Explain the methods, show your working and interpret results in context.
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Revise the key ideas
Time series and rates
Time-series display — Record the same measure at successive times and put time on the horizontal axis. Check equal time spacing and consistent definitions. A trend is the underlying longer-term direction; individual increases or decreases can reflect short-term noise or seasonality.
Seasonal and cyclic — Seasonal variation repeats on a fixed calendar cycle, such as higher summer sales each year. Cyclic variation repeats over longer or less fixed intervals, such as business cycles. Irregular variation includes unusual events; a single spike does not establish a seasonal pattern.
Four-point moving average — Average four successive values, then move one observation forward and repeat. For 10,14,18,22,30 the averages are 16 and 21. Plot each halfway between its middle two time points; use a common period length so regular seasonal changes are smoothed.Moving averages smooth a time series. Original illustrative diagram; numerical datasets are fictional worked examples.Enlarge diagram
Trend interpretation — A rising smoothed line suggests an increasing underlying measure; its gradient is change per time unit. Smoothing loses endpoint information and can hide abrupt changes. Draw a trend line consistently and state whether a prediction assumes that the trend persists.
Simple index — Set the base period to 100. Index = current value / base value ×100. If a price rises from £40 to £50 its index is 125, meaning 25% above the base, not £125. Compare indices only after checking their bases and definitions.
Change between indices — Percentage change from index 120 to 132 is (132−120)/120 ×100 = 10%, not 12%. The 12 is an index-point change. Distinguish percentage change from percentage-point change in proportions, and use the earlier value as the denominator.
RPI, CPI and GDP — RPI and CPI track prices through defined baskets and methods; their values need not be identical. GDP describes economic output, and index series can express change relative to a base. Check whether values are nominal or adjusted for price changes before comparing purchasing power or growth.
Rates per population — A context formula such as births / population ×1000 standardises for different population sizes. With 240 births in 20,000 people the crude birth rate is 12 per thousand. A higher count need not mean a higher rate; use comparable time periods and definitions.
Forecasts from rates — If an annual rate of 8 per thousand applies to 50,000 people, expected count is 400. Such a prediction assumes the rate and population basis are suitable. Demographic differences can make a crude-rate comparison misleading even when arithmetic is correct.
Higher — seasonal estimates and indices
Other periods and centring — Choose a moving-average length to match the seasonal period. Odd lengths have a central observation; even lengths fall between observations and can be centred by averaging adjacent moving averages. A quarterly four-point average removes a typical annual seasonal cycle; a monthly three-point average does not.
Mean seasonal effect — In an additive model calculate actual minus trend for matching times, then average these deviations for each season across years. Effects should balance around zero for a complete cycle; rounding or imperfect trends can require adjustment. Seasonal variation is a difference in the measure's units.
Seasonal prediction — Add a season's mean effect to the projected trend. If trend predicts 120 sales and the winter effect is −18, forecast 102. Explain that the forecast assumes stable trend and seasonal behaviour; unusual events or extrapolation can make it inaccurate.
Weighted index — Multiply each component index by its weight, add the products and divide by total weight. Indices 110 and 130 with weights 3 and 1 give 115. Weights reflect expenditure or importance; an unweighted mean changes the meaning of the basket.
Chain-based index — Each link compares a period with the preceding period. Multiply link factors to obtain overall change: links 110 then 120 give factor 1.1×1.2=1.32 and overall index 132 from the original base. Do not add the percentage increases.
Standardisation — Follow any supplied rate-standardisation formula and identify its output and denominator. Applying group-specific rates to a common standard population can make comparisons fairer when age structures differ. Do not confuse a predicted count from a rate with a rate per thousand; report the formula's stated units.
Test yourself
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Mind map
Use the branches to recall the ideas and explain their connections. Check the revision notes for the full detail.
ST8 · Time / rates 1 / Time / rates 2 / Time / rates 3
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Time-series display: Time on x; separate trend from noise.
Seasonal and cyclic: Seasonal calendar pattern; cycles less fixed.
Four-point moving average: Average successive fours at half-time positions.
Trend interpretation: Smoothing helps show underlying change.
Time / rates 2
Simple index: Base 100; index expresses relative change.
Change between indices: Percentage change uses the earlier denominator.
RPI, CPI and GDP: Check indicator, basket, base and price adjustment.
Rates per population: Standardise counts using the relevant population.
Time / rates 3
Forecasts from rates: Rate-based predictions depend on appropriate populations.
H: extensions 1
Higher: Other periods and centring: Match period; centre even moving averages if needed.
Higher: Mean seasonal effect: Average actual − trend for each season.
Higher: Seasonal prediction: Forecast = trend + mean seasonal effect.
Higher: Weighted index: Weighted index = ΣwI ÷ Σw.
H: extensions 2
Higher: Chain-based index: Multiply link factors across successive periods.
Higher: Standardisation: Common population basis improves rate comparisons.
Connections
Time / rates 1 → Time / rates 2: A smoothed time trend and relative indices describe changes from an explicit reference.
Time / rates 2 → Time / rates 3: Population rates and stated time periods provide the basis for expected-count forecasts.
Part connections
ST8 · Time / rates 1 / Time / rates 2 / Time / rates 3: Time / rates 1 → Time / rates 2 — A smoothed time trend and relative indices describe changes from an explicit reference.
ST8 · Time / rates 1 / Time / rates 2 / Time / rates 3: Time / rates 2 → Time / rates 3 — Population rates and stated time periods provide the basis for expected-count forecasts.
ST8 · H: extensions 1 / H: extensions 2: H: extensions 1 → H: extensions 2 — Additive seasonal forecasts and chain indices use distinct models whose assumptions must be checked.
Mr Kidman's Classroom
