Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Probability and Statistics → Statistics ↓


Correlation and Regression


Introduction

In statistics, it is important to understand the relationship between two variables. This can reveal how one variable can affect another. Two key concepts that help us understand these relationships are "correlation" and "regression". These concepts allow us to investigate whether variables are related to each other and how strongly. Let's discuss these interesting topics in depth!

Correlation

Correlation is a statistical measure that describes the size and direction of the relationship between two variables, usually denoted as X and Y. It tells us whether variables move together (and if they do, whether they move in the same or opposite directions) without implying a cause-effect relationship.

Understanding correlation

When two variables are correlated, it means there is a predictable pattern in the changes that occur between them. The correlation can be positive, negative, or zero.

  • Positive correlation: As one variable increases, the other increases as well. For example, the relationship between the amount of time studied and the score obtained on an exam might exhibit a positive correlation.
  • Negative correlation: As one variable increases, the other decreases. An example of this could be the relationship between the number of movies watched per week and time spent studying.
  • No correlation (zero correlation): No predicted change connects the variables. For example, the relationship between eye color and intelligence level is expected to show no correlation.

Visual example of correlation

In a scatter plot, the correlation between two variables is displayed visually:

Positive correlation Negative correlation No correlation

Expressing correlation mathematically

The most commonly used correlation coefficient is the Pearson correlation coefficient, denoted by r. The formula to calculate it is as follows:

R = Σ((X_i - X̄)(Y_i - Ȳ)) / (√(Σ(X_i - X̄)² * Σ(Y_i - Ȳ)²))

Where:

  • X_i and Y_i are different data points.
  • X̄ is the mean of the X values and Ȳ is the mean of the Y values.
  • The range of r is from -1 to +1.

If r = 1, it indicates a perfect positive linear relationship. If r = -1, it is a perfect negative linear relationship. When the value of r is close to 0, it means that no linear relationship exists.

Example

Consider a simple dataset with two variables:

  • X: 1, 2, 3, 4, 5
  • Y: 2, 4, 5, 4, 5

To determine the correlation between X and Y, you need to apply the formula specified above.

Regression

While correlation measures the strength and direction of the relationship between two variables, regression is about predicting one variable based on another. It predicts the dependent variable (often denoted as Y) using the independent variable (X).

Understanding regression

Regression helps to understand how a specific value of a dependent variable changes when one of the independent variables is changed while the other independent variables remain constant. Its simplest form is linear regression, which is represented as a line.

Linear regression

Linear regression attempts to model the relationship between two variables by fitting a linear equation to the observed data. The equation of a line is usually presented as:

y = a + bx

Where:

  • Y is the dependent variable we are trying to predict.
  • X is the independent variable that we are using for prediction.
  • a is the intercept, the value of Y when X=0.
  • b is the slope, which represents the change in Y for a one unit change in X.

Visual example of regression

Drawing a line back across data points can often be seen in a scatter plot as follows:

Line of best fit

The red line is called the line of best fit or regression line. It minimizes the distance of all points from the line which is known as the least squares method.

Finding the regression line mathematically

The formulas to calculate the slope b and intercept a are given as:

B = Σ((X_i - X̄)(Y_i - Ȳ)) / Σ((X_i - X̄)²)
a = Ȳ − bx̄

These formulas arise from minimizing the squared difference of the observed values from the line.

Example

Using the first dataset with variables X: [1, 2, 3, 4, 5] and Y: [2, 4, 5, 4, 5].

  • First calculate X̄ and Ȳ.
  • Then, using the above formula, determine b and a.

After calculation:

b = 0.6
a = 2.2
Y = 2.2 + 0.6X

Thus, your regression equation becomes Y = 2.2 + 0.6X.

Key differences and summary

  • Purpose: Correlation measures the direction and strength of a relationship. However, regression models and predicts one variable from another.
  • Dependence: Correlation does not depend on cause and effect. Regression, theoretically, assumes a dependent direction.
  • Symmetry: The correlation is symmetric because corr(X, Y) = corr(Y, X). The regression changes direction because Y = a + bX is not identical to X = c + dY.

In conclusion, correlation and regression provide valuable insight into the relationships between variables. Understanding these concepts is crucial for data analysis in many fields and provides an important foundation for advanced statistical modeling.


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Correlation and Regression

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