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Statistics and Probability - Graphical Representation of Data (Bar Graphs, Histograms, Pie Charts)

Grade 8IB

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

🔑Concepts

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Categorical and Discrete Data Representation: Bar graphs are used to represent categorical data (labels like 'Red', 'Blue') or discrete numerical data (countable values like 'Number of pets'). Visually, the graph consists of rectangular bars of equal width. A defining characteristic is the uniform gap between bars, which signals that the categories are separate and distinct rather than continuous.

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Continuous Data and Histograms: For data that can take any value within a specific range (like height, mass, or time), histograms are the appropriate tool. Unlike bar graphs, the vertical bars in a histogram are adjacent to one another with no gaps, visually representing the continuous nature of the data along the horizontal axis.

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Class Intervals and Boundaries: In a histogram, data is grouped into ranges called class intervals. For example, an interval labeled 10≤x<2010 \le x < 20 means all values from 1010 up to, but not including, 2020 are counted in that bar. The x-axis scale marks these boundaries accurately to show the spread of the data.

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Pie Charts and Sector Proportions: A pie chart represents data as a circle divided into sectors (slices). The entire circle represents the total sum of all data points (100%100\% or 360∘360^{\circ}). The size of each sector—its area and the angle at the center—is directly proportional to the frequency of the category it represents.

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Frequency and Axis Scaling: On bar graphs and histograms, the vertical axis (y-axis) indicates the frequency (how often a value occurs). It is crucial that the scale on the y-axis is linear and consistent, usually starting at 00, to ensure the heights of the bars accurately reflect the differences in data values.

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Mode and Distribution Trends: Graphical representations allow for immediate visual analysis. In a bar graph or histogram, the tallest bar represents the 'modal class' or the most frequent value. In a pie chart, the largest sector identifies the dominant category. The overall shape of a histogram (e.g., symmetrical or skewed) provides insight into how data is distributed across the range.

📐Formulae

Sector Angle=Frequency of CategoryTotal Frequency×360∘\text{Sector Angle} = \frac{\text{Frequency of Category}}{\text{Total Frequency}} \times 360^{\circ}

Percentage of Total=FrequencyTotal Frequency×100%\text{Percentage of Total} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 100\%

Relative Frequency=FrequencyTotal Frequency\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Frequency}}

Frequency Density (for advanced histograms)=FrequencyClass Width\text{Frequency Density (for advanced histograms)} = \frac{\text{Frequency}}{\text{Class Width}}

💡Examples

Problem 1:

A group of 4040 students was surveyed about their favorite sports. 1010 students chose Football, 1414 chose Basketball, 88 chose Swimming, and 88 chose Tennis. Calculate the central angles required to represent this data in a pie chart.

Solution:

  1. Identify the total frequency: 10+14+8+8=4010 + 14 + 8 + 8 = 40.
  2. Calculate the angle for Football: 1040×360∘=0.25×360∘=90∘\frac{10}{40} \times 360^{\circ} = 0.25 \times 360^{\circ} = 90^{\circ}.
  3. Calculate the angle for Basketball: 1440×360∘=0.35×360∘=126∘\frac{14}{40} \times 360^{\circ} = 0.35 \times 360^{\circ} = 126^{\circ}.
  4. Calculate the angle for Swimming: 840×360∘=0.20×360∘=72∘\frac{8}{40} \times 360^{\circ} = 0.20 \times 360^{\circ} = 72^{\circ}.
  5. Calculate the angle for Tennis: 840×360∘=0.20×360∘=72∘\frac{8}{40} \times 360^{\circ} = 0.20 \times 360^{\circ} = 72^{\circ}.
  6. Verification: 90∘+126∘+72∘+72∘=360∘90^{\circ} + 126^{\circ} + 72^{\circ} + 72^{\circ} = 360^{\circ}.

Explanation:

Each frequency is converted into a fraction of the total population (4040). This fraction is then multiplied by 360∘360^{\circ} to find the portion of the circle the category occupies.

Problem 2:

The heights (in cm) of 1515 seedlings are recorded: 2,5,7,12,14,15,18,21,23,25,26,31,33,34,382, 5, 7, 12, 14, 15, 18, 21, 23, 25, 26, 31, 33, 34, 38. Create a frequency table with class intervals of width 1010 and describe the resulting histogram.

Solution:

  1. Group the data into intervals:
    • 0≤h<100 \le h < 10: 2,5,72, 5, 7 (Frequency = 33)
    • 10≤h<2010 \le h < 20: 12,14,15,1812, 14, 15, 18 (Frequency = 44)
    • 20≤h<3020 \le h < 30: 21,23,25,2621, 23, 25, 26 (Frequency = 44)
    • 30≤h<4030 \le h < 40: 31,33,34,3831, 33, 34, 38 (Frequency = 44)
  2. The x-axis will be labeled with the boundaries 0,10,20,30,400, 10, 20, 30, 40.
  3. The y-axis will show frequencies from 00 to 55.
  4. Four bars will be drawn with heights 3,4,4,43, 4, 4, 4 respectively, touching each other to show continuity.

Explanation:

Since height is continuous data, a histogram is used. Data is sorted into 'bins' or intervals. The equal width of the intervals (1010) ensures that the height of each bar accurately represents the frequency of seedlings in that height range.