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 ↓


Mean Median and Mode


In the world of statistics, understanding data is an important part of statistics. Whenever we encounter a large set of numbers, it is helpful to have some way to summarize all those numbers into a single, representative value. This summary of the data helps us understand general patterns. The three most common measures of central tendency are the mean, median, and mode. These measures help us describe a data set with single numbers that summarize different aspects of the data.

Meaning

The mean is what most people commonly refer to as the "average." When we talk about the average of a set of numbers, we add up all the values and then divide by the number of values. The mean provides the balance or central point of the data.

Calculating the mean

Follow this formula to calculate the mean:

Mean = (Sum of all data values) / (Number of data values)

For example, consider the data set: 2, 4, 6, 8, 10.

Step 1: Add up all the numbers:

2 + 4 + 6 + 8 + 10 = 30

Step 2: Divide by the number of values (in this case, 5):

Mean = 30 / 5 = 6

Therefore, the mean of the data set is 6.

Visualizing the mean

Imagine each number as a weight spread evenly on the scale. The mean is the point where the scale will balance.

Meaning

Median

The median is the middle value in a data set that is ranked in order. Half the numbers will be above this number and half will be below. The median is especially useful when the data set contains outliers or is skewed, because it is not affected by extreme values like the mean.

Calculating the median

To find the median:

  1. Arrange the data values from smallest to largest.
  2. If the number of data values is odd, the median is the middle value.
  3. If the number of data values is even, the median is the average of the two middle values.

Example of odd numbered values:

Data set: 3, 1, 9, 12, 7

Step 1: Arrange the data in this order: 1, 3, 7, 9, 12

Step 2: Identify the middle value: 7

Example of even numbered values:

Data set: 8, 3, 5, 10

Step 1: Arrange the data in this order: 3, 5, 8, 10

Step 2: Find the average of the two middle numbers (5 and 8):

Median = (5 + 8) / 2 = 6.5

So, when the number of values in a data set is odd, the median is one of the actual data points, but when it is even, the median will be between two data points.

Median visualization

An example may help to understand how the median divides the data into two equal parts.

Median

Method

The mode is the value that appears most often in a data set. A set of data may have one mode, more than one mode, or no mode. The mode is especially useful when the data is not numerical because it can be a good way to summarize categorical data.

Finding the mode

To find the mode, look at the number that occurs most often.

For example, in the data set: 4, 5, 4, 6, 7, 4, 9,

The number 4 occurs most often, so the mode is 4.

In some data sets, you may find two or more modes:

Data set: 2, 3, 5, 3, 5, 8

Both 3 and 5 appear twice, which is more often than any other value, so there are two modes: 3 and 5. This type of data set is called bimodal.

If no numbers are repeated, then the data set has no mode.

Viewing mode

This illustration highlights the mode by showing which value appears most often.

Method

Comparing the mean, median, and mode

These measures provide different information about the data, and which measure you choose depends on the particular characteristics of your data set.

  • Mean: Best for data sets that do not contain extreme values (outliers) and are not skewed, as it considers every value in the data set.
  • Median: This is useful for skewed data sets or when there are outliers, as it only considers the middle position, not the actual values.
  • Mode: Useful for identifying the most common items in a data set and effective with categorical data.

Practical example

Imagine you are looking at the test marks of a class of 10 students: 75, 80, 90, 100, 85, 95, 70, 60, 85, 100. Let's find the mean, median, and mode of these marks.

Meaning:

Total: 75 + 80 + 90 + 100 + 85 + 95 + 70 + 60 + 85 + 100 = 840
Number of students: 10
Mean = 840 / 10 = 84

The average mark is 84.

Median:

Arrange in order: 60, 70, 75, 80, 85, 85, 90, 95, 100, 100

There are even number of students, so find the average of the middle marks (85, 85):

Median = (85 + 85) / 2 = 85

The average mark is 85.

Method:

The numbers 85 and 100 both occur twice and are modes.

Conclusion

Measures of central tendency – mean, median, and mode – provide important insights into a data set. By understanding these measures, we can summarize and make sense of large amounts of data. Each measure has its own merits and is better suited in certain scenarios, helping us draw accurate conclusions in statistical analysis.


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Measures of Central Tendency
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Mean Median and Mode

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