Time Series, Index Numbers and Rates of Change

Time Series and Trend Lines

A time series is a set of values recorded at regular points in time — monthly sales, quarterly unemployment figures, or annual birth rates. Plotting these values against time on a graph reveals patterns that a table of numbers alone would hide.

The trend is the long-term direction the data is heading in, once you ignore the short-term wobbles. A trend line drawn through the data shows this direction: the steeper its gradient, the faster the value is rising or falling. A roughly flat trend line means the underlying value is staying about the same.

Calculating a Moving Average

Rather than drawing a trend line purely by eye, the exam board expects you to calculate a moving average. For a 4-point moving average, you add four consecutive values and divide by four:

$$\text{4-point moving average} = \frac{x_1+x_2+x_3+x_4}{4}$$

Each moving average value is plotted at the midpoint of the four values it was calculated from. Once you have several moving average points, join them to form a trend line. The gradient of this trend line tells you how quickly the underlying quantity is changing — a business owner looking at a steep upward trend line knows the quantity is rising quickly; growth is only accelerating if the trend line becomes steeper over time.

Seasonal and Cyclic Trends, and Making Predictions

A seasonal trend is a predictable pattern tied to the calendar — ice cream sales peak every summer, heating bills peak every winter, retail sales peak every December. These patterns repeat at the same point in every cycle.

A cyclic trend is similar but is not tied to a fixed calendar period — economic cycles of boom and slowdown, for example, might last several years rather than repeating every twelve months exactly.

To predict a future value, take the trend line's value at that point in time, then adjust it using the average seasonal effect for that period. If winter sales are typically £3,000 below the trend line, subtract £3,000 from the trend value to estimate the actual winter figure.

Predictions made this way rely on extrapolation — extending a pattern beyond the data you have. The exam board expects you to flag this as a source of unreliability: the pattern observed so far might not continue once you move far enough beyond the collected data.

Index Numbers

An index number expresses a value relative to a chosen base period, usually set to 100:

$$\text{index number} = \frac{\text{current value}}{\text{base year value}} \times 100$$

If the base year has an index of 100 and a later year has an index of 115, the value has risen by 15% since the base year. Three named indices you should recognise are:

• RPI (Retail Price Index): tracks changes in the average price of a basket of household goods and services.
• CPI (Consumer Price Index): a similar price measure using a different basket and method; it is the Bank of England’s inflation target.
• GDP (Gross Domestic Product): tracks the total economic output of a country over time.

Rates of Change: Percentage Change and Birth Rates

A rate of change puts a raw count into proportion so fair comparisons can be made. For house prices, unemployment, or similar figures, this is often a straightforward percentage change:

$$\text{percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100$$

For population statistics, the exam board uses the crude birth rate. Learn this formula — it may not be given to you in the question:

$$\text{crude birth rate} = \frac{\text{number of births}}{\text{total population}} \times 1000$$

A country with 60,000 births and a population of 2,000,000 has a crude birth rate of (60,000 ÷ 2,000,000) × 1000 = 30 births per 1,000 people. Using the rate rather than the raw count of 60,000 lets you compare fairly against a country with a very different population size.