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 ↓


Mean Median and Mode


In statistics, mean, median, and mode are measures of central tendency, used to analyze data. These concepts help us understand the center or typical value in a set of numbers. Let’s take a deeper look at each of these terms and explore them through various examples and visual aids.

Meaning

The mean is more commonly referred to as the "average." To find the mean of a set of numbers, you add up all the numbers and then divide the total by the count of the numbers. The mean is useful because it provides a measure of the central tendency of the data.

How to calculate the mean

Suppose you have a set of numbers: 3, 7, 7, 19.

Follow these steps to find the mean:

  1. Add up all the numbers: 3 + 7 + 7 + 19 = 36
  2. Count the numbers: There are 4 numbers in this example.
  3. Divide the total by the count of the numbers: 36 / 4 = 9

Thus, the mean of the numbers 3, 7, 7 and 19 is 9.

3 7 7 19 Average: 9

Median

The median is the middle value in a list of numbers, which separates the upper half from the lower half. If the number of observations is even, the median is calculated by averaging the two middle numbers.

How to calculate the median

Let's find the median of the following set of numbers: 12, 3, 5, 8, 7.

  1. First, arrange the numbers in ascending order: 3, 5, 7, 8, 12.
  2. Now find the middle number. Since there are 5 numbers in total, the middle number is the third number: 7.

Thus, the mean of the numbers 12, 3, 5, 8 and 7 is 7.

3 5 7 8 12 Median: 7

If the list has an even number of elements, such as 2, 4, 6, 8, then the median is the average of 4 and 6, so the calculation will be as follows:

Median = (4 + 6) / 2
Median = 10 / 2
Median = 5

Method

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

How to calculate the mode

Let's consider the set of numbers: 2, 4, 4, 6, 8, 8, 8, 10.

  1. Identify the number or numbers that appear most often. In this set:
    • 2 appears once
    • 4 appears twice
    • 6 appears once
    • 8 appears three times
    • 10 appears once
  2. The number 8 occurs most frequently, so it is the mode of this group.
2 4 4 6 8 8 8 10 Modes: 8

No mode

In some cases, a data set may not have a mode if no numbers are repeated. Consider the set of numbers: 1, 2, 3, 4, 5. Since all numbers occur only once, this set has no mode.

Multiple modes

A set of numbers can have more than one mode if several numbers occur with the same highest frequency. For example, in the set 1, 1, 2, 3, 3, both 1 and 3 appear twice. Therefore, this set is bimodal, with modes being 1 and 3.

Visualization of mean, median and mode

To understand these concepts better, let's look at them through another example. Consider the following data set: 1, 4, 4, 6, 7, 8, 9.

Meaning:

Mean = (1 + 4 + 4 + 6 + 7 + 8 + 9) / 7 = 39 / 7 ≈ 5.57

Median:

The numbers are already presented in ascending order: 1, 4, 4, 6, 7, 8, 9. The middle number is 6, because there are 7 numbers in total and the fourth number is in the middle.

Method:

The number 4 occurs twice, more often than any other number, making it the mode.

1 4 4 6 7 8 9 Mean ≈ 5.57 Median: 6 Modes: 4

Choosing the right solution

Understanding the difference between the mean, median, and mode is important in choosing the right measure of central tendency depending on the type of data and the analysis you want to perform.

Using the mean

The mean is very effective for data with no significant outliers or a skewed distribution because it uses all the values in the data set. However, it can be misleading with skewed data sets or outliers. For example, the average salary in a company may be high simply because the CEO's salary is much higher than other employees.

Using the median

The median is a better choice when there are outliers in the data. It is not affected by extreme values, making it a more accurate reflection of a typical price in the set. For example, in a neighborhood where most homes are priced the same, but one mansion is very expensive, the median home price will provide a better sense of the typical home price.

Using the mode

Mode is especially useful when the most common or popular items are needed. It works well with categorical data where we want to know which category is most frequently used. For example, knowing the mode can help us understand the most popular color, type of food, or mode of transportation among people in a survey.

All together: an example

Let's examine a scenario where we use all three measures. Consider a class of 10 students with the following test scores: 70, 80, 85, 85, 90, 100, 100, 100, 105, 110.

  1. Meaning: 
        Mean = (70 + 80 + 85 + 85 + 90 + 100 + 100 + 100 + 105 + 110) / 10 = 925 / 10 = 92.5
  2. Median:

    Arrange the marks in this order: 70, 80, 85, 85, 90, 100, 100, 100, 105, 110. Since there are 10 marks in total, the median will be the average of the 5th and 6th marks, which are 90 and 100:

        Median = (90 + 100) / 2 = 190 / 2 = 95
  3. Method:

    The score 100 occurs most often, so it is the mode.

Each measure gives us different information about the test scores. The mean provides the average score, the median indicates the middle score in the order, and the mode shows us the most frequently obtained scores.

Conclusion

Mean, median, and mode are fundamental concepts in statistics used to describe the central tendency of a data set. Each measure has its own strengths and weaknesses and is chosen based on the specific characteristics of the data and the information we wish to extract. Understanding the proper context and application is essential for accurate data analysis and interpretation.


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Measures of Central Tendency
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Mean Median and Mode

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