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 ↓


Descriptive Statistics


Descriptive statistics is a branch of statistics that aims to summarize a set of data. It provides simple summaries about samples and measures. Such summaries can be either quantitative, using numerical calculations, or visual, through various charts and graphs. Descriptive statistics help simplify large amounts of data in an understandable way. Each descriptive statistic reduces a lot of data to a simple summary.

Types of descriptive statistics

Descriptive statistics are divided into measures of central tendency and measures of variability or dispersion.

Measures of central tendency

Measures of central tendency describe the center point of a data set. There are three main measurements:

  • Meaning
  • Median
  • Method

Meaning

The mean is the average of the data set. It is calculated by adding up all the numbers and dividing by the count of the numbers.

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

Example:

Data: 2, 3, 5, 7, 11

Mean = (2 + 3 + 5 + 7 + 11) / 5 = 5.6

Average (5.6)

Median

The median is the middle value of an ordered data set. If the number of data points is odd, the median is the middle number. If even, it is the average of the two middle numbers.

Example of odd number of data points:

Data: 3, 5, 7, 9, 11

Median = 7

Example of even number of data points:

Data: 3, 5, 7, 9

Median = (5 + 7)/2 = 6

Method

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

Example:

Data: 4, 4, 6, 8, 2, 4, 10

Mode = 4

Measures of variability

Measures of variability describe the spread of data within a data set. Key measurements include:

  • Category
  • Quarrel
  • Standard Deviation

Category

The range is the difference between the highest and lowest values in a data set.

Range = (Maximum value) - (Minimum value)

Example:

Data: 3, 7, 8, 15, 20

Range = 20 – 3 = 17

Quarrel

Variance measures how far each number in the set is from the mean and thus how far it is from every other number in the set. It is calculated by taking the average of the squared deviations from the mean.

Variance = (Σ (xi - Mean)^2) / N

Example:

Data: 3, 7, 7, 19

Mean = (3 + 7 + 7 + 19) / 4 = 9

Variance = [(3-9)^2 + (7-9)^2 + (7-9)^2 + (19-9)^2] / 4 = 30

Standard deviation

The standard deviation is the square root of the variance and provides a measure of the average distance from the mean.

Standard Deviation = √Variance

Example:

Data: 3, 7, 7, 19

Variance = 30

Standard deviation = √30 ≈ 5.48

Visualization of descriptive statistics

Descriptive statistics can be represented using a variety of graphical techniques. These include histograms, bar charts, pie charts, box plots, and scatter plots.

Bar chart

Bar charts are used to display categorical data, with rectangular bars indicating the frequency of each category. The length of the bars is proportional to the number of cases in each category.

A B C

Histogram

Histograms are used to display continuous data and it shows the frequency distribution of a set of continuous data points.

Sketch

Box plots are used to display the distribution of data based on a five-point summary: minimum, first quartile, median, third quartile, and maximum.

Pie chart

Pie charts display proportional data and each slice represents a part of the whole. It is particularly effective for showing part-to-whole relationships.

Scatter plot

Scatter plots are used to determine the relationship between two variables. Data are plotted as a collection of points, each of which has the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis.

Importance of descriptive statistics

Descriptive statistics are incredibly useful because they provide a simple summary of samples and measurements, giving a quick overview of a data set. They also provide the basis for further statistical analysis, including inferential statistics, which helps ensure accurate and reliable research findings.

Visual examples such as charts and plots not only make data understandable at a glance but also provide insightful tools that can highlight important characteristics of a data set such as trends, fluctuations, and relationships between variables.

In practice, these tools are invaluable in various fields such as science, finance, business analysis, and economics, where having a summary and good understanding of datasets can guide important decision-making processes.

This comprehensive investigation of descriptive statistics highlights its critical role in simplifying and communicating complex data in an understandable and actionable way. By translating large amounts of numbers into digestible insights, descriptive statistics provides the lens through which we can view, understand, and analyze the world through data.


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Descriptive Statistics

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