PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Probability and Statistics → Probability Theory ↓


Central Limit Theorem


The central limit theorem (CLT) is one of the most important results in the field of probability theory and statistics. It explains why many distributions are approximately normal under certain conditions and provides a basis for making inferences about the population from sample data. The beauty and simplicity of the theorem have made it a cornerstone of statistical theory and application.

Understanding the central limit theorem

Simply put, the central limit theorem states that the distribution of the sample mean will look like a normal distribution (or Gaussian distribution) when the sample size is large, regardless of the shape of the population distribution. This emerges regardless of whether the population is normal or skewed; as long as the sample size is large enough, the sampling distribution of the mean will be approximately normal.

If X₁, X₂, ..., Xₙ are independent random variables, from any distribution with a finite mean μ and a finite variance σ², the sample mean (X̄ = (X₁ + X₂ + ... + Xₙ) / n) will be approximately normally distributed with mean μ and variance σ²/n for a large n.

Formal definition

Let us explore a more formal definition. Consider a random sample of size n drawn from a population with known population mean μ and finite standard deviation σ. The sample mean X̄ is given by:

X̄ = (1/n) * Σ Xᵢ (i = 1 to n)

According to the Central Limit Theorem, as n gets larger, the distribution of X̄ will approach a normal distribution with mean μ and variance σ²/n.

Why is CLT important?

  • Basis for inference: The CLT allows statisticians to make inferences about population parameters even when the population distribution is not normal.
  • Simplifies analysis: It simplifies the mathematical modeling of data, especially when dealing with large samples.
  • Justification of standardization: It justifies the use of standard normal distribution tables to estimate probabilities of sample means.

Visual example of the central limit theorem

Suppose we have a population that is uniformly distributed over the values from 1 to 6, just like rolling a fair six-sided dice. After many trials of taking samples and then calculating their mean, according to the CLT, these means will form a distribution that approximates a normal curve as our sample size increases.

Sample Instrument

In this SVG illustration, we see that the sample means from different groups as the experiment progresses give a rough normal distribution shape. The results from most samples average closer to the mean than the extremes, creating a bell-shaped curve.

Historical perspective and development

The CLT originated in the 18th century from the work of Abraham de Moivre, who showed that binomial distributions approximate normal distributions as the number of trials becomes large. Pierre-Simon Laplace also made significant contributions by extending de Moivre's work into more general forms. The theorem took its more modern form through the works of Carl Friedrich Gauss and soon became an integral tool in the field of statistics, thanks to the Russian mathematician Aleksandr Lyapunov in 1901.

Application of CLT: An example

Let's consider how the central limit theorem can be applied in real-world scenarios. Suppose a company wants to know the average length of time its employees take for a lunch break. This company has hundreds of employees, and measuring the length of every employee's lunch break would be impractical. Instead, they decide to take samples.

By selecting a sample of, say, 50 employees and measuring the time of their lunch breaks, the company can calculate the sample mean. Provided that the sample size is large enough and random, CLT assures that this sample mean will be a good estimate of the true population mean, and the sample mean aggregated over many such samples will form a normal distribution.

More mathematical insights

The beauty of the CLT lies not only in its application but also in its mathematical insights. Convergence towards the normal distribution is a cornerstone of understanding statistical variability and uncertainty.

Law of large numbers vs. central limit theorem

The Law of Large Numbers (LLN) and the Central Limit Theorem may sound similar but are fundamentally different. While the LLN states that sample means will converge to the expected value as the sample size increases, it does not specify the distribution shape of these means. The CLT, on the other hand, is specifically concerned with the distribution shape, predicting a normal distribution as the number of observations increases.

Terms and limitations

CLT comes with some conditions and potential limitations. It generally applies when:

  • The sample size is large enough. Although there is no set number, a general rule of thumb is to have at least 30 samples.
  • The samples are selected at random and are independent.
  • The mean and variance of the population from which samples are taken must be finite.

Non-independence and other distributions

When the samples are not independent, or when other distributional characteristics are at play, the usefulness of the CLT can be challenged or adjusted to fit different contexts. For example, when dealing with distributions with heavy tails or infinite variance, the CLT may not be directly applicable, or we may need to apply appropriate variations or generalizations for specific cases.

Conclusion

The central limit theorem is not only a key theoretical concept, but a practical tool that forms the basis of many statistical methods in use today. It convinces us that understandable and intuitive normality often emerges from randomness, allowing for a high level of implementation in other fields, including scientific, economic, engineering, and social science research.

As we conclude our exploration of the Central Limit Theorem, it is important to keep its assumptions and conditions in mind while appreciating its power and utility. As a fundamental bridge between probability theory and applied statistics, CLT transforms collections of real-world data into powerful predictive models and insights that drive decision-making and understanding in complex systems.


PHD → 8.1.3


U
username
0%
completed in PHD

← Prev (8.1.2)
Expectation and Variance
Next (8.2) →
Stochastic Processes

Comments 



Central Limit Theorem

https://www.buddymath.com/phd/8.1.3?bmal=en