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 ↓


Histogram


In the world of statistics, understanding data is very important. We often need a way to visually represent complex data sets so that trends and patterns can be quickly understood. An effective way to achieve this is to use a histogram. A histogram is a graphical representation that organizes a group of data points into user-specified categories.

Before delving into histograms, it is important to understand some key concepts like data distribution and frequency. Data distribution refers to how the values are spread in a data set. On the other hand, frequency refers to how often a data value occurs.

What is a histogram?

A histogram is a type of bar chart that shows the frequency distribution of numerical data. Unlike bar charts that display categorical data, histograms are used to present continuous data or data that comes in an ordered sequence. Each bar in a histogram, also known as a bin, shows the frequency of data within certain intervals.

Consider this simple data set: 
4, 5, 5, 6, 9, 9, 10, 10, 10, 11

Compartments: 
3-5, 6-8, 9-11

frequency:
3-5 => 3 data points (4, 5, 5)
6-8 => 1 data point (6)
9-11 => 6 data points (9, 9, 10, 10, 10, 11)

Structure of the histogram

A histogram consists of contiguous (adjacent) rectangles. It is important to note that in a histogram, the bars touch each other, indicating that the original variable is continuous. The main elements of a histogram are:

  • Axis: The x-axis usually represents intervals or bins, while the y-axis shows the frequency of data points within each bin.
  • Bars: Each bar represents a bin that contains a certain range of data. The height of the bar represents the number of data points or frequency in that range.
frequency
3
1
6
3-5
6-8
9-11
Data Bins

Creating a histogram

To create a histogram you need to follow several steps:

  1. Collect the data: First, collect the numerical data that you will display in the histogram.
  2. Decide the number of bins: Choose how many bins to use. Common methods for determining this include the square root method, where the number of bins is roughly equal to the square root of the number of data.
  3. Determine the bin width: It's important for your bins to have non-overlapping intervals. If you have n data points and k bins, a general formula for the bin width is: 
     width = (max(data) - min(data)) / k
  4. Count data points in each bin: Count the number of data points that fall into each bin.
  5. Create a histogram: Choose an appropriate scale for the axes and draw the bars accordingly.
Example dataset:
7, 8, 8, 8, 9, 10, 11, 11, 11, 12, 13, 14, 14, 15, 15

phase:
1. Number of compartments: 4
2. Width: (max - min) / bins = (15 - 7) / 4 = 2
3. Cans: 7-8.5, 8.5-10, 10-11.5, 11.5-13

frequency:
7-8.5 => 4 data points (7, 8, 8, 8)
8.5-10 => 2 data points (9, 10)
10-11.5 => 5 data points (11, 11, 11)
11.5-13 => 2 data points (12, 13)
frequency
4
2
5
2
7-8.5
8.5-10
10-11.5
11.5-13
Data Bins

Interpreting the histogram

The histogram provides a snapshot of the data distribution. The shape of the histogram can tell us a lot about the underlying data distribution. Here are some common patterns that can be observed:

  • Symmetrical distribution: The histogram is approximately the same on either side of the center. A classic bell-shaped curve is known as a normal distribution.
  • Skewed distribution: The histogram is tilted to one side. If it is tilted to the left, it is positively skewed; if it is tilted to the right, it is negatively skewed.
  • Uniform distribution: All bars are approximately the same height; the data has no obvious mode.
  • Multimodal distribution: More than one peak in a histogram, indicating several major groups of data.

Advantages and disadvantages of histogram

Benefit

  • Gives a clear visual representation of the data distribution.
  • It helps us to identify the shape of the data, whether it is normal, skewed or uniform.
  • Very useful for large data sets.

Loss

  • Not suitable for small data sets as it may not accurately reflect the distribution.
  • The choice of bin number can affect the interpretation of the data.

Conclusion

In summary, histograms are a powerful tool in statistics for visually representing continuous data. They provide insight into distributions and can efficiently summarize large data sets. Understanding how to read and interpret histograms can help perform detailed data analysis and make informed decisions based on statistical data.


Grade 10 → 7.3.2.2


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Bar Graph
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Frequency Polygon

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Histogram

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