Charts for Categorical and Discrete Data

Extracting Frequencies from a Bar Chart Scale

A bar representing "Green Bicycles" reaches a height on the $y$-axis scale where the number $50$ is marked. Four small squares above this, the number $70$ is marked. The bar ends $3$ small squares above the $50$ line. Calculate the frequency of Green Bicycles.

Step 1: Calculate the value of one small square on the bar chart scale.

• $\text{Value per division} = \frac{\text{Upper Label} - \text{Lower Label}}{\text{Number of Squares}}$

• $\text{Value per division} = (70 - 50) \div 4 = 5$ units per square

Step 2: Read the bar height.

• $\text{Frequency} = \text{Base Label} + (\text{Number of Squares} \times \text{Scale Unit})$

• $\text{Frequency} = 50 + (3 \times 5) = 65$ bicycles

Representing Data with Pie Charts

• 3D pie charts are often used in media, but they are statistically misleading because the 3D perspective distorts the size of the sectors.
• In a standard 2D pie chart, the size of each slice is directly proportional to the frequency of the category.
• A complete pie chart represents a total frequency corresponding to $360^\circ$.
• The central angle of each slice is called the sector angle.

Calculating Pie Chart Sector Angles

In a survey of $90$ people, $15$ chose "Tennis" as their favorite sport. Calculate the sector angle for Tennis.

Step 1: State the formula.

Step 2: Substitute the values.

• $\text{Sector Angle} = \frac{15}{90} \times 360^\circ$

Step 3: Calculate the final answer.

• $\text{Sector Angle} = \frac{1}{6} \times 360^\circ = 60^\circ$

Representing Data with Pictograms

• Imagine tracking the number of cars sold by a dealership over a week using a simple circle symbol for every 8 sales.
• A pictogram is a statistical diagram that uses symbols or pictures to represent the frequency of data.
• Every pictogram must include a pictogram key defining exactly what one whole symbol represents.
• If a whole symbol represents $n$ items, a half-symbol represents $n \div 2$, and a quarter-symbol represents $n \div 4$.

Determining Values in a Pictogram

A pictogram shows the number of pizzas sold. The key states that one pizza symbol ($\bigcirc$) represents $12$ pizzas. On Friday, the pictogram shows $3$ whole pizza symbols and one quarter of a pizza symbol. Calculate the total number of pizzas sold on Friday.

Step 1: State the logic for the calculation.

Step 2: Substitute the values.

• $\text{Total Frequency} = (3 \times 12) + (\frac{1}{4} \times 12)$

Step 3: Calculate the final answer.

• $\text{Total Frequency} = 36 + 3 = 39$ pizzas

Constructing Vertical Line Charts

• Unlike standard bar charts that use wide rectangles, vertical line charts use thin lines with no width to show that data exists only at specific points, not on a continuum.
• A vertical line chart is used exclusively for discrete numerical data (e.g., test scores, goals scored).
• The lines must not touch each other, making them ideal for discrete data with many outcomes where a standard bar chart would appear too cluttered.
• To ensure full marks on OCR constructions, remember the SALT mnemonic:
  • Scale: The vertical scale must start at $0$ and increase in equal increments.
  • Axes: Must be drawn straight using a ruler. The vertical lines must be perpendicular lines (drawn at $90^\circ$) to the horizontal $x$-axis.
  • Labels: Proper axis labeling is mandatory (e.g., the $y$-axis must be explicitly labeled "Frequency").
  • Title: Must describe the data shown.

Constructing a Vertical Line Chart and Calculating the Mode

A teacher records the number of absences per student in a term: $0$ absences ($4$ students), $1$ absence ($7$ students), $2$ absences ($3$ students). Construct a vertical line chart and calculate the modal value and total frequency.

Step 1: Identify and perform axis labeling.

• The horizontal axis ($x$) is labeled "Number of Absences" with discrete values $0, 1, 2$.
• The vertical axis ($y$) is labeled "Frequency" and starts at $0$.

Step 2: Choose scale and plot.

• The maximum frequency is $7$. Create a linear scale on the $y$-axis from $0$ to $7$.
• At $x = 0$, draw a thin vertical line up to $4$ on the $y$-axis using a ruler.
• At $x = 1$, draw a vertical line up to $7$.
• At $x = 2$, draw a vertical line up to $3$.

Step 3: Calculate the modal value.

• The mode is the value on the $x$-axis corresponding to the tallest line.
• The tallest line is at $x = 1$.
• $\text{Modal Value} = 1$ absence

Step 4: Calculate the total frequency.

• Sum the heights of all the lines.
• $\text{Total Frequency} = 4 + 7 + 3 = 14$ students