Graphical Representation of Data: Basic Charts

Pictograph: Definition and Construction

Definition and Purpose

A pictograph, also known as a pictogram or picture graph, is a simple and intuitive way to represent data using relevant pictures or symbols. Each chosen symbol is assigned a fixed value, representing a certain quantity or frequency of the item being depicted. The data is then illustrated by drawing a number of these symbols proportional to the values they represent.

Pictographs are particularly effective for:

• Representing data in a visually appealing and easy-to-understand manner.
• Comparing discrete quantities across different categories.
• Communicating simple statistical information to a general audience, including children.

However, pictographs have limitations, especially when dealing with large numbers or values that require precise fractional symbols, which can make them less accurate compared to other graphical methods like bar graphs.

Key Elements of a Pictograph

A well-constructed pictograph includes the following essential elements:

• Title:

  A clear and concise title that describes the data being presented.

• Categories:

  The different groups or classifications into which the data falls (e.g., days of the week, types of fruits, names of cities).

• Symbol:

  The picture or icon chosen to represent the data. The symbol should be relevant to the context of the data (e.g., using a car symbol to represent car sales).

• Key or Legend:

  This is a crucial part that explicitly states the quantity or value that each single symbol represents. Choosing an appropriate value for the symbol (scale) is important to keep the number of symbols manageable.

• Rows or Columns of Symbols:

  Arrangement of symbols corresponding to each category. The number of symbols in a row or column indicates the frequency or value for that category.

Construction of a Pictograph

Follow these steps to construct a pictograph from a given dataset:

1. Understand the Data: Look at the data provided. Identify the categories and the corresponding numerical values or frequencies.
2. Choose a Suitable Symbol: Select a picture or symbol that is relevant to the data you are representing. For example, if the data is about students, a symbol of a person is suitable.
3. Determine the Value of Each Symbol (Set the Key/Scale): Decide on a value that each single symbol will represent. This value should be chosen carefully to make the pictograph easy to read and draw. Look for common factors in the data values. If the values are large, choose a larger value per symbol (e.g., 10, 100, 1000). If the values are small, a smaller value per symbol might be appropriate (e.g., 1, 5). State this value clearly in the key.
4. Calculate the Number of Symbols Required: For each category, divide the frequency or value of that category by the value assigned to one symbol.

   $\text{Number of Symbols for a Category} = \frac{\text{Value/Frequency of the Category}}{\text{Value represented by one symbol}}$

   If the result is not a whole number, you may need to use parts of a symbol (e.g., half or quarter of a symbol) to represent the fractional part, provided the chosen value allows for easy division (e.g., choosing 10 allows for representing 5 using half a symbol).

5. Draw the Pictograph: Draw the axes (optional but can make it clearer) or simply list the categories. Draw the calculated number of symbols next to each category label. Ensure the symbols are aligned and equally spaced within each row or column to facilitate comparison.
6. Add Title and Key: Place a suitable title at the top of the pictograph. Create a key, usually at the bottom or side, showing the symbol and the exact value it represents. If using partial symbols, explain what each partial symbol represents.

Example

Answer:

Bar Graphs: Definition and Construction (Single Bar Graphs)

Definition and Purpose

A bar graph or bar chart is a visual representation of data using rectangular bars. These bars have a uniform width, and their lengths (or heights, if vertical) are proportional to the values or frequencies they represent. Bar graphs are one of the most common types of graphs used because they are easy to create and interpret.

Bar graphs are particularly useful for:

• Comparing the magnitude of different categories at a glance.
• Showing discrete data clearly.
• Highlighting differences between groups.
• Tracking changes over discrete periods of time (though line graphs are often preferred for continuous time series).

The bars can be drawn either vertically or horizontally. A single bar graph represents one set of data for different categories.

Construction of a Vertical Bar Graph

To construct a vertical bar graph (the most common type), follow these steps:

1. Draw the Axes: Draw two perpendicular lines, one horizontal (the x-axis) and one vertical (the y-axis), intersecting at a point called the origin (usually labelled 0).
2. Label the Axes:
   • The horizontal axis (x-axis) is typically used for the categories or classes of the data (e.g., names of items, months, student names).
   • The vertical axis (y-axis) is typically used for the numerical values or frequencies corresponding to each category.
   Label each axis clearly with what it represents.
3. Choose a Scale for the Vertical Axis: This is a crucial step. Select an appropriate scale for the y-axis based on the range of your data values (the minimum and maximum frequencies). The scale should start from 0. Choose equal increments (intervals) along the y-axis such that the largest value in the data fits comfortably on the graph. The smaller the interval, the more precise the reading, but too small an interval might make the graph too large. The increments must be uniform (e.g., 0, 10, 20, 30... or 0, 5, 10, 15...).
4. Draw the Bars: For each category marked on the x-axis, draw a rectangular bar.
   • The width of all bars must be the same.
   • The height of each bar must be proportional to the frequency or value of the corresponding category, as determined by the scale on the y-axis.
   • Leave equal space between consecutive bars. This space should ideally be uniform and less than the width of the bars themselves. The first bar might start a small distance away from the y-axis.
5. Add a Title: Give the bar graph a suitable title that clearly indicates what the graph represents. The title is usually placed at the top of the graph.

Note: For a horizontal bar graph, the categories are placed on the y-axis, the numerical values on the x-axis, and the bars are drawn horizontally.

Example

Answer:

Given: Data showing the count of students in five different school clubs.

To Represent: Data using a vertical bar graph.

Solution:

We will construct a vertical bar graph following the steps:

• X-axis: We will label the horizontal axis "School Clubs" and mark points for each club: Music, Dance, Drama, Art, Debate.
• Y-axis: We will label the vertical axis "Number of Students".
• Scale: The highest frequency is 70. We can choose a scale with intervals of 10 on the y-axis, going from 0 up to 80 (or at least 70). Let's use intervals: 0, 10, 20, 30, 40, 50, 60, 70, 80.
• Bars: We draw bars of equal width for each club, with heights corresponding to the number of students:
  • Music: Bar height up to 60.
  • Dance: Bar height up to 40.
  • Drama: Bar height up to 50.
  • Art: Bar height up to 70.
  • Debate: Bar height up to 30.
  Ensure equal spacing between bars.
• Title: The title will be "Number of Students in School Clubs".

Title: Number of Students in School Clubs

Double Bar Graphs: Construction and Comparison

Definition and Purpose

A double bar graph, also known as a comparative bar graph, is a graphical tool used to display and compare two sets of related data simultaneously across the same categories. It consists of pairs of rectangular bars, typically placed side-by-side for each category. Each bar within a pair represents a value from one of the two datasets.

The key purpose of a double bar graph is to facilitate direct visual comparison:

• Comparing the two values within a single category (e.g., comparing performance in Subject A in Term 1 vs. Term 2).
• Comparing values across different categories within the same dataset (e.g., comparing Term 1 performance in English vs. Maths).
• Observing trends or differences between the two datasets across all categories.

Like single bar graphs, double bar graphs use bars of uniform width, and their heights (or lengths, if horizontal) are proportional to the values they represent. A clear legend is essential to distinguish between the two sets of data.

Construction of a Double Bar Graph

Constructing a double bar graph follows similar steps to a single bar graph, with the added complexity of representing two datasets:

1. Draw and Label Axes: Draw two perpendicular axes, the horizontal (x-axis) and the vertical (y-axis), intersecting at the origin (0). The x-axis will typically represent the categories, and the y-axis will represent the numerical values or frequencies. Label both axes clearly.
2. Choose a Scale for the Value Axis: Select an appropriate scale for the axis representing values (usually the y-axis for vertical bars). The scale should start at 0 and cover the range of values from both datasets. Choose equal increments along the axis to ensure accurate representation.
3. Draw Paired Bars: For each category on the category axis (x-axis), draw a pair of rectangular bars side-by-side.
   • One bar in the pair represents the value from the first dataset for that category.
   • The other bar represents the value from the second dataset for the same category.
   All bars in the graph should have the same uniform width. The two bars within a pair are usually drawn touching or very close together.
4. Maintain Spacing: Leave an equal amount of space between different pairs of bars (i.e., between one category's pair of bars and the next category's pair of bars). The width of these spaces should be consistent throughout the graph.
5. Create a Key/Legend: Since there are two sets of data, a key (or legend) is necessary to identify which bar in each pair corresponds to which dataset. This is usually done by using different colours, shading patterns, or textures for the bars of each dataset and explaining what each colour/pattern signifies.
6. Add a Title: Give the double bar graph a descriptive title that clearly indicates the comparison being made.

Note: Double bar graphs can also be drawn horizontally by swapping the roles of the x and y axes.

Example

Answer:

Given: Marks of a student in four subjects for Term 1 and Term 2.

To Represent: Data using a double bar graph and interpret the results.

Solution:

We will construct a vertical double bar graph:

• X-axis: Subjects (English, Maths, Science, History).
• Y-axis: Marks Obtained.
• Scale: The marks range from 60 to 85. We can choose a scale with increments of 10 on the y-axis, starting from 0 up to 90 or 100. Let's use: 0, 10, 20, 30, 40, 50, 60, 70, 80, 90. (Note: While we can start the scale from a higher value like 50 and use a zig-zag line on the axis to indicate that the axis is broken, starting from 0 is generally preferred for better comparison of magnitudes).
• Bars: For each subject, we draw two adjacent bars: one for Term 1 marks and one for Term 2 marks. Let's use blue for Term 1 bars and green for Term 2 bars.
  • English: Blue bar height 65, Green bar height 70.
  • Maths: Blue bar height 80, Green bar height 85.
  • Science: Blue bar height 75, Green bar height 70.
  • History: Blue bar height 60, Green bar height 68.
  Ensure uniform bar width and equal space between the pairs of bars for different subjects.
• Key: A key indicating "Blue bar = Term 1 Marks" and "Green bar = Term 2 Marks" is required.
• Title: A suitable title is "Comparison of Student's Term 1 and Term 2 Marks".

Interpretation:

The double bar graph allows for easy comparison of the student's performance between the two terms for each subject. By looking at the paired bars:

• We can see that the student's marks increased in English, Maths, and History from Term 1 to Term 2 (the green bars are taller than the blue bars).
• The most significant improvement appears to be in History (from 60 to 68).
• The marks decreased slightly in Science (from 75 to 70).
• Maths remained the student's highest scoring subject in both terms.

We can also compare the relative performance across subjects within each term by looking at the heights of the blue bars (Term 1) or the green bars (Term 2) independently.

Pie Chart: Construction and Interpretation

Definition and Purpose

A pie chart, also known as a circle graph, is a circular statistical graphic that is divided into slices, illustrating numerical proportion. In a pie chart, the entire circle represents the whole (100% or the total sum of all values), and each slice (sector) represents a part of that whole. The size of each slice – measured by its central angle and its area – is directly proportional to the quantity or frequency it represents.

Pie charts are most effective for:

• Showing the composition of a whole, i.e., how different parts contribute to the total.
• Displaying percentage distribution clearly.
• Comparing the relative size of each component to the whole and to each other.

Pie charts are best used when the number of categories is small (ideally not more than 5-7), and when you want to emphasize the relationship of each part to the whole rather than precise comparisons between parts.

Construction of a Pie Chart

To construct a pie chart, you need to convert the data values or frequencies into corresponding angles of the sectors in a circle. A circle consists of $360^\circ$.

Follow these steps:

1. Calculate the Total: Sum up the values of all the categories. This sum represents the total value or the whole (which corresponds to $360^\circ$ in the circle).
2. Calculate the Proportion or Percentage for Each Category: For each category, determine its proportion of the total:

   $\text{Proportion for Category} = \frac{\text{Value for the Category}}{\text{Total Value}}$

   You can optionally calculate the percentage for each category by multiplying the proportion by 100%.

   $\text{Percentage for Category} = \left(\frac{\text{Value for the Category}}{\text{Total Value}}\right) \times 100\%$

   The sum of all proportions should be 1, and the sum of all percentages should be 100%.

3. Calculate the Central Angle for Each Category: Since the total circle angle is $360^\circ$, the angle for each sector is the proportion of that category multiplied by $360^\circ$.

   $\text{Central Angle} = \left(\frac{\text{Value for Category}}{\text{Total Value}}\right) \times 360^\circ$

   ... (1)

   Alternatively, using percentages:

   $\text{Central Angle} = \left(\frac{\text{Percentage}}{100}\right) \times 360^\circ$

   ... (2)

   The sum of all calculated central angles must be exactly $360^\circ$.

4. Draw the Circle: Draw a circle of a convenient radius using a compass.
5. Draw the Sectors:
   • Draw a radius from the center of the circle to the circumference (usually starting vertically upwards or horizontally to the right for consistency).
   • Using a protractor, measure the calculated central angle for the first category from this radius and draw the second radius to form the first sector.
   • From the new radius drawn, measure the central angle for the second category and draw the next radius.
   • Continue this process for all categories. The last sector should naturally complete the circle.
6. Label the Slices: Label each sector clearly with the name of the category it represents. It is also common practice to write the actual value or the percentage for that category within or next to the slice. Use different colors or shading for different slices to enhance clarity.
7. Add a Title: Give the pie chart a descriptive title that clearly indicates the data being represented.

Example

Answer:

Given: Number of students favouring different sports.

To Represent: Data using a pie chart.

Solution:

We follow the steps for constructing a pie chart:

1. Calculate the Total:

   Total number of students = Number of students in Football + Cricket + Basketball + Tennis

   Total = $12 + 9 + 8 + 7 = 36$ students.

   This total of 36 students corresponds to $360^\circ$ of the circle.

2. Calculate the Proportion/Percentage and Central Angles:

   We calculate the central angle for each sport using the formula: Central Angle $= \left(\frac{\text{Number of Students for Sport}}{\text{Total Number of Students}}\right) \times 360^\circ$

   • Football:

     Angle for Football $= \left(\frac{12}{36}\right) \times 360^\circ = \left(\frac{1}{3}\right) \times 360^\circ = 120^\circ$

   • Cricket:

     Angle for Cricket $= \left(\frac{9}{36}\right) \times 360^\circ = \left(\frac{1}{4}\right) \times 360^\circ = 90^\circ$

   • Basketball:

     Angle for Basketball $= \left(\frac{8}{36}\right) \times 360^\circ = \left(\frac{2}{9}\right) \times 360^\circ = 80^\circ$

   • Tennis:

     Angle for Tennis $= \left(\frac{7}{36}\right) \times 360^\circ = \left(\frac{7}{36}\right) \times (36 \times 10)^\circ = 7 \times 10^\circ = 70^\circ$

   Check: Sum of angles $= 120^\circ + 90^\circ + 80^\circ + 70^\circ = 360^\circ$. This confirms our calculations are correct.

3. Draw Circle and Sectors: Draw a circle. Using a protractor, draw sectors with these calculated angles ($120^\circ$, $90^\circ$, $80^\circ$, $70^\circ$) starting from a reference radius.
4. Label Slices: Label each sector with the sport name and the number of students (or percentage). For example, the $120^\circ$ sector is labelled "Football (12)".
5. Add a Title: Add the title "Favourite Sports of 36 Students".

Interpretation:

The pie chart visually represents how the preference for different sports is distributed among the 36 students. The size of each sector clearly shows the popularity of each sport relative to the total. We can easily see that Football is the most popular sport (largest sector), followed by Cricket, then Basketball, and Tennis is the least favourite sport among the four. The pie chart effectively illustrates the proportion of students who prefer each sport out of the total class.