Data Science Mathematics and Statistics Syllabus

Mathematics and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics and Statistics in Data Science, not a summary of it.

1. 1

   Linear Algebra

   5 topics

   • Vectors
   • Matrices
   • Determinants
   • Eigenvalues and Eigenvectors
   • Matrix Decomposition (LU, QR, SVD)
2. 2

   Calculus

   6 topics

   • Limits and Continuity
   • Differentiation
   • Integration
   • Partial Derivatives
   • Gradient, Divergence, and Curl
   • Optimization (Gradient Descent, Lagrange Multipliers)
3. 3

   Probability

   7 topics

   • Basic Probability Principles
   • Random Variables
   • Probability Distributions
   • Expectation and Variance
   • Bayes' Theorem
   • Law of Large Numbers
   • Central Limit Theorem
4. 4

   Discrete Mathematics

   5 topics

   • Set Theory
   • Combinatorics
   • Graph Theory
   • Boolean Algebra
   • Logic and Proof Techniques
5. 5

   Descriptive Statistics

   4 topics

   • Measures of Central Tendency (Mean, Median, Mode)
   • Measures of Dispersion (Range, Variance, Standard Deviation)
   • Skewness and Kurtosis
   • Data Visualization (Histograms, Box Plots, Scatter Plots)
6. 6

   Inferential Statistics

   6 topics

   • Sampling Methods
   • Estimation (Point and Interval Estimates)
   • Hypothesis Testing
   • Confidence Intervals
   • P-values and Significance Levels
   • ANOVA (Analysis of Variance)
7. 7

   Regression Analysis

   6 topics

   • Simple Linear Regression
   • Multiple Linear Regression
   • Logistic Regression
   • Polynomial Regression
   • Ridge and Lasso Regression
   • Assumptions and Diagnostics of Regression Models
8. 8

   Multivariate Statistics

   5 topics

   • Principal Component Analysis (PCA)
   • Factor Analysis
   • Cluster Analysis
   • Discriminant Analysis
   • Multivariate Analysis of Variance (MANOVA)
9. 9

   Time Series Analysis

   5 topics

   • Components of Time Series
   • Smoothing Techniques
   • ARIMA Models
   • Seasonal Decomposition
   • Forecasting Methods
10. 10

    Non-Parametric Methods

    4 topics

    • Chi-Square Tests
    • Mann-Whitney U Test
    • Kruskal-Wallis Test
    • Spearman's Rank Correlation

Mathematics and Statistics flashcards for Data Science

19 of 52 cards from the Mathematics and Statistics deck — real questions with worked answers.

1. What is a vector, and how is it commonly represented in $n$-dimensional space?

   A vector is a quantity with both magnitude and direction, represented as an ordered $n$-tuple of components: $\vec{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^{n}$. It can be written as a column $\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$.

2. How do you compute the dot (inner) product of two vectors, and what does it equal geometrically?

   $\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i = \|\vec{a}\|\,\|\vec{b}\|\cos\theta$, where $\theta$ is the angle between them. It is a scalar; $\vec{a}\cdot\vec{b}=0$ means the vectors are orthogonal.

3. What is the Euclidean norm (length) of a vector $\vec{v}=(v_1,\ldots,v_n)$?

   $\|\vec{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^{2}} = \sqrt{\vec{v}\cdot\vec{v}}$.

4. State the formula for the cosine similarity between two nonzero vectors.

   $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}$. It ranges in $[-1, 1]$ and measures directional similarity independent of magnitude.

5. What is a matrix, and what does the notation $A \in \mathbb{R}^{m \times n}$ mean?

   A matrix is a rectangular array of numbers arranged in rows and columns. $A \in \mathbb{R}^{m \times n}$ denotes a matrix with $m$ rows and $n$ columns, with entries $a_{ij}$ ($i$-th row, $j$-th column).

6. What is the rule for matrix multiplication $C = AB$, and the dimensional requirement?

   If $A$ is $m\times n$ and $B$ is $n\times p$, then $C=AB$ is $m\times p$ with $c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$. The inner dimensions must match ($n=n$).

7. What is the transpose of a matrix, and what defines a symmetric matrix?

   The transpose $A^{T}$ swaps rows and columns: $(A^{T})_{ij} = a_{ji}$. A matrix is symmetric if $A = A^{T}$ (so $a_{ij}=a_{ji}$).

8. Define the identity matrix and the inverse of a square matrix.

   The identity $I_n$ has $1$s on the diagonal and $0$s elsewhere, with $AI = IA = A$. The inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$; it exists only if $\det(A) \neq 0$.

9. What is the formula for the determinant of a $2\times 2$ matrix?

   For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\det(A) = ad - bc$.

10. What does the determinant of a matrix tell you, and what does $\det(A)=0$ imply?

    The determinant measures the scaling factor of the volume under the linear transformation. $\det(A)=0$ means the matrix is singular (non-invertible), its columns are linearly dependent, and it collapses space to a lower dimension.

11. State two key multiplicative properties of determinants.

    $\det(AB) = \det(A)\det(B)$ and $\det(A^{-1}) = \dfrac{1}{\det(A)}$. Also $\det(A^{T}) = \det(A)$.

12. Define an eigenvalue and eigenvector of a square matrix $A$.

    A nonzero vector $\vec{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ if $A\vec{v} = \lambda \vec{v}$. The transformation $A$ only scales $\vec{v}$ (no change in direction).

13. What is the characteristic equation used to find eigenvalues?

    $\det(A - \lambda I) = 0$. Its roots $\lambda$ are the eigenvalues; the corresponding eigenvectors solve $(A - \lambda I)\vec{v} = \vec{0}$.

14. What do the trace and determinant of a matrix equal in terms of its eigenvalues?

    $\operatorname{tr}(A) = \sum_{i} \lambda_i$ (sum of eigenvalues) and $\det(A) = \prod_{i} \lambda_i$ (product of eigenvalues).

15. For a real symmetric matrix, what special properties do its eigenvalues and eigenvectors have?

    All eigenvalues are real, and eigenvectors for distinct eigenvalues are orthogonal. It is orthogonally diagonalizable: $A = Q \Lambda Q^{T}$ with $Q$ orthogonal and $\Lambda$ diagonal.

16. What is the LU decomposition, and what is its main use?

    LU decomposition factors a square matrix as $A = LU$, where $L$ is lower-triangular and $U$ is upper-triangular. It is used to solve linear systems $A\vec{x}=\vec{b}$ efficiently via forward and back substitution.

17. What is the QR decomposition and a key property of its factors?

    $A = QR$, where $Q$ has orthonormal columns ($Q^{T}Q = I$) and $R$ is upper-triangular. It is used for least-squares problems and computing eigenvalues (the QR algorithm).

18. State the Singular Value Decomposition (SVD) of a matrix $A$.

    $A = U \Sigma V^{T}$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal with non-negative singular values $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$. It applies to any $m\times n$ matrix.

19. How does SVD relate to PCA and dimensionality reduction?

    The singular values quantify the variance captured along each principal direction (columns of $V$). Keeping the largest $k$ singular values gives the best rank-$k$ approximation (Eckart–Young theorem), enabling dimensionality reduction.

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