What you'll learn
Time series and index numbers form a crucial component of AQA GCSE Statistics, allowing you to analyse data collected over time and measure changes in quantities like prices or performance. You'll learn to identify patterns, calculate moving averages, predict future values, and interpret index numbers used in real-world contexts such as retail price indices and stock market indicators.
Key terms and definitions
Time series — a set of observations or measurements taken at regular time intervals (e.g., daily, monthly, quarterly, yearly).
Trend — the underlying long-term movement or direction in a time series, showing the general pattern when seasonal variations are removed.
Seasonal variation — regular, predictable patterns that repeat over fixed periods (e.g., ice cream sales rising every summer).
Moving average — a calculated mean using a fixed number of consecutive values that 'moves' through the time series to smooth out short-term fluctuations and highlight the trend.
Index number — a measure showing the relative change in a variable compared to a base value, typically expressed as a percentage with the base period set at 100.
Base period — the reference time point against which all other values in an index are compared.
Weighted index — an index number that takes into account the relative importance of different items by applying weights to each component.
Chain base index — an index where each period is compared to the immediately preceding period rather than to the original base period.
Core concepts
Understanding time series data
Time series data displays how a variable changes over time. When presented with time series data, you must:
- Identify the time intervals used (daily, weekly, monthly, quarterly, annually)
- Recognise the pattern type: increasing trend, decreasing trend, cyclical pattern, or seasonal variation
- Distinguish between short-term fluctuations and long-term trends
On a time series graph:
- Time is always plotted on the horizontal (x) axis
- The measured variable is plotted on the vertical (y) axis
- Points are typically connected with straight lines to show progression
Seasonal variation occurs when values regularly increase or decrease at specific times. For example, umbrella sales peak in winter months every year, whilst garden furniture sales peak in spring/summer. At GCSE level, you must identify when seasonal effects are present and describe them clearly.
Calculating moving averages
Moving averages smooth out fluctuations to reveal the underlying trend. The order of a moving average tells you how many values to include in each calculation.
For odd-numbered moving averages (e.g., 3-point, 5-point):
- Select the appropriate number of consecutive values
- Calculate their mean
- Plot this value at the middle time point of the group
- Move forward one position and repeat
For even-numbered moving averages (e.g., 4-point for quarterly data):
Because there's no middle value with an even number of points, you must centre the moving average:
- Calculate the first 4-point mean (for quarters 1-4)
- Calculate the second 4-point mean (for quarters 2-5)
- Find the mean of these two means
- Plot this value at quarter 3 (the middle point)
- Continue this process throughout the series
The centred moving average eliminates seasonal variation and reveals the trend line.
Identifying and measuring seasonal variation
Once you've calculated the moving average (trend), you can find the seasonal effect for each period:
For additive models: Seasonal effect = Actual value - Trend value
For multiplicative models: Seasonal effect = Actual value ÷ Trend value
At GCSE level, you typically work with additive models. The seasonal effect shows how much higher or lower the actual value is compared to the trend.
To find the seasonal component for a particular period (e.g., Quarter 1):
- Calculate the seasonal effect for all Quarter 1 values in the data
- Find the mean of these seasonal effects
- This mean is the seasonal component for Quarter 1
Making predictions using time series
To predict future values:
- Extend the trend line: Continue the pattern established by the moving averages
- Read the trend value at the required future time point
- Apply the seasonal component: Add (or multiply) the appropriate seasonal component to the trend value
When extending trend lines on graphs:
- Use a ruler for linear trends
- Follow the established curve pattern for non-linear trends
- Clearly label any predictions as estimates
- Understand that predictions become less reliable further into the future
Understanding index numbers
An index number provides a standardised way to compare values over time. The base period is set at 100, and other periods are expressed relative to this base.
To calculate a simple index:
Index = (Current value ÷ Base value) × 100
For example, if house prices in 2020 were £200,000 (base year) and £240,000 in 2024:
Index for 2024 = (240,000 ÷ 200,000) × 100 = 120
This means house prices in 2024 are 120% of the 2020 value, representing a 20% increase.
Interpreting index numbers:
- Index = 100: no change from base period
- Index > 100: increase from base period
- Index < 100: decrease from base period
- Index = 110: 10% increase from base
- Index = 95: 5% decrease from base
Calculating weighted index numbers
A weighted index reflects the relative importance of different items. This is essential for measures like the Retail Price Index (RPI) or Consumer Price Index (CPI).
Method for weighted index calculation:
- For each item, multiply its index by its weight: (Index × Weight)
- Sum all these products: Σ(Index × Weight)
- Sum all the weights: ΣWeight
- Divide: Weighted index = Σ(Index × Weight) ÷ ΣWeight
The weights often represent how much a typical household spends on each category or how important each component is to the overall measure.
Chain base indices compare each period only to the previous period. To convert back to the original base:
If Year 1 (base) = 100, Year 2 chain index = 105, Year 3 chain index = 110:
- Year 2 on original base = 105
- Year 3 on original base = 105 × (110/100) = 115.5
Worked examples
Example 1: Calculating a 4-point moving average
Question: The table shows quarterly sales (£000s) for a company. Calculate the 4-point moving average for Quarter 3, 2023.
| Year | Quarter | Sales (£000s) |
|---|---|---|
| 2023 | Q1 | 45 |
| 2023 | Q2 | 62 |
| 2023 | Q3 | 58 |
| 2023 | Q4 | 51 |
| 2024 | Q1 | 48 |
Solution:
For a 4-point moving average centred on Q3 2023:
Step 1: Calculate first 4-point mean (Q1-Q4 2023): (45 + 62 + 58 + 51) ÷ 4 = 216 ÷ 4 = 54
Step 2: Calculate second 4-point mean (Q2 2023 - Q1 2024): (62 + 58 + 51 + 48) ÷ 4 = 219 ÷ 4 = 54.75
Step 3: Centre the moving average: (54 + 54.75) ÷ 2 = 54.375
The 4-point moving average for Q3 2023 = 54.375 (or £54,375) ✓ [3 marks]
Example 2: Calculating and interpreting a weighted index
Question: A school canteen monitors the cost of lunch ingredients. Calculate the weighted price index for 2024 using 2023 as the base year.
| Item | Weight | Price 2023 (£) | Price 2024 (£) |
|---|---|---|---|
| Bread | 20 | 1.20 | 1.32 |
| Chicken | 35 | 4.00 | 4.40 |
| Vegetables | 25 | 2.40 | 2.52 |
| Fruit | 20 | 1.80 | 1.98 |
Solution:
Step 1: Calculate simple index for each item:
Bread: (1.32 ÷ 1.20) × 100 = 110 Chicken: (4.40 ÷ 4.00) × 100 = 110 Vegetables: (2.52 ÷ 2.40) × 100 = 105 Fruit: (1.98 ÷ 1.80) × 100 = 110
Step 2: Create calculation table:
| Item | Index | Weight | Index × Weight |
|---|---|---|---|
| Bread | 110 | 20 | 2,200 |
| Chicken | 110 | 35 | 3,850 |
| Vegetables | 105 | 25 | 2,625 |
| Fruit | 110 | 20 | 2,200 |
| Total | 100 | 10,875 |
Step 3: Calculate weighted index: Weighted index = 10,875 ÷ 100 = 108.75 ✓ [4 marks]
Interpretation: The overall cost of lunch ingredients has increased by 8.75% from 2023 to 2024. [1 mark]
Example 3: Predicting using trend and seasonal components
Question: A café's quarterly ice cream sales show the following trend values and seasonal components:
Trend equation: T = 150 + 10q (where q = quarter number, starting from Q1 2023 as q = 1)
Seasonal components: Q1 = -40, Q2 = +15, Q3 = +60, Q4 = -35
Predict sales for Q2 2024 (quarter 6).
Solution:
Step 1: Calculate trend value for q = 6: T = 150 + 10(6) = 150 + 60 = 210
Step 2: Apply seasonal component for Q2: Predicted sales = Trend + Seasonal component = 210 + 15 = 225 units ✓ [3 marks]
Common mistakes and how to avoid them
Plotting moving averages at the wrong position: For odd-numbered moving averages (3-point, 5-point), always plot at the middle value. For even-numbered moving averages, you must centre them by averaging two consecutive means. Never plot a 4-point moving average at a quarter position without centring.
Confusing index values with percentage changes: An index of 120 means 20% increase from base (not 120% increase). To find percentage change: subtract 100 from the index value. Index 95 represents a 5% decrease, not a 95% decrease.
Forgetting to multiply by 100 in index calculations: The formula is (Current ÷ Base) × 100. Students often forget the × 100 step, giving answers like 1.15 instead of 115.
Mixing up weights and index values: In weighted index calculations, ensure you multiply each index by its correct weight. Create a clear table showing Item, Index, Weight, and Index × Weight to avoid errors.
Extending trend lines without a ruler: Always use a ruler for linear trends. Freehand extensions introduce inaccuracy and lose marks.
Not showing working for moving averages: Examiners need to see your calculation steps. Show each mean calculation clearly, especially when centring 4-point moving averages.
Exam technique for "Time Series and Index Numbers"
Command word awareness: "Calculate" requires showing full working with numerical answers. "Describe" the trend means stating whether it's increasing/decreasing and mentioning the rate (e.g., "steady increase" or "sharp decline"). "Predict" requires extending the trend and applying seasonal components with clear method shown.
Graph work: Use a sharp pencil and ruler. Plot points accurately (±½ small square tolerance). When drawing trend lines, ensure your line follows the general pattern of moving averages. Label axes fully with units. For predictions, clearly indicate these with different notation (dotted line or labelled 'P').
Index number context: Read questions carefully to identify the base period. When comparing index numbers, calculate the difference and interpret in context (e.g., "Prices increased 15% more in Region A than Region B"). For weighted indices, always show the table with Index × Weight column.
Mark allocation guides detail: 1 mark typically for identifying correct values, 2-3 marks for calculation process, 1 mark for final answer. On 4-mark questions, expect to show setting up formula, substitution, working, and answer. Always write your final answer clearly, underlining or boxing it.
Quick revision summary
Time series analysis reveals patterns in data collected over time. Calculate moving averages to identify trends: use the mean of consecutive values, centring for even-numbered periods. Seasonal variation shows regular fluctuations; find seasonal components by comparing actual values to trend values. Predict future values by extending the trend and applying appropriate seasonal adjustments. Index numbers measure relative change with base period = 100; use (Current ÷ Base) × 100. Weighted indices multiply each index by its importance weight before averaging. Always show clear working, use rulers for graph work, and interpret results in context.