Articles

Introduction to Statistics

October 25, 2026
Statistics Notes
EN

Statistics is the science of learning from data. It provides methods for collecting, summarizing, analyzing, and interpreting observations so that we can describe patterns, quantify uncertainty, and make informed decisions...

F Distribution

October 25, 2026
Statistics Notes
EN

The F-distribution, also known as the Fisher-Snedecor distribution, is a continuous probability distribution of the ratio F = (U/d₁)/(V/d₂) of two independent chi-square variables U ~ χ²(d₁) and V ~ χ²(d₂), each divided by its degrees of freedom. It arises in hypothesis testing when comparing the va...

Student T Distribution

October 25, 2026
Statistics Notes
EN

Student's t-distribution is a continuous probability distribution that arises when a normally distributed quantity is standardized using an estimated standard deviation rather than a known population standard deviation. Formally, if Z ~ N(0,1) and V ~ χ²(ν) are independent, then T = Z/√(V/ν) ~ t(ν)...

Sampling Distributions

October 25, 2026
Statistics Notes
EN

A statistic is computed from a sample, but before the sample is observed the statistic is itself a random variable. Its probability distribution over hypothetical repeated samples is called its sampling distribution...

Chi Square Distribution

October 25, 2026
Statistics Notes
EN

A chi-square distribution is a continuous probability distribution of the sum of the squares of k independent standard normal random variables. The chi-square distribution is denoted as X ~ χ²(k), where k is the number of degrees of freedom...

Central Limit Theorem

October 25, 2026
Statistics Notes
EN

The central limit theorem describes the fluctuations of a sum or average around its expectation. For independent, identically distributed observations with finite, positive variance, the centered and standardized mean approaches a standard normal distribution as sample size grows. The population can...

Standard Error and Lln

October 25, 2026
Statistics Notes
EN

A standard deviation describes variability of a specified random quantity. When that quantity is an estimator, its standard deviation is called a standard error. The law of large numbers explains why certain averages concentrate as the amount of suitable information grows. These ideas are connected...

Bayesian vs Frequentist

October 25, 2026
Statistics Notes
EN

Bayesian and frequentist statistics are two major approaches to statistical inference. Both use sample data to learn about an underlying population or data-generating process, but they differ in how they interpret probability, represent uncertainty, and draw conclusions about unknown parameters...

Multiple Regression

October 25, 2026
Statistics Notes
EN

Multiple linear regression is a statistical technique used to model the relationship between a single dependent variable and two or more independent variables. It extends the concept of simple linear regression by incorporating multiple predictors to explain the variability in the dependent variable...

Logistic Regression

October 25, 2026
Statistics Notes
EN

Logistic regression is a statistical method used for modeling the probability of a binary outcome based on one or more predictor variables. It is widely used in various fields such as medicine, social sciences, and machine learning for classification problems where the dependent variable is dichotom...

Simple Linear Regression

October 25, 2026
Statistics Notes
EN

Simple linear regression is a statistical method used to model the relationship between a single dependent variable and one independent variable. It aims to find the best-fitting straight line through the data points, which can be used to predict the dependent variable based on the independent varia...

Analysis of Variance

October 25, 2026
Statistics Notes
EN

Does peer assessment enhance student learning...

Statistical Moments

October 25, 2026
Statistics Notes
EN

A statistical moment is a probability-weighted average of a power of a value or deviation. The chosen reference point matters. Squared distance from zero gives a raw second moment; squared distance from the mean gives variance. Higher powers put progressively more weight on observations far from tha...

Normal Curve and z Score

October 25, 2026
Statistics Notes
EN

A z-score expresses a value's distance from a mean in standard-deviation units. It turns a height difference in inches or a score difference in points into a dimensionless comparison. When the original variable is normal, this transformation also gives a shared probability scale: the standard normal...

Introduction to Distributions

October 25, 2026
Statistics Notes
EN

A probability distribution assigns probabilities to the possible values of a random quantity and to events involving those values. It describes a model for what could happen, rather than merely listing the values already observed. A dataset is one realization of that model; its relative frequencies ...

Validation and Model Selection

October 25, 2026
Statistics Notes
EN

A fitted model is optimized using observed data. Model assessment asks a different question: how well will the modeling procedure perform on new data generated under comparable conditions...

Metrics

October 25, 2026
Statistics Notes
EN

An evaluation metric turns predictions and observed outcomes into a numerical answer to a particular question. Accuracy measures correct decisions at a threshold; AUC measures ordering; log loss measures probabilistic predictions; absolute and squared errors assign different costs to numerical misse...

Resampling

October 25, 2026
Statistics Notes
EN

Simulation approximates a probability calculation by repeatedly generating data under a specified mechanism. A parametric Monte Carlo experiment samples from an explicit model; a nonparametric bootstrap samples from the empirical distribution of the observed data; a permutation test rearranges data ...

Geostatistics

October 25, 2026
Statistics Notes
EN

Geostatistics is used when observations are tied to locations and nearby values may be more similar than distant values. Typical examples are...

Spatial Weights

October 25, 2026
Statistics Notes
EN

A spatial weights matrix turns a qualitative statement such as...

Point Processes

October 25, 2026
Statistics Notes
EN

A spatial point process is used when the observed data are the event locations themselves. This differs from geostatistics. In geostatistics, locations are usually fixed and a value is observed at each location...

Spatial Autocorrelation

October 25, 2026
Statistics Notes
EN

Spatial autocorrelation asks whether values attached to locations show a systematic spatial pattern. The idea is...

Kriging

October 25, 2026
Statistics Notes
EN

Kriging is a method for predicting a spatial variable at unsampled locations using a model of spatial dependence. Kriging is often introduced as the best linear unbiased predictor (BLUP). The phrase is useful when each part is interpreted carefully...

Spatial Data and Distance

October 25, 2026
Statistics Notes
EN

A map supplies coordinates, but coordinates alone do not define a statistical observation. A dot might represent a sensor reading, an event location, or a district total plotted at its centroid. Those three cases require different probability models. Before calculating Moran's I, fitting a variogram...

Spatial Regression

October 25, 2026
Statistics Notes
EN

Regression models the mean relationship between an outcome and its predictors. A basic linear model is...

Spatial Validation

October 25, 2026
Statistics Notes
EN

Spatial model performance depends not only on which observations are held out, but also on where those observations are located relative to the training data. This makes spatial validation fundamentally different from ordinary random train/test splitting. A random split often leaves test locations s...

Covariance

October 25, 2026
Statistics Notes
EN

Covariance averages products of deviations from two means. When both deviations have the same sign, their product is positive; when their signs differ, it is negative. The average therefore indicates the direction of linear co-variation, while its numerical size also reflects the units and variabili...

Correlation

October 25, 2026
Statistics Notes
EN

Correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a fundamental concept in statistics, enabling researchers and analysts to understand how one variable may predict or relate to another. The most commonly used corre...

Joint Distributions

October 25, 2026
Statistics Notes
EN

A univariate distribution describes a single random variable. A joint distribution describes several random variables simultaneously and contains the information needed to reason about dependence, conditional behavior, and functions of multiple variables...

Negative Binomial Distribution

October 25, 2026
Statistics Notes
EN

The negative binomial distribution extends geometric waiting from the first success to the r-th success. Repeat independent Bernoulli trials with a common success probability p and stop when the required positive integer number r of successes has occurred...

Binomial Distribution

October 25, 2026
Statistics Notes
EN

A binomial random variable counts successes in a fixed number of independent trials. Each trial has two possible labels, success and failure, and the same success probability p. “Success” names the event being counted; it need not be desirable. Counting defective products can use exactly the same ma...

Geometric Distribution

October 25, 2026
Statistics Notes
EN

The geometric distribution counts trials up to and including the first success. Unlike a binomial experiment, it does not fix the number of trials in advance. Repeat independent Bernoulli trials with a common success probability p until the first success, and call its trial number X...

Poisson Distribution

October 25, 2026
Statistics Notes
EN

A Poisson random variable counts events in a specified interval of time, distance, area, or other exposure. Write X ~ Poisson(λ), where λ is the expected count in that interval. Its support is {0,1,2,…}: there is no fixed number of trials that caps the count...

Log Normal Distribution

October 25, 2026
Statistics Notes
EN

A positive random variable is log-normal when its natural logarithm is normal...

Beta Distribution

October 25, 2026
Statistics Notes
EN

The beta distribution models a continuously varying quantity confined to the unit interval, such as an uncertain probability or an underlying proportion. Write X ~ Beta(α,β), where both shape parameters are positive. Unlike a normal model, it assigns no probability outside the permitted range...