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 → Presentation of Data → Graphical Form ↓


Ogive


In statistics, an ogive is a graphical representation of the cumulative distribution function or cumulative frequency distribution. It is a curve that shows the cumulative sum of frequencies at the end of each class. Ogives are useful in statistics because they allow us to see the total number of observations lying below or above a particular value in a data set.

Understanding ogives

An ogive is constructed by plotting the cumulative frequencies against the upper class limits and then connecting these points with a smooth curve. There are two types of ogives:

  • Least Ogive: It is plotted using the least cumulative frequency distribution.
  • Greater than Ogive: It is plotted using the greater than cumulative frequency distribution.

By constructing both types of ogives on the same graph, the median of the data can be found at the point where the two ogives intersect.

Step-by-step construction of ogive

Let's build "Less Than Ogive" step-by-step. We'll use a simple example dataset to illustrate this:

Suppose we have the following frequency distribution table:

Class intervalFrequency
0-105
10-208
20-3012
30-4010
40-505

Step 1: Calculate the cumulative frequency

To calculate the cumulative frequency for each class, we add the frequency of the current class to the cumulative frequency of the previous class.

Class intervalFrequencyCumulative frequency
0-1055
10-20813
20-301225
30-401035
40-50540

Step 2: Set upper class boundaries

The upper class limit is slightly higher than the upper limit of each class interval. Here it coincides with the upper limit.

  • 0-10: Upper limit = 10
  • 10-20: Upper limit = 20
  • 20-30: Upper limit = 30
  • 30-40: Upper limit = 40
  • 40-50: Upper limit = 50

Now, we plot the cumulative frequency against these upper limits.

Step 3: Plot the points

Next, plot these points using the x-axis for the upper limit and the y-axis for the cumulative frequency.

Here are the points you will mark:

  • (10, 5)
  • (20, 13)
  • (30, 25)
  • (40, 35)
  • (50, 40)

Connect these points with a smooth curve. The graph obtained is "less than ogive".

1020304050513253540upper class limitcumulative frequency

Applications and interpretations

Understanding ogives is particularly helpful in quickly identifying the median, quartiles, and percentiles of a dataset. These elements provide important insights into a data set, which can be essential in fields such as economics, business, and the social sciences.

Finding the median

To find the median using ogive, we identify the point where 50% (half) of the observations are located. This will be at the cumulative frequency position of N/2 where N is the total number of observations.

In our example, N = 40. Therefore, N/2 = 20. From the graph, find the point on the x-axis corresponding to the cumulative frequency of 20. This gives the median value.

Benefits of using toran

There are several benefits of using pylons:

  • Easy visualization of cumulative data.
  • Helps in finding the median, quartiles and percentiles.
  • Provides a quick way to compare different data sets.

While this is important, remember that ogives are primarily used for quick, visual insights. They cannot provide detailed analytical information like other statistical methods.

Conclusion

Ogives are an essential tool for graphically representing cumulative frequency distributions in statistics. By understanding how to construct and interpret ogives, you gain the power to easily analyze and draw important conclusions from statistical data. As you work with more complex datasets, ogives will become indispensable in visualizing and drawing initial insights.

In conclusion, mastering the art of drawing and interpreting ogives can greatly improve your statistical analysis abilities, laying a strong foundation for more complex statistical methods. With practice, you will find that ogives provide a simple but powerful means of understanding and presenting data.


Grade 10 → 7.3.2.4


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Frequency Polygon
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Measures of Central Tendency

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Ogive

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