Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Statistics ↓


Measures of Dispersion


In statistics, the concept of “measures of dispersion” is very important. It tells us how much a data set spreads out or varies. To put it simply, while you may know the average of a group of numbers, measures of dispersion help you understand how far other numbers typically fall from this average. Let’s look at this in more detail.

Understanding dispersion with an example

Imagine we have two small sets of numbers representing the marks obtained by two students in five different subjects:

  • Student A: 80, 85, 78, 90, 82
  • Student B: 60, 95, 70, 100, 55

Both Student A and Student B have an average score (mean) of 83. However, there is some difference in their performance, which we can understand using measures of dispersion. While Student A's scores are close to each other, Student B's scores vary widely. This difference in dispersion is exactly what measures of dispersion measure.

Types of measures of dispersion

There are several ways to measure the dispersion of data. The most common measure of dispersion is:

  1. Category
  2. Quartile and Inter Quartile Range
  3. Quarrel
  4. Standard Deviation
  5. Mean absolute deviation

Category

The simplest measure of dispersion is the range. The range measures the difference between the maximum and minimum values in a data set. The formula for calculating the range is:

Range = Maximum Value - Minimum Value

Use our example:

  • Student A: Range = 90 – 78 = 12
  • Student B: Range = 100 – 55 = 45

As you can see, Student B's marks are more spread out, indicating that the marks are more spread out than Student A's.

55 60 70 95 100

Quartile and inter quartile range

Quartiles divide our data set into four equal parts. This helps us understand the spread in more detail by focusing not only on the extremes but also on the middle values. The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). It is a measure of the spread in the middle 50% of the data:

IQR = Q3 - Q1

For each student, arrange the marks in ascending order to find the quartiles:

  • Student A: 78, 80, 82, 85, 90
  • Student B: 55, 60, 70, 95, 100

For Student A:

  • Q1 = 80, Q2 (median) = 82, Q3 = 85

For Student B:

  • Q1 = 60, Q2 (median) = 70, Q3 = 95

Therefore, IQR for student A: 85 - 80 = 5 and for student B: 95 - 60 = 35.

Quarrel

Variance is a measure of how much each number in a data set varies from the mean square. It is more complex, but very useful. The formula to calculate variance is:

Variance (σ²) = Σ (xᵢ - x̄)² / N

Where:

  • Σ means the sum...
  • xᵢ is each individual number in the set
  • x̄ is the mean of the numbers
  • N is the number of observations in the set

For a practical demonstration, let's calculate the variance for Student A's marks:

  • Mean = 83
  • (80-83)² + (85-83)² + (78-83)² + (90-83)² + (82-83)² = 9 + 4 + 25 + 49 + 1 = 88
  • Variance = 88 / 5 = 17.6
Meaning

Standard deviation

The standard deviation is simply the square root of the variance and gives us a clear, scale-equivalent measure of spread. The formula is:

Standard Deviation (σ) = √Variance

Using Student A's variance, the standard deviation is √17.6 ≈ 4.2.

This number gives a good sense of how spread out the data set is in terms of the original units, in this case numbers.

Mean absolute deviation

Another simple measure of dispersion is the mean absolute deviation (MAD). This is the average of the absolute differences between each data point and the mean:

MAD = Σ |xᵢ - x̄| / N

Using Student A:

  • Mean = 83
  • |80-83| + |85-83| + |78-83| + |90-83| + |82-83| = 3 + 2 + 5 + 7 + 1 = 18
  • MAD = 18 / 5 = 3.6

Why measures of dispersion are important

Understanding measures of dispersion provides essential insight into a data set:

  • Risk Assessment: In finance, understanding the risk (variance) of an investment.
  • Quality Assurance: In manufacturing, ensuring that products meet a certain standard, and that variations are kept to a minimum.
  • Comparative Analysis: In education, comparing performance across different assessments.

Conclusion

Measures of dispersion help us better understand data beyond just knowing about the average. They show how spread out the data is, the risks involved or the volatility. From simple calculations like range to complex calculations like variance and standard deviation, each measure provides unique insights, leading to a richer analysis of the data.


Grade 10 → 7.5


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Quartiles and Percentiles
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Range and Interquartile Range

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Measures of Dispersion

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