Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Statistics ↓


Measures of Central Tendency


Measures of central tendency are statistics that describe the center or average of a dataset. In simple terms, they tell us where most of the values in a dataset are concentrated. Generally, we discuss three main measures of central tendency: mean, median, and mode.

Meaning

The mean is what most people commonly refer to as the "average." It is calculated by adding up all the numbers in a dataset and then dividing by the number of values in that dataset.

Formula of mean

Mean = (Sum of all values) / (Total number of values)

Consider the dataset: 3, 5, 7, 9, 11.

To find the mean:

Total = 3 + 5 + 7 + 9 + 11 = 35
Number of values = 5
Mean = 35 / 5 = 7
3 5 7 9 11 Average (7)

In the above figure, each blue circle represents a data point and the green circle represents the position of the mean (average).

Median

When the numbers in a data set are arranged in order, the median is the middle value, either from lowest to highest or highest to lowest. If the number of values is odd, the median is the middle number. If the number of values is even, the median is the average of the two middle numbers.

Example 1 (odd number of values): Consider the dataset: 5, 3, 8, 1, 7.

First, order the dataset: 1, 3, 5, 7, 8.

The median value is the third number:

Median = 5

Example 2 (even number of values): Consider the dataset: 22, 15, 30, 17.

First, sort the dataset: 15, 17, 22, 30.

The median is the average of the two middle numbers, 17 and 22:

Median = (17 + 22) / 2 = 19.5
1 3 5 7 8

In the above figure, the red circle represents the median, which shows its central position in the ordered data set.

Mode

The mode is the number that appears most often in the dataset. A dataset can have one mode, more than one mode, or no mode if no numbers are repeated.

Consider the dataset: 4, 1, 2, 4, 3, 4, 5.

In this dataset, the number 4 appears most often:

Mode = 4

For the dataset: 6, 2, 6, 3, 5, 5, 7:

The numbers 6 and 5 both appear twice:

Mode = 6, 5 (bimodal)
2 3 4 5 6 7

The red circles indicate the modes of sample datasets with more than one mode.

Comparison of mean, median and mode

Each measure of central tendency provides different insights and works better in different situations.

  • Mean: Best for datasets without outliers (extreme values) as it considers all values.
  • Median: Useful for skewed datasets or when outliers are present because it indicates the middle of the dataset.
  • Mode: Valued for determining the most common value, especially in categorical data.

Working example

Let's compare the mean, median, and mode with a more complex example:

Consider the dataset: 2, 3, 5, 7, 10, 3, 9, 2, 3, 11.

First, order the dataset: 2, 2, 3, 3, 3, 5, 7, 9, 10, 11.

Mean:

Total = 2 + 2 + 3 + 3 + 3 + 5 + 7 + 9 + 10 + 11 = 55
Total number of values = 10
Mean = 55 / 10 = 5.5

Median:

Median = (5 + 7) / 2 = 6

Mode:

Mode = 3 (appears most frequently)

So, for this example:

The mean is 5.5, the median is 6, and the mode is 3.

Conclusion

Measures of central tendency are key components of descriptive statistics. By understanding the difference between the mean, median, and mode, we can better analyze datasets to find patterns and make predictions. Practice with different datasets to see how these calculations can differ and what they reveal about the data.


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Measures of Central Tendency

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