Random sampling ensures that each member of the population has an equal chance of being selected, reducing bias.
• Stratified sampling divides the population into subgroups (strata) and selects samples from each stratum.
• Cluster sampling divides the population into clusters and randomly selects entire clusters to sample.
• Sample size is an important factor in statistical inference; larger samples generally lead to more accurate estimates.