Charts for Discrete and Time Series Data

Constructing a Vertical Line Chart for Test Scores

Score (out of 10)	5	6	7	8	9
Frequency	2	5	8	4	1

Step 1: Label Axes.

• Horizontal ($x$) = "Score"; Vertical ($y$) = "Frequency".

Step 2: Set Scale.

• Number the $y$-axis from 0 to 10 in increments of 1. Mark the $x$-axis with equal spacing for values 5 through 9.

Step 3: Plot.

• Use a ruler to draw a single thin line at $x=5$ up to $y=2$. Repeat for all scores (e.g., at $x=7$, the line reaches a height of 8).

Step 4: Analysis.

• The mode is 7. The total frequency is $2 + 5 + 8 + 4 + 1 = 20$.

Time Series Graphs

Tracking how temperature changes throughout the day requires a different approach to simply counting totals. A time series graph is a line graph showing how a quantity changes over a specific period of time. Time is always plotted on the horizontal ($x$) axis, while the quantitative variable is plotted on the vertical ($y$) axis.

Data points must be plotted at regular time intervals and connected with straight line segments using a ruler. If the data values are far from zero, a zigzag (break) should be used on the $y$-axis to avoid a misleading representation.

Plotting Temperature over 24 Hours

Data: (00:00, 8°C), (04:00, 6°C), (08:00, 11°C), (12:00, 18°C), (16:00, 20°C), (20:00, 14°C), (24:00, 9°C).

Step 1: Label Axes and Scale.

• $x$-axis = Time (0 to 24, steps of 4); $y$-axis = Temp °C (0 to 20).

Step 2: Plotting.

• Mark each coordinate with a small cross ($\times$).

Step 3: Joining.

• Connect $(00:00, 8)$ to $(04:00, 6)$ with a straight ruler line, continuing for all points.

Step 4: Result.

• The graph shows a "trough" (lowest point) at 04:00 and a "peak" (highest point) at 16:00.

Interpreting Time Series Trends

A single unusually high sales week does not necessarily mean a business is growing. Once a time series graph is drawn, it is used to identify long-term patterns and separate them from short-term noise. A trend is the long-term direction of data, ignoring short-term fluctuations, and can be described as rising, falling, or level.

Seasonal variation refers to predictable patterns that repeat at regular intervals, such as daily or quarterly. The difference between a specific data point and the general trend is found using the formula $\text{Seasonal Variation} = \text{Actual Value} - \text{Trend Line Value}$. To smooth out short-term fluctuations and reveal the underlying trend, a moving average is calculated. This is the average of a fixed number of consecutive data points.

The calculated average must be plotted exactly at the midpoint of the time intervals used. Extending the trend line to predict future values is called extrapolation. This technique is explicitly considered unreliable because established patterns may change unexpectedly.

Analysing a Sales Trend and Making a Prediction

A shop's umbrella sales peak every Q4 (Winter) and trough every Q2 (Summer).

Step 1: Identify Trend.

• If the trend line rises from 100 to 150 units over 3 years, describe it as a "general rising trend".

Step 2: Identify Seasonal Pattern.

• Note the context: "Sales peak in the 4th quarter each year".

Step 3: Predicting.

• To predict Year 4, Q4: find the value on the trend line for that time and add the mean seasonal variation for all Q4s.

Choosing the Right Chart

With so many ways to display data, how do you select the most appropriate format? The choice of statistical diagram depends entirely on whether your data is qualitative or quantitative. Exams often ask you to justify your choice, so you must know the specific purpose of each chart type.

For qualitative data (non-numerical categories like colours or car types), use pie charts to compare proportions or bar charts to compare specific totals. Pictograms can also be used, but they must always include a key. For discrete quantitative data (counted numerical values), use vertical line charts or bar charts, ensuring equal widths and equal gaps between bars.

For continuous numerical data, use line graphs to show change over time, or histograms where the bars must touch. Always evaluate charts for misleading features. A graph is misleading if it is missing titles or labels, if the $y$-axis does not start at zero (which exaggerates differences), or if bar widths and gaps are unequal.

Calculating an Angle for a Pie Chart

$$\text{Angle} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ$$

If 15 out of 60 people choose "Apples":

Step 1: Substitute values into the formula.

• $\text{Angle} = \frac{15}{60} \times 360^\circ$

Step 2: Calculate.

• $\text{Angle} = 0.25 \times 360^\circ = 90^\circ$