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How Quantities Combine: Understanding Data - An Alternative to Pie Chart

Grade 9CBSE

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

🔑Concepts

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When comparing parts of a whole, a Pie Chart uses circular sectors where the central angle is proportional to the quantity. An alternative is the Segmented Bar Chart or Stacked Bar Chart, which represents the whole as a single bar of 100% height, divided into segments based on proportions.

A segmented bar chart showing proportions of a whole.
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The central angle θ\theta for a component in a pie chart is calculated by the formula: Central Angle=Value of ComponentTotal Value×360∘\text{Central Angle} = \frac{\text{Value of Component}}{\text{Total Value}} \times 360^\circ. This helps in visualizing the relative size of data points.

A circle showing a 120 degree sector.
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Data can also be visualized using Frequency Polygons. This is an alternative to histograms where mid-points of the upper sides of the rectangles are joined by line segments. It provides a clearer view of the 'shape' or 'distribution' of the data.

A frequency polygon showing data distribution.
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To compare two sets of data relative to a total, we use a Double Bar Graph. This allows side-by-side comparison of quantities across categories without losing the individual context of each data set.

📐Formulae

Central Angle of a Component=(Value of the componentTotal value×360)∘\text{Central Angle of a Component} = \left( \frac{\text{Value of the component}}{\text{Total value}} \times 360 \right)^\circ

Percentage of a Component=Value of the componentTotal value×100%\text{Percentage of a Component} = \frac{\text{Value of the component}}{\text{Total value}} \times 100\%

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

💡Examples

Problem 1:

A student spends Rs 3600 per month. The expenses are: Food (Rs 1200), Rent (Rs 900), Education (Rs 600), and Others (Rs 900). Calculate the central angle for each category to represent this in a pie chart.

A pie chart showing central angles for expenditure.

Solution:

Total Expenditure = Rs 3600. Food: 12003600×360∘=120∘\frac{1200}{3600} \times 360^\circ = 120^\circ Rent: 9003600×360∘=90∘\frac{900}{3600} \times 360^\circ = 90^\circ Education: 6003600×360∘=60∘\frac{600}{3600} \times 360^\circ = 60^\circ Others: 9003600×360∘=90∘\frac{900}{3600} \times 360^\circ = 90^\circ Sum: 120∘+90∘+60∘+90∘=360∘120^\circ + 90^\circ + 60^\circ + 90^\circ = 360^\circ.

Explanation:

To find the angle, we divide the specific expense by the total expense and multiply by the total degrees in a circle (360).

Problem 2:

Represent the following percentage data of a family's budget as a segmented bar chart: Savings (20%), Clothing (10%), Food (40%), Others (30%).

Segmented bar chart for family budget.

Solution:

Draw a vertical bar of total length representing 100 units. Segment 1 (Savings): 0 to 20 units. Segment 2 (Clothing): 20 to 30 units (10%). Segment 3 (Food): 30 to 70 units (40%). Segment 4 (Others): 70 to 100 units (30%).

Explanation:

In a segmented bar chart, the height of each segment corresponds to its percentage of the total.