Charts for Discrete and Time Series Data

Constructing a Vertical Line Chart

When drawing a vertical line chart, accuracy is essential. The vertical frequency scale must start at zero and use equal, consistent intervals. A broken axis (starting higher than 0) is generally not permitted.

Step 1: Create a frequency table from your raw data to organise the outcomes. Step 2: Using a pencil and ruler, draw and label the x-axis with the outcomes and the y-axis with "Frequency". Step 3: Ensure your y-axis starts at zero, and leave equal gaps between the outcome values on the x-axis. Step 4: Draw a single, straight vertical line for each outcome up to the required frequency height.

Interpreting Data from a Vertical Line Chart

You can extract all key statistical measures directly from the chart. The total frequency is found by summing the heights of all the vertical lines: $\sum \text{frequencies} = \text{Total}$.

The mode is the most common outcome, which is simply the value on the x-axis that has the tallest line. The range is calculated as the highest outcome value minus the lowest outcome value that has a drawn line.

To find the mean, you multiply each outcome by its frequency, sum the results, and divide by the total frequency.

$$\text{Mean} = \frac{\sum (\text{Outcome} \times \text{Frequency})}{\text{Total Frequency}}$$

Worked Example: Calculating the Mean

A vertical line chart shows the number of goals scored by a team. There are 3 lines at '0 goals' (height 3), '1 goal' (height 4), and '2 goals' (height 8).

Step 1: Multiply each outcome by its frequency.

• $(0 \times 3) + (1 \times 4) + (2 \times 8) = 0 + 4 + 16 = 20$

Step 2: Find the total frequency.

• $3 + 4 + 8 = 15$

Step 3: Divide the total from Step 1 by the total frequency.

• $\text{Mean} = \frac{20}{15} = 1.33$ goals.

What is Time Series Data?

How do supermarkets know when to stock up on barbecue charcoal? They track sales data over time to spot repeating summer peaks. This is an example of a time series.

A time series graph plots time on the horizontal x-axis (such as days, months, or years) and the measured variable on the vertical y-axis. It is used to identify a long-term trend, predictable seasonal variation, or irregular short-term fluctuations.

Constructing Tables and Time Series Graphs

Before drawing a graph, data must be organised in a table with uniform time intervals. When designing a table, every column and row must have clear headings, and units must be placed in the heading only (e.g., "Time ($s$)"), never in the data cells.

When plotting a time series graph, data points must be marked with small crosses ($\times$). If the data values are very far from zero, you can use a zigzag line (an axis break) on the y-axis to focus on the relevant range.

Crucially, points must be joined chronologically using straight line segments drawn with a ruler. Do not draw freehand curves unless specifically asked for a curve of best fit.

Worked Example: Plotting and Interpreting a Time Series Graph

A table shows ice cream sales over two years:

• 2022: Q1 (£10k), Q2 (£35k), Q3 (£50k), Q4 (£15k)
• 2023: Q1 (£12k), Q2 (£40k)

Step 1: Label the x-axis "Time (Year/Quarter)" and the y-axis "Sales (£1000s)". Step 2: Scale the y-axis from 0 to 60 in increments of 10, and mark equal intervals for each Quarter on the x-axis. Step 3: Plot small crosses at the coordinates: (2022 Q1, 10), (2022 Q2, 35), (2022 Q3, 50), etc. Step 4: Connect the points chronologically from left to right using a ruler.

By looking at the connected lines, we can interpret the data. There is clear seasonal variation because peaks occur every Q3 (summer). There is also a slight upward trend, as Q1 and Q2 sales in 2023 are higher than the corresponding quarters in 2022.