Sample Notes: Time Series

O Level and IGCSE Statistics – Detailed Notes: 10. Time Series

10.1 Understanding of Trend

Definition of Trend

• A trend is the general direction in which data points are moving over time.
• It reveals the long-term increase, decrease, or stability in the data.
• Identified by plotting data on a time graph or line chart.

Types of Trends

• Upward Trend: Values are generally increasing over time.
• Downward Trend: Values are generally decreasing over time.
• No Trend: Values fluctuate but show no consistent direction.
• Cyclical/Seasonal Pattern with Trend: Regular fluctuation in data, but with a visible long-term movement.

Example:

• Here, there is an upward trend in sales.

10.2 Understanding of Seasonal Variation

Definition

• Seasonal variation refers to periodic fluctuations in data that recur at regular intervals due to seasonal effects (e.g. months, quarters).
• These variations are superimposed on the trend.

Sources of Seasonal Variation

• Weather: e.g. higher ice cream sales in summer.
• Festivals/Events: e.g. higher retail sales in December.
• School terms/Academic calendar
• Economic cycles

Graphical Representation

• On a time series graph, seasonal variations appear as regular spikes and drops around the trend line.

Moving Averages

Purpose

• To smooth out short-term fluctuations in data to better observe the underlying trend.

Simple Moving Average (SMA)

• Calculated by taking the average of a fixed number of consecutive values in the series.

Example: 3-point moving average

Centered Moving Average

• Used when the number of periods in the moving average is even.
• Averages two consecutive moving averages to center them.

Example of Centering:

Suppose the 4-point moving averages are:

• (Q1 to Q4) = 28
• (Q2 to Q5) = 30
  Then centered moving average at Q2.5 = (28 + 30) / 2 = 29

Seasonal Variation Analysis

Calculating Mean Seasonal Variation

1. Determine trend values using moving averages.
2. Subtract trend values from actual data to get seasonal components.
   • Seasonal Variation = Actual Data – Trend
3. Group seasonal variations by period (e.g. Q1, Q2…)
4. Calculate the average for each period over multiple years.

Example:

If Q1 data over 4 years:
Actual: 110, 115, 120, 125
Trend values: 108, 112, 118, 123
Seasonal components: 2, 3, 2, 2 → Mean seasonal variation for Q1 = (2+3+2+2)/4 = 2.25

Using Trend and Seasonal Component in Prediction

Forecasting Future Values

• Predicted value = Trend value + Seasonal variation

Steps to Predict:

1. Calculate the trend (using moving average or regression).
2. Identify seasonal variation for the desired period.
3. Add the seasonal component to the trend forecast.

Example:

If the forecasted trend for Q2 next year = 160
And mean seasonal variation for Q2 = +5
Then forecasted value = 165

Time Series Graphs

Components

• X-axis: Time (e.g., months, quarters, years)
• Y-axis: Measured variable (e.g., sales, temperature)
• Plotted points show the actual data
• Trend line can be drawn manually or calculated via moving averages
• Seasonal fluctuations are seen as regular up/down deviations from the trend

Benefits of Time Series Analysis

• Identifies long-term direction of data.
• Highlights repeating patterns in the short-term.
• Allows business forecasting (sales, production, inventory).
• Helps in resource planning and strategic decisions.

Common Mistakes

• Misinterpreting random fluctuations as trends.
• Ignoring the need for centering in even-numbered moving averages.
• Applying wrong seasonal values to wrong quarters.
• Using outdated seasonal patterns for current forecasts.

Exam-Ready Pointers

• Always state whether a trend is upward, downward or none.
• If asked to calculate moving averages: show clear working.
• Use correct units and time intervals.
• For seasonal variation: always mention which quarter/month you’re referring to.
• When drawing time series graphs: label axes clearly and show trend line distinctly.

Key Definitions Recap