Solutions: Time-Series Graphs and Trends

1. Trend: Upward. Each quarter in Year 2 is higher than the corresponding quarter in Year 1 (e.g. Q1: $85,000 → $91,000; Q4: $110,000 → $118,000), indicating a gentle upward trend in revenue.

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   Compare matching quarters across years: Q1 rose by $6,000; Q2 rose by $7,000; Q3 rose by $6,000; Q4 rose by $8,000. Every quarter shows a year-on-year increase, confirming the trend is upward overall despite within-year fluctuations.

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   Yes, seasonal variation is present. The pattern Q4 (highest) → Q1 (drops) → Q2 (rises) → Q3 (falls to lowest) → Q4 repeats in both years: Q2 peaks at 22 (Y1) and 23 (Y2); Q4 is the lowest at 12 (Y1) and 13 (Y2). A regular, repeating within-year pattern is the defining feature of seasonal variation.

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   Axes: Horizontal axis: Month (Jan – Dec). Vertical axis: Electricity consumption (MWh).

   Highest consumption: January (480 MWh).

   Lowest consumption: May (370 MWh).

   The pattern reflects Queensland’s climate: high electricity use in summer (Jan, Feb, Dec) for air-conditioning, a mid-year dip in autumn, and a secondary rise in winter (Jun–Jul) for heating. May sits in the mild transition period with minimal climate control needed.

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   Irregular variation refers to unpredictable, random fluctuations in a time series that are not due to trend or seasonal patterns. These arise from one-off, unexpected events.

   Example: Australian retail sales in March–April 2020 showed an extraordinary spike in food and grocery sales (panic buying) followed by a sharp drop when lockdowns closed non-essential retail. This could not have been forecast from historical data — it is classic irregular variation driven by the COVID-19 pandemic.

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   (a) The overall trend is upward. Despite a slight dip in Year 3 (142 c/L, down from 145 c/L), the general direction is upward from 138 c/L (Year 1) to 179 c/L (Year 8), a rise of 41 c/L over 8 years.

   (b) The Year 3 dip represents irregular variation. It is a one-off deviation from the upward trend, not part of a repeating annual pattern. Possible causes include a temporary fall in global oil prices or a government fuel tax reduction.

   (c) Annual averages would not show seasonal variation. Averaging all months within each year smooths out within-year seasonal fluctuations. To observe seasonal patterns, monthly (not annual) data would be needed.

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   (a) The peak months are 7, 8, and 9 (July, August, September), with arrivals of 60,000; 72,000; and 68,000 respectively. This represents seasonal variation — the Australian school holiday period (July) drives peak tourism to the island each year, and this pattern is expected to repeat.

   (b) Month 1 (Jan Year 1): 52,000 arrivals. Month 13 (Jan Year 2): 58,000 arrivals. The 6,000 increase in the same calendar month of the following year suggests an upward trend in overall tourist numbers, separate from the seasonal fluctuation.

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   (a) A series that fluctuates around a roughly constant mean with no long-term upward or downward direction is called stationary (or “no trend”).

   (b) The consistently higher December values compared to June each year represent seasonal variation. The pattern repeats regularly each year, which is the defining characteristic.

   (c) The Year 3 factory closure represents irregular variation. It was an unexpected, one-off event — not predictable from the trend or seasonal pattern.

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   (a) Q3 (winter) sales increased each year: $95,000 → $104,000 → $112,000. The upward trend in Q3 averages approximately $8,500 per year.

   (b) Q3 (winter) consistently has the highest sales; Q1 (summer) consistently has the lowest. This is seasonal variation — demand for winter clothing peaks in the cooler months and falls in summer, and this pattern repeats every year.

   (c) Year 1 average: (42 + 78 + 95 + 55) ÷ 4 = 270 ÷ 4 = $67,500. Year 2: (48 + 85 + 104 + 62) ÷ 4 = 299 ÷ 4 = $74,750. Year 3: (53 + 91 + 112 + 68) ÷ 4 = 324 ÷ 4 = $81,000. Yes, the annual averages increase each year, confirming the upward trend across all seasons.

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   (a) The overall trend is downward. Year 2 unemployment rates are consistently lower than corresponding Year 1 months. The rate fell from 7.2% (Month 1, Y1) to 5.3% (Month 12, Y2).

   (b) The mid-year rise in Months 6–8 represents seasonal variation. A likely explanation: school leavers and university graduates enter the job market in mid-year (June–July), temporarily increasing the unemployment rate before employers absorb them through August–September.

   (c) Year 1 average: (7.2 + 7.0 + 6.8 + 6.5 + 6.3 + 6.6 + 7.0 + 7.1 + 6.9 + 6.4 + 6.1 + 5.8) ÷ 12 = 79.7 ÷ 12 ≈ 6.64%. Year 2 average: (6.9 + 6.7 + 6.4 + 6.1 + 5.9 + 6.2 + 6.6 + 6.7 + 6.5 + 6.0 + 5.7 + 5.3) ÷ 12 = 75.0 ÷ 12 = 6.25%. The average fell by approximately 0.39 percentage points.

   (d) This would initially appear as an irregular component. The announcement is a specific one-off event. However, if it leads to sustained job creation over subsequent months, the effect would gradually become visible as part of the downward trend.

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    (a) Year 1: (210 + 140 + 90 + 165) ÷ 4 = 605 ÷ 4 = $151,250. Year 2: (228 + 152 + 98 + 180) ÷ 4 = 658 ÷ 4 = $164,500. Year 3: (245 + 165 + 108 + 196) ÷ 4 = 714 ÷ 4 = $178,500. The annual averages increase each year, confirming an upward trend. The director is correct on this point.

    (b) The managing director is incorrect about seasonal patterns. The data shows a strong, consistent seasonal pattern: Q1 (summer) is always the highest and Q3 (winter) is always the lowest, in every year. This is the definition of seasonal variation. For example: Year 1 Q1 = $210,000 vs Q3 = $90,000; Year 2 Q1 = $228,000 vs Q3 = $98,000; Year 3 Q1 = $245,000 vs Q3 = $108,000. Summer sales are more than double winter sales each year.

    (c) Year 1: 90 ÷ 210 × 100 = 42.9%. Year 2: 98 ÷ 228 × 100 = 43.0%. Year 3: 108 ÷ 245 × 100 = 44.1%. Winter (Q3) sales are consistently about 43% of summer (Q1) sales. This remarkably stable ratio across all three years confirms the seasonal pattern is regular and proportional — characteristic of true seasonal variation, not random fluctuation.