The bike shop that compared December with January
A Wellington bike shop's monthly repair-job counts over 36 months (January 2022 to December 2024) are being reviewed. The manager says repairs are in decline because December 2024 had 415 jobs but January 2025 had only 340, a fall of 18%. The shop's analyst has estimated seasonal indices from the three years of data: January 0.78, February 0.72, March 0.95, and so on through to December 1.24, with the indices averaging 1.00 over the year. January is affected by the two-week summer shutdown and fewer workshop days; December includes pre-holiday servicing. A three-month moving mean of the raw counts is also available and drifts upward from about 330 to about 390 over the three years. The shop has three full-time mechanics and one part-timer, and the manager has asked whether to cut staff hours in the workshop from March.
1. Use the seasonal indices to test the manager's claim that repairs fell by 18%.
Read the worked answer
Deseasonalise each month by dividing the raw count by its seasonal index:
- December 2024: 415 ÷ 1.24 = 334.7
- January 2025: 340 ÷ 0.78 = 435.9
After removing the normal seasonal pattern, January was about 100 jobs higher than December, not 75 lower. The raw fall is almost entirely seasonal: January's index (0.78) is only 63% of December's (1.24), so a January with the same underlying demand as December would naturally record about 0.63 × 415 ≈ 261 jobs. January's 340 is well above that.
Conclusion: the manager's comparison is invalid - it compares two months from different points in the seasonal cycle. A like-for-like comparison is January 2025 with January 2024, or deseasonalised values for any pair of months.
Note also that a fall of 75 jobs from 415 to 340 is an 18.1% decrease, so the arithmetic is fine; it is the interpretation that is wrong.
2. Explain the difference between the trend and seasonal components here, and evaluate the three-month moving mean as a tool.
Read the worked answer
Seasonal component: the repeating within-year pattern - January low because of the shutdown and lost workshop days, December high because people service bikes before summer holidays. It is roughly the same shape each year and is captured by the indices.
Trend component: the longer-term underlying level of demand, after the seasonal ups and downs are removed. This is what the manager actually needs, because it is what staffing decisions should follow.
Three-month moving mean: it smooths some short-term noise but it does not remove a 12-month seasonal pattern. A three-month window spans a quarter of the cycle, so a low-season quarter still looks low. To eliminate seasonality from monthly data you need a 12-month moving mean (or a centred 12-month mean), or explicit deseasonalisation using the indices. Averaging over only three months simply moves the seasonal problem around.
That said, the upward drift in the three-month moving mean is suggestive: it rises from about 330 to about 390 across three years, which is consistent with the deseasonalised January figure. A proper 12-month moving mean or a deseasonalised series should be plotted before anything is concluded, and the analyst should check whether the seasonal pattern itself is stable across the three years - if December has become more or less busy relative to other months, the fixed indices are misleading.
3. The February index is 0.72 and the deseasonalised trend estimate for February 2025 is about 430. Produce a forecast, and advise the manager on staffing.
Read the worked answer
Forecast = deseasonalised trend × seasonal index = 430 × 0.72 = 309.6, about 310 repair jobs.
So February is expected to be the quietest month of the year not because demand is collapsing but because of the seasonal pattern. Reporting the raw January-to-February change (340 → 310, a further fall) would repeat the manager's error.
Advice on staffing:
- Compare the forecast 310 with the same month in previous years and with the trend, not with December. A trend of about 430 deseasonalised is consistent with the upward drift in the moving mean; there is no evidence of decline in the data described.
- Cutting hours from March would take effect just as demand begins to recover seasonally (March index 0.95, so expected jobs ≈ 0.95 × trend ≈ 410). Cutting permanent staff is hard to reverse; a better response to a seasonal trough is to schedule leave, reduce casual hours, or defer non-urgent work.
- Treat 310 as a central estimate, not a certainty. Forecast error grows with the horizon and with the variability of the residuals; give the manager a range, and say plainly what assumptions the forecast rests on - a stable seasonal pattern and a continuing trend.
- If the manager wants a real early-warning system, monitor the deseasonalised series monthly and compare each month with the same month a year earlier.