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 ↓


Measures of Central Tendency


Measures of central tendency are statistical tools used to measure the central or typical behavior of a data set. These measures provide a way to summarize a large data set with a single, central value, which represents the entire distribution. The three most common measures of central tendency are the mean, median, and mode. Each of these measures provides a different type of information about a data set, and their utility varies depending on the nature of the data.

Meaning

The mean, often referred to as the average, is the sum of all data points divided by the number of data points. It provides a measure of the center of the data by balancing the deviations from this central value. The formula to calculate the mean is:

Mean (μ) = (ΣX) / N

Where:

  • ΣX is the sum of all data values
  • N is the number of data values

Example

Consider a set of exam scores: 70, 85, 78, 92, 88.

Mean = (70 + 85 + 78 + 92 + 88) / 5 = 413 / 5 = 82.6

Visual example

70 85 78 92 88 82.6 (average)

Here, the scores are shown as lines and the mean is marked by a red circle at 82.6 on the number line.

Median

The median is the middle value in a data set when it is arranged in ascending or descending order. When the number of observations (N) is odd, it is the middle value. If N is even, the median is the average of the two middle numbers. When N is odd, the formula for the median is:

Median = X(N+1)/2

For even number of observations:

Median = (X(N/2) + X(N/2 + 1)) / 2

Example

Consider the following age list: 22, 27, 25, 24, 23.

First, arrange them in order: 22, 23, 24, 25, 27. Middle value or Median = 24.

Suppose the list is extended to 22, 27, 25, 24, 23, 30.

In sequence: 22, 23, 24, 25, 27, 30.

Median = (24 + 25) / 2 = 24.5

Visual example

22 23 24 (median) 25 27

For the initial odd-member list, the third point symbolized the median.

Mode

The mode is the value that occurs most often in a data set. If a number does not repeat, the data set has no mode. A set may have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

Example

Consider the data set: 2, 4, 4, 6, 8.

Its mode is 4, because it occurs twice as often as any other number.

Visual example

2 4(mode) 4(mode) 6 8

Features of central tendency measures

Mean

  • Sensitive to extreme values or exceptions.
  • Best for symmetric distributions without outliers.
  • Often this represents the 'balance point' of the data set.

Median

  • Robust to outliers.
  • Useful for skewed distributions.
  • Central reason only when the data are ordinal.

Mode

  • Can be used with categorical data.
  • Indicating the most common value in the dataset.
  • Insensitive to external elements.

Which solution should be adopted?

Choosing the right measure of central tendency depends largely on the nature of the data set and the research question. Here are some general guidelines:

  • Use the mean for data that is symmetrically distributed with no outliers, as it considers all data points.
  • If the data is skewed or has outliers then choose the median for better central value representation.
  • Consider the mode when identifying the most common range or score in a distribution.

Conclusion

Measures of central tendency are fundamental concepts in statistics that help describe a data set as a whole and allow comparisons to be made between different data sets. By understanding the mean, median, and mode, one can effectively analyze data and communicate findings in a meaningful way. Whether you are dealing with large data sets in business analysis or smaller sets in academic research, it is important to understand these concepts. As always, the right choice of measure depends on the context and the specific nature of the data involved.


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Measures of Central Tendency

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