The time series plot and its four components

What makes data a time series

• A time series is a set of measurements of one variable, recorded at regular intervals over time — every month, every quarter, every year.
• The order matters. Unlike most data sets, you cannot shuffle a time series without destroying the information in it.
• The intervals must be regular, and the variable must be measured the same way throughout.

The plot comes first, always

• Your first step is a time series plot: time on the horizontal axis, the variable on the vertical axis, with the points joined in time order.
• Joining the points is not decoration — it is what makes the trend and the seasonal pattern visible.
• Read the plot before you fit anything. Everything you say later comes from what you notice here.

The four components

• The additive model — the one this standard names — says every value is the sum of four parts:

$$\text{data} = \text{trend} + \text{seasonal} + \text{cyclical} + \text{irregular}$$

• In practice you work with three, since cyclical effects are rarely separable in a school data set.

Component	What it is	How to spot it
Trend	The long-term direction — rising, falling or flat	The overall slope once the zigzag is ignored
Seasonal	A pattern that repeats over a fixed period — every 4 quarters, every 12 months	The regular zigzag
Cyclical	Longer swings of no fixed length, often economic	Multi-year waves; usually not separated at this level
Irregular	What is left — random variation and one-off events	Scatter around the fitted pattern

• Seasonal and cyclical are different, and the distinction is examinable. Seasonal has a fixed, known period; cyclical does not.

Describing the trend

• Say direction, strength and whether it changes:
  • "The series shows a steady upward trend across the six years, rising from about 42 000 to about 61 000."
  • "The trend is upward until 2021, flat from 2021 to 2023, then upward again."
• Always quote values from the data. A trend statement without numbers is Achieved-level at best; with a start value, an end value and the change, it becomes evidence-based.

Describing the seasonal pattern

• Identify which period is high and which is low, and by roughly how much:
  • "Quarter 1 is consistently the highest and Quarter 3 the lowest, with a gap of about 15 000 between them."
• Then explain it from the context. This is the sentence that earns Merit:
  • "Quarter 1 is highest because it covers the New Zealand summer and the school holidays, when both domestic and overseas travel peak."

Identifying unusual features

• Look for and name any of these, then explain them:
  • Outliers — a single point far from the pattern
  • Level shifts — the series jumps to a new level and stays there
  • Changes in the seasonal pattern — the zigzag gets bigger or smaller over time
  • Changes in variability — the series becomes more or less erratic
• Real New Zealand series contain real events. A tourism series covering 2020–21 shows a collapse from border closures; an electricity series shows a spike in a cold winter. Naming the event is exactly the contextual integration Excellence asks for.

Additive or multiplicative?

• The additive model assumes the seasonal swing is roughly the same size each year — the zigzag has constant amplitude.
• If the seasonal swing grows as the series grows — the zigzag fans out — the pattern is multiplicative rather than additive.
• Check this from the plot and say what you found. Noticing that the additive model is or is not appropriate, and saying why, is a genuine Excellence observation about the adequacy of the model.