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Data Handling - Bar Graphs and Histograms

Grade 8ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Data Representation: Raw data is often disorganized. To make sense of it, we use Frequency Distribution Tables. For discrete categories (like favorite colors), we use Bar Graphs. For continuous grouped data (like height or weight intervals), we use Histograms.

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Bar Graphs: These consist of rectangular bars of equal width with uniform spacing between them. The length or height of each bar is proportional to the frequency it represents. Visually, the bars are separate entities standing on a horizontal (x-axis) or vertical (y-axis) line, representing distinct categories.

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Histograms: Unlike bar graphs, histograms are used for continuous class intervals. The bars are drawn adjacent to each other with no gaps between them. The width of the rectangle represents the class interval, and the height represents the frequency. Visually, it looks like a solid block of bars of varying heights connected at their boundaries.

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Class Intervals and Limits: Grouped data is divided into intervals like 10−20,20−3010-20, 20-30. In the interval 10−2010-20, 1010 is the Lower Class Limit and 2020 is the Upper Class Limit. In the exclusive method, the value 2020 is included in the next class (20−3020-30) rather than the current one.

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The Kink (Broken Line): If the class intervals do not start from zero, a 'kink' or a 'broken line' (represented as a zig-zag line) is drawn on the horizontal axis between the origin (0,0)(0,0) and the first class limit. This indicates that the scale on the axis is not continuous from zero.

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Class Mark (Mid-value): This is the central value of a class interval. It is used in various statistical calculations and represents the entire class. On a graph, the class mark would be the point exactly in the middle of the width of a histogram bar.

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Frequency: This refers to the number of times a particular observation or data point occurs within a specific category or class interval. In both Bar Graphs and Histograms, frequency is typically plotted on the vertical y-axis.

📐Formulae

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Class Mark=Upper Limit+Lower Limit2\text{Class Mark} = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}

Class Size=Upper Limit−Lower Limit\text{Class Size} = \text{Upper Limit} - \text{Lower Limit}

Frequency Density=Frequency of the classClass Width\text{Frequency Density} = \frac{\text{Frequency of the class}}{\text{Class Width}}

💡Examples

Problem 1:

Given the class intervals 10−20,20−30,30−40,40−5010-20, 20-30, 30-40, 40-50, find the class mark for the third class and the class size.

Solution:

  1. Identify the third class interval: 30−4030-40.
  2. Identify limits: Lower Limit (LL) = 3030, Upper Limit (UU) = 4040.
  3. Calculate Class Mark: Class Mark=30+402=702=35\text{Class Mark} = \frac{30 + 40}{2} = \frac{70}{2} = 35.
  4. Calculate Class Size: Class Size=U−L=40−30=10\text{Class Size} = U - L = 40 - 30 = 10.

Explanation:

The class mark is the average of the boundaries, and the class size is the uniform width of the intervals used for the histogram base.

Problem 2:

The weights (in kg) of 5 students are: 45,52,48,60,5545, 52, 48, 60, 55. Calculate the range of this data.

Solution:

  1. Find the Maximum value: 6060.
  2. Find the Minimum value: 4545.
  3. Apply Range formula: Range=Max−Min\text{Range} = \text{Max} - \text{Min}.
  4. Calculate: 60−45=1560 - 45 = 15. Result: The range is 1515 kg.

Explanation:

Range gives the spread of the data, which helps in deciding the scale for the axes when drawing a bar graph or histogram.