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Data Handling - Bar Graphs: Interpretation and Drawing

Grade 6ICSE

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

🔑Concepts

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A bar graph is a pictorial representation of numerical data using rectangular bars (columns) of uniform width. The height or length of each bar is proportional to the value it represents. Visually, these bars stand side-by-side with equal spacing between them on a horizontal or vertical baseline.

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Every bar graph consists of two perpendicular lines called axes: the horizontal axis (xx-axis) and the vertical axis (yy-axis). Usually, the xx-axis represents the categories or items being compared, while the yy-axis represents the numerical values or frequencies. The intersection of these axes is the starting point, often marked as 00.

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The width of the bars must be uniform throughout the graph to ensure a fair visual comparison. Similarly, the gaps between any two consecutive bars must be equal. This maintains the mathematical integrity and clarity of the data representation.

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Choosing a scale is a vital step in drawing a bar graph. A scale is the ratio between the length of the bar on paper and the actual value it represents, such as 1 unit length=10 units1\text{ unit length} = 10\text{ units} of data. The scale should be chosen such that the tallest bar fits within the available space on the graph paper.

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Data interpretation involves reading the heights of the bars against the marked scale on the yy-axis. By looking at the relative heights, one can quickly identify the maximum value (the tallest bar), the minimum value (the shortest bar), and the total sum of all observations by adding the values of all bars.

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Each bar graph must have a clear title at the top describing what the data represents, and both the xx-axis and yy-axis must be clearly labeled with the names of the categories and the units of measurement used.

📐Formulae

Length of a bar=Value of the observationValue represented by 1 unit length\text{Length of a bar} = \frac{\text{Value of the observation}}{\text{Value represented by 1 unit length}}

Scale=1 unit length=n items (where n is a constant chosen based on the data range)\text{Scale} = 1\text{ unit length} = n\text{ items (where } n \text{ is a constant chosen based on the data range)}

Actual Value=Length of bar in units×Scale factor\text{Actual Value} = \text{Length of bar in units} \times \text{Scale factor}

Difference between values=(Height of Bar A−Height of Bar B)×Scale factor\text{Difference between values} = (\text{Height of Bar A} - \text{Height of Bar B}) \times \text{Scale factor}

💡Examples

Problem 1:

The following data shows the number of students who joined different hobby clubs in a school: Music: 4040, Dance: 2525, Art: 3535, Drama: 1515. Choose a suitable scale and determine the length of the bars for each category if 1 unit=5 students1\text{ unit} = 5\text{ students}.

Solution:

  1. Choose the Scale: Given 1 unit length=5 students1\text{ unit length} = 5\text{ students}.
  2. Calculate Bar Heights:
    • For Music: 405=8 units\frac{40}{5} = 8\text{ units}
    • For Dance: 255=5 units\frac{25}{5} = 5\text{ units}
    • For Art: 355=7 units\frac{35}{5} = 7\text{ units}
    • For Drama: 155=3 units\frac{15}{5} = 3\text{ units}
  3. Drawing: Draw the xx-axis for 'Clubs' and yy-axis for 'Number of Students'. Mark points at 5,10,15,…,405, 10, 15, \dots, 40 on the yy-axis. Draw rectangular bars of the calculated heights with equal width and spacing.

Explanation:

To represent the data correctly, we divide each actual value by the scale factor to find the corresponding height of the bar in units. This ensures the graph is proportional.

Problem 2:

A bar graph shows the sale of cars in four months. The scale is 1 cm=100 cars1\text{ cm} = 100\text{ cars}. The heights of the bars are: Jan: 4 cm4\text{ cm}, Feb: 2.5 cm2.5\text{ cm}, Mar: 5 cm5\text{ cm}, Apr: 3 cm3\text{ cm}. Find the total number of cars sold in these four months.

Solution:

  1. Find individual values using the scale:
    • Jan: 4×100=400 cars4 \times 100 = 400\text{ cars}
    • Feb: 2.5×100=250 cars2.5 \times 100 = 250\text{ cars}
    • Mar: 5×100=500 cars5 \times 100 = 500\text{ cars}
    • Apr: 3×100=300 cars3 \times 100 = 300\text{ cars}
  2. Calculate Total Sale: 400+250+500+300=1450 cars400 + 250 + 500 + 300 = 1450\text{ cars} Total cars sold = 14501450.

Explanation:

Interpretation involves converting the visual bar lengths back into actual numerical data using the multiplication rule of the scale, then performing the required arithmetic operation (addition in this case).