Practice Maths

Smoothing Time Series

Key Terms

Moving average
Replaces each data value with the average of n surrounding values to reduce short-term fluctuations.
3-point moving average
(yt−1 + yt + yt+1) / 3 at time t; first and last values cannot be calculated.
Centred moving average (CMA)
Used for even-period data (e.g. quarterly) to align the average with an actual time point.
4-point CMA
Compute 4-point totals, then average pairs of consecutive 4-point totals and divide by 8.
Purpose of smoothing
Removes seasonal and irregular variation to reveal the underlying trend; CMA values are then used for trend-line fitting.
Endpoints
The first and last n/2 values (approx.) cannot have a moving average computed; they are left blank.

Why Smooth a Time Series?

Raw time series data contains trend, seasonal variation, and irregular variation all mixed together. Smoothing removes short-term fluctuations (seasonal and irregular) to make the underlying trend more visible. The main smoothing method in General Maths is the moving average.

Moving Averages

n-point moving average: replace each value with the average of n consecutive values centred on that point.

A 3-point moving average at time t = (yt−1 + yt + yt+1) ÷ 3

A 5-point moving average at time t = (yt−2 + yt−1 + yt + yt+1 + yt+2) ÷ 5

Note: the first and last few values cannot be smoothed (there are no neighbours on one side).
Hot Tip: For a 4-point (quarterly) moving average, you CANNOT simply take a 4-point average — this would fall between time periods. Instead, you must use the centred moving average (CMA): compute pairs of 4-point totals, then average them. The CMA is what you plot and use for trend-line fitting.

Centred Moving Averages (Even-Period Data)

For quarterly data (4-point) or monthly data (12-point), a simple moving average does not centre on an actual time point. We first compute a 4-point moving average, then average pairs of those to get a centred moving average (CMA) that aligns with the original time points.

4-point CMA at position t:
Step 1: Compute 4-point totals (add 4 consecutive values).
Step 2: Add consecutive pairs of 4-point totals and divide by 8.

Worked Example — 3-Point Moving Average

Monthly sales figures: 12, 15, 18, 14, 20, 22, 19.

Month Sales 3-pt MA
112—
215(12+15+18)÷3 = 15.0
318(15+18+14)÷3 = 15.7
414(18+14+20)÷3 = 17.3
520(14+20+22)÷3 = 18.7
622(20+22+19)÷3 = 20.3
719—

The 3-point moving averages (15.0, 15.7, 17.3, 18.7, 20.3) reveal a clear upward trend that was masked by the irregular fluctuations in the raw data.

Full Lesson: Smoothing Time Series

1. The Problem with Raw Time Series Data

When we plot raw time series data, we see all four components at once: trend, seasonal variation, cyclical variation, and irregular variation. These are superimposed on each other, making the underlying trend difficult to see. For a retailer trying to decide whether overall sales are growing, the seasonal Christmas spike every December is not helpful — it masks the year-on-year growth rate. Smoothing techniques separate the signal (trend) from the noise (seasonal and irregular variation).

2. Moving Averages — The Core Idea

A moving average replaces each data value with the average of a group of values surrounding it. Because positive and negative seasonal fluctuations cancel out when averaged, the result is a smoother series that better reflects the trend.

The number of values averaged is the order of the moving average. For quarterly data, a 4-point moving average averages exactly one full year of data at a time, which completely removes any seasonal effect that repeats annually. For monthly data, a 12-point moving average serves the same purpose.

Trade-off: Higher-order moving averages are smoother but lose more data points at the ends. A 3-point MA loses 1 point at each end; a 5-point MA loses 2 points at each end.

3. Odd-Period Moving Averages (Centring Naturally)

When the order is odd (3, 5, 7, ...), each moving average value naturally centres on the middle data point. For example, the 3-point MA for Month 4 is the average of Months 3, 4, and 5. This aligns perfectly with the original data points, so no further adjustment is needed.

4. Even-Period Moving Averages and Centring

When the order is even (4, 12, ...), the moving average falls between two time points. For example, the first 4-point total for quarterly data covers Q1, Q2, Q3, Q4 — its midpoint is between Q2 and Q3, not at either. To align with the actual data, we compute a second layer of 2-point averages from the 4-point averages. This process is called centring and produces Centred Moving Averages (CMA).

The formula for the centred 4-point moving average at quarter t is:

CMAt = (yt−2 + 2yt−1 + 2yt + 2yt+1 + yt+2) ÷ 8

In practice, it is computed in two steps: first compute all 4-point sums, then add consecutive pairs of sums and divide by 8.

5. Interpreting Smoothed Data

Once moving averages are computed:

  • Plot the original data and the smoothed values on the same graph.
  • The smoothed values show the trend direction more clearly.
  • Fit a least-squares regression line to the smoothed values to model the trend mathematically.
  • Use this trend line for forecasting (extended in the Forecasting lesson).

6. Choosing the Order of Moving Average

The order should match the seasonal period of the data. Quarterly data → 4-point CMA. Monthly data → 12-point CMA. Daily data with weekly seasonality → 7-point MA. If no clear seasonal period exists, 3- or 5-point MAs are common choices.

Mastery Practice

  1. Fluency

    The monthly sales figures ($000) for a hardware store over 7 months are: 23, 27, 31, 25, 29, 33, 28. Calculate the 3-point moving averages for all possible positions.

  2. Fluency

    The following data shows the quarterly rainfall (mm) for a region over two years. Calculate the 4-point moving averages.

    Quarter Q1 Y1 Q2 Y1 Q3 Y1 Q4 Y1 Q1 Y2 Q2 Y2 Q3 Y2 Q4 Y2
    Rainfall (mm) 120 80 40 60 130 90 50 70
  3. Fluency

    Using the data from Q2, calculate the centred 4-point moving averages (CMA) for all possible positions.

  4. Fluency

    The weekly visitor numbers to a museum over 9 weeks are: 310, 290, 330, 280, 320, 300, 340, 285, 325. Calculate the 5-point moving averages for all possible positions.

  5. Understanding

    The quarterly revenue ($000) for a resort over 3 years is shown below.

    Year Q1 Q2 Q3 Q4
    1 280 190 410 240
    2 300 210 440 260
    3 315 225 465 275
    1. Calculate the 4-point CMAs for the 10 central data points.
    2. What trend does the CMA reveal? Describe it.
    3. Why are the Q3 raw values so much higher than the CMAs for those same periods?
  6. Understanding

    A student computes a 3-point moving average for a dataset of 12 values.

    1. How many smoothed values will the student obtain?
    2. Why is a 3-point moving average better than a 2-point moving average for centring purposes?
    3. If the data has a 12-month seasonal cycle, what order moving average would best eliminate this seasonality?
  7. Understanding

    The following are monthly electricity bills ($) for a household over 9 months: 180, 155, 130, 110, 125, 145, 170, 190, 165.

    1. Calculate the 3-point moving averages.
    2. Describe the trend revealed by the moving averages.
    3. In which month does electricity use appear to reach a minimum, according to the smoothed data?
  8. Understanding

    A 5-point moving average is calculated for a dataset with values at time points 1 through 10. The 5-point MA at t = 5 is 42.4.

    1. Which five values were used to calculate this moving average?
    2. The values at t = 3, 4, 5, 6 are 38, 44, 40, 46. Find the value at t = 7.
  9. Problem Solving

    The quarterly sales data for an electrical retailer over 3 years is shown below (units: $000).

    Year Q1 Q2 Q3 Q4
    1 320 260 295 430
    2 340 275 315 455
    3 360 290 330 480
    1. Calculate all possible 4-point centred moving averages.
    2. By fitting a trend line to the CMAs (number the data quarters 1–12), estimate the trend equation ŷ = a + bt.
    3. Describe the trend in terms of average quarterly growth.
  10. Problem Solving

    A time series analyst states: “A 3-point moving average and a 5-point moving average both smooth the same dataset, but they produce different results. The 5-point MA is always the better choice.”

    1. Explain one advantage of a 5-point MA over a 3-point MA.
    2. Explain one disadvantage of a 5-point MA compared to a 3-point MA.
    3. Is the analyst’s conclusion that “5-point is always better” correct? Justify with reference to the purpose of smoothing.
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