Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Probability and Statistics ↓


Statistics


Statistics is a branch of mathematics that deals with collecting, analyzing, interpreting, presenting, and organizing data. In the context of probability and statistics, it is primarily concerned with making predictions or inferences about the entire population based on a sample of data. Statistics is used in every field from business to science to make informed decisions.

What is data?

Data refers to information collected for reference or analysis. In statistics, data is essential because it represents the observations or measurements one is interested in. Data can be classified into:

  • Qualitative data: These are descriptive items such as names, labels, or categories. Common examples are colors, city names, and types of music.
  • Quantitative data: This type of data refers to numerical information that can be measured or counted. Examples include height, weight, and temperature.

Types of statistics

In statistics, there are two main types:

  • Descriptive Statistics: This involves summarizing and presenting data. Measures such as mean, median, mode, and standard deviation are used to provide an overview of the main characteristics of the dataset.
  • Inferential Statistics: This is about making generalisations or predictions about the population based on sample data. This includes using techniques such as hypothesis testing, confidence intervals and regression analysis.

Descriptive statistics

Meaning

The mean is the average of a set of values. It is calculated by adding up all the numbers and then dividing by the number of those numbers.

Mean = (Sum of all data points) / (Number of data points)

For example, if you have a dataset of numbers: 2, 4, 6, 8, and 10, the mean would be:

Mean = (2 + 4 + 6 + 8 + 10) / 5 = 6

Median

The median is the middle value when the data points are arranged in ascending order. If the number of observations is even, the median is the average of the two middle numbers.

For example, in the dataset: 3, 5, 7, 9, the median is:

Median = (5 + 7) / 2 = 6

Method

The mode is the number that appears most often in the dataset. A dataset may have one mode, more than one mode, or no mode.

For example, the dataset: 3, 3, 5, 7, 9 has the mode:

Mode = 3

Standard deviation

Standard deviation is a measure of the range of variation or dispersion in a dataset. A low standard deviation means that the data points are close to the mean, while a high deviation means that the data points are spread across a larger range.

Standard Deviation (σ) = √[(Σ(Xi - μ)²) / N]

Here, Σ denotes the sum, Xi are the individual data points, μ is the mean of the data, and N is the number of data points.

Visualization of standard deviation

Inferential statistics

Inferential statistics uses sample data to make inferences, decisions, predictions, or other generalizations about a population. Some of the primary methods include:

Hypothesis testing

It is a statistical method used to make a decision about a hypothesis. The null hypothesis represents the status quo, and the alternative hypothesis represents what we want to prove. The main components are:

  • Null hypothesis ( H0 ): It is assumed that there is no significant difference or effect.
  • Alternative Hypothesis ( H1 ): It is assumed that there is a significant difference or effect.
  • p-value: Determines the probability of observing the sample results if the null hypothesis is true. A small p-value (< 0.05) indicates strong evidence against the null hypothesis.

Confidence interval

A confidence interval is a range of values that is used to estimate the true value of a population parameter. It is an interval estimate that gives us a range where we expect the true parameter to be.

Confidence Interval = Statistic ± Margin of Error

For a 95% confidence level, there is a 95% probability that the interval calculated from the sample contains the true population parameter.

Regression analysis

Regression analysis is a statistical method for examining the relationship between two or more variables. It allows us to understand how a specific value of a dependent variable changes when any of the independent variables are changed.

y = mx + c

Here, y denotes the dependent variable, m is the slope, x is the independent variable, and c is the y-intercept.

regression line

Applications of statistics

The use of statistics is widespread. Some of the major areas where it plays a vital role include:

  • Business: Decision making, market analysis, and financial performance measurement rely heavily on statistical methods such as trends and regression analysis.
  • Health care: Predicting disease outbreaks, treatment effectiveness, and patient demographics.
  • Science and research: analysis of experimental data, survey sample analysis, and research study design.
  • Sports: Performance analysis, improving sports tactics and training techniques.

Conclusion

Statistics is a powerful tool used to extract information from data. Its vast range of techniques and methods is useful in a wide variety of fields as it helps summarize, analyze, and draw conclusions about information. The basics of statistics, such as measures of central tendency, dispersion, and inferential techniques, prepare individuals to understand the story of data and make informed decisions.


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