Solutions: Smoothing Time Series

3-point moving averages for months 2–6:

Data: 23, 27, 31, 25, 29, 33, 28

MA(2) = (23 + 27 + 31) ÷ 3 = 81 ÷ 3 = 27.0

MA(3) = (27 + 31 + 25) ÷ 3 = 83 ÷ 3 = 27.7

MA(4) = (31 + 25 + 29) ÷ 3 = 85 ÷ 3 = 28.3

MA(5) = (25 + 29 + 33) ÷ 3 = 87 ÷ 3 = 29.0

MA(6) = (29 + 33 + 28) ÷ 3 = 90 ÷ 3 = 30.0

Months 1 and 7 cannot be smoothed. The moving averages reveal a gentle upward trend from $27,000 to $30,000.
Data: 120, 80, 40, 60, 130, 90, 50, 70

4-point sums (these fall between quarters):

After Q2 Y1: 120 + 80 + 40 + 60 = 300

After Q3 Y1: 80 + 40 + 60 + 130 = 310

After Q4 Y1: 40 + 60 + 130 + 90 = 320

After Q1 Y2: 60 + 130 + 90 + 50 = 330

After Q2 Y2: 130 + 90 + 50 + 70 = 340

There are 5 four-point sums, each positioned between quarters. (These are not yet centred — that is done in Q3.)
Using the 4-point sums from Q2: 300, 310, 320, 330, 340.

Centred 4-point moving averages (add consecutive pairs of sums, divide by 8):

CMA at Q3 Y1: (300 + 310) ÷ 8 = 610 ÷ 8 = 76.25

CMA at Q4 Y1: (310 + 320) ÷ 8 = 630 ÷ 8 = 78.75

CMA at Q1 Y2: (320 + 330) ÷ 8 = 650 ÷ 8 = 81.25

CMA at Q2 Y2: (330 + 340) ÷ 8 = 670 ÷ 8 = 83.75

The CMAs rise from 76.25 to 83.75, confirming a gentle upward trend in annual rainfall.
Data: 310, 290, 330, 280, 320, 300, 340, 285, 325

5-point MA at week 3: (310 + 290 + 330 + 280 + 320) ÷ 5 = 1530 ÷ 5 = 306.0

5-point MA at week 4: (290 + 330 + 280 + 320 + 300) ÷ 5 = 1520 ÷ 5 = 304.0

5-point MA at week 5: (330 + 280 + 320 + 300 + 340) ÷ 5 = 1570 ÷ 5 = 314.0

5-point MA at week 6: (280 + 320 + 300 + 340 + 285) ÷ 5 = 1525 ÷ 5 = 305.0

5-point MA at week 7: (320 + 300 + 340 + 285 + 325) ÷ 5 = 1570 ÷ 5 = 314.0

Weeks 1, 2, 8, 9 cannot be smoothed. The smoothed values hover around 305–314, suggesting a roughly stationary trend with some irregular fluctuation.
(a) Data sequence (quarters 1–12): 280, 190, 410, 240, 300, 210, 440, 260, 315, 225, 465, 275

4-point sums (positioned between quarters):

After Q2 Y1: 280+190+410+240 = 1120

After Q3 Y1: 190+410+240+300 = 1140

After Q4 Y1: 410+240+300+210 = 1160

After Q1 Y2: 240+300+210+440 = 1190

After Q2 Y2: 300+210+440+260 = 1210

After Q3 Y2: 210+440+260+315 = 1225

After Q4 Y2: 440+260+315+225 = 1240

After Q1 Y3: 260+315+225+465 = 1265

After Q2 Y3: 315+225+465+275 = 1280

CMAs (add consecutive pairs, divide by 8):

Q3 Y1: (1120+1140)÷8 = 2260÷8 = 282.5

Q4 Y1: (1140+1160)÷8 = 2300÷8 = 287.5

Q1 Y2: (1160+1190)÷8 = 2350÷8 = 293.8

Q2 Y2: (1190+1210)÷8 = 2400÷8 = 300.0

Q3 Y2: (1210+1225)÷8 = 2435÷8 = 304.4

Q4 Y2: (1225+1240)÷8 = 2465÷8 = 308.1

Q1 Y3: (1240+1265)÷8 = 2505÷8 = 313.1

Q2 Y3: (1265+1280)÷8 = 2545÷8 = 318.1

(b) The CMAs increase steadily from 282.5 (Q3 Y1) to 318.1 (Q2 Y3). The trend is upward, with an approximate increase of about $35,000 revenue per year (or roughly $8.75k per quarter).

(c) Q3 (winter) is the peak season for a ski resort — raw Q3 values of $410k, $440k, $465k are far above the CMAs (~$282k, $304k) because the CMA averages across all four seasons. The CMA removes the seasonal effect, leaving only the trend.
(a) With 12 data values and a 3-point MA, we lose 1 point at each end. The student obtains 12 − 2 = 10 smoothed values.

(b) A 2-point MA averages only two values and falls between them (like a 4-point MA). This does not centre on any actual data point, making it awkward to plot and interpret. A 3-point MA averages three values and centres naturally on the middle point (which corresponds to an actual observation), so no additional centring step is needed.

(c) A 12-point moving average (or 12-point centred moving average) would best eliminate a 12-month seasonal cycle, as it averages exactly one full year of data at each step.
(a) Data: 180, 155, 130, 110, 125, 145, 170, 190, 165

MA(2) = (180+155+130)÷3 = 465÷3 = 155.0

MA(3) = (155+130+110)÷3 = 395÷3 = 131.7

MA(4) = (130+110+125)÷3 = 365÷3 = 121.7

MA(5) = (110+125+145)÷3 = 380÷3 = 126.7

MA(6) = (125+145+170)÷3 = 440÷3 = 146.7

MA(7) = (145+170+190)÷3 = 505÷3 = 168.3

MA(8) = (170+190+165)÷3 = 525÷3 = 175.0

(b) The moving averages show a clear U-shaped pattern: electricity bills decrease from $155.0 in Month 2 to a minimum of $121.7 in Month 4, then increase steadily to $175.0 in Month 8. The trend is downward then upward, reflecting seasonal climate changes.

(c) The minimum smoothed value ($121.7) occurs at Month 4, suggesting electricity use is lowest in April (autumn), when neither air-conditioning nor heating is heavily needed.
(a) The 5-point MA at t = 5 uses the values at t = 3, 4, 5, 6, and 7.

(b) MA at t = 5 = (y₃ + y₄ + y₅ + y₆ + y₇) ÷ 5 = 42.4

So y₃ + y₄ + y₅ + y₆ + y₇ = 42.4 × 5 = 212

38 + 44 + 40 + 46 + y₇ = 212

168 + y₇ = 212

y₇ = 44
(a) Data sequence (quarters 1–12): 320, 260, 295, 430, 340, 275, 315, 455, 360, 290, 330, 480

4-point sums: Q1–Q4 Y1: 1305; Q2 Y1–Q1 Y2: 1325; Q3 Y1–Q2 Y2: 1340; Q4 Y1–Q3 Y2: 1360; Q1 Y2–Q4 Y2: 1385; Q2 Y2–Q1 Y3: 1400; Q3 Y2–Q2 Y3: 1435; Q4 Y2–Q3 Y3: 1455; Q1 Y3–Q4 Y3: 1460

CMAs (pairs of consecutive 4-pt sums ÷ 8):

t=3 (Q3 Y1): (1305+1325)÷8 = 2630÷8 = 328.75

t=4 (Q4 Y1): (1325+1340)÷8 = 2665÷8 = 333.13

t=5 (Q1 Y2): (1340+1360)÷8 = 2700÷8 = 337.50

t=6 (Q2 Y2): (1360+1385)÷8 = 2745÷8 = 343.13

t=7 (Q3 Y2): (1385+1400)÷8 = 2785÷8 = 348.13

t=8 (Q4 Y2): (1400+1435)÷8 = 2835÷8 = 354.38

t=9 (Q1 Y3): (1435+1455)÷8 = 2890÷8 = 361.25

t=10 (Q2 Y3): (1455+1460)÷8 = 2915÷8 = 364.38

(b) Fitting a trend line to the 8 CMA values (at t = 3 to 10): The CMAs increase approximately linearly from 328.75 to 364.38 over 7 intervals. Gradient ≈ (364.38 − 328.75) ÷ 7 ≈ 5.09. Using CAS to perform linear regression on (t, CMA): ŷ ≈ 313.5 + 5.1t (approximate; CAS will give a precise equation).

(c) The trend shows approximately $5,100 average quarterly growth in revenue, or roughly $20,400 per year, after seasonal effects are removed.
(a) Advantage of 5-point MA: Greater smoothing. It averages more values, so it more effectively removes irregular fluctuations and seasonal effects, making the underlying trend clearer.

(b) Disadvantage of 5-point MA: More data is lost at the ends (2 values at each end, versus 1 for a 3-point MA). This matters most for short datasets. Also, the 5-point MA reacts more slowly to genuine changes in trend direction.

(c) The analyst’s conclusion is not always correct. The best choice depends on the purpose and the data. If the data has a 4-quarter seasonal cycle, neither a 3- nor a 5-point MA is ideal — a 4-point CMA is better because it exactly spans one seasonal cycle. If there is no specific seasonal period to eliminate, a 5-point MA may indeed smooth more effectively than a 3-point MA, but the cost is losing more end values. For short datasets (e.g. 6–8 values), a 3-point MA preserves more information. Neither is “always” superior.