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 Central Tendency ↓


Quartiles and Percentiles


Introduction

In statistics, central tendency is a way of describing the center of a data set. Two important measures for understanding the spread of data are quartiles and percentiles. Both are part of descriptive statistics. These measures help you understand the way data can be divided into parts, each of which represents a specific proportion of the data set.

Quartile

Quartiles are values that divide your data into quarters. There are three points called the first quartile (Q1), the second quartile (Q2), which is also the median, and the third quartile (Q3). These points divide your data into four equal parts.

First quartile (Q1)

The first quartile, or Q1, is the mean of the bottom half of your data set (excluding the median if the number of observations is odd). It represents the 25th percentile. This means that 25% of the data is below this point.

Q1 = (n + 1) / 4th term

Second quartile (Q2)

The second quartile, or Q2, is simply the median. It represents the 50th percentile, meaning that 50% of the data is below this point.

Third quartile (Q3)

The third quartile, or Q3, is the mean of the upper half of your data set. It represents the 75th percentile, which indicates that 75% of the data is below this point.

Q3 = 3(n + 1) / 4th term

Visual example of quartiles

Question 1 Question 2 (median) Question 3 25% 25% 25% 25%

Example of calculating quartiles

Suppose you have the following data set of 9 numbers:

Data: 3, 7, 8, 5, 12, 14, 21, 13, 18

First, arrange it in ascending order:

Sorted data: 3, 5, 7, 8, 12, 13, 14, 18, 21

Question 1: Since there are 9 numbers in our data, (9 + 1) / 4 = 2.5, so Question 1 is the average of the second and third value: (5 + 7) / 2 = 6.

Q2: The median, or Q2, is the 5th number of the data set, which is: 12.

Question 3: 3(9 + 1) / 4 = 7.5, so the average of question 3, 7th and 8th values is: (14 + 18) / 2 = 16.

Percentile

Percentiles indicate the relative position of a value within a data set. They divide your data into 100 equal parts. The nth percentile is the value below which n percent of the data falls. Common percentiles include the 25th percentile (Q1), the 50th percentile (median or Q2), and the 75th percentile (Q3).

Percentage formula: Pn = (n/100) * (total number of values in the data set)

Visual example of percentiles

0% 25% 50% (median) 75% 100%

Example of calculating percentiles

Suppose you have the same data set as before:

Data: 3, 7, 8, 5, 12, 14, 21, 13, 18

Calculate the rank position for the 70th percentile:

Rank position = (70 / 100) * 9 = 6.3

Since 6.3 is between the 6th and 7th number in the sorted data, take the weighted average:

0.7 * 14 + 0.3 * 13 = 13.7

So, the 70th percentile is about 13.7.

Interpreting quartiles and percentiles

Quartiles and percentiles help to understand data, showing where a particular data point lies relative to the rest of the data. They can help to identify outliers and are often used in box plots.

Understanding these concepts can give you a better picture of your data, especially when you're trying to determine variability, identify potential biases, or make predictions.

Conclusion

Quartiles and percentiles are important tools in statistics for dividing data into parts and understanding the distribution of data. They provide information about the spread, central tendency, and relative position of data points and are useful in a variety of real-world applications.


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Quartiles and Percentiles

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