Calgary data lessonFitting a sine curve to a Calgary rooftop’s solar output
The City of Calgary publishes how much electricity each of its solar installations produces every month. Math 30-1 students take one site, find its amplitude, midline, period and phase shift, and test how well a sinusoidal function describes a real year. Every figure below is our own calculation.
The dataset is Monthly Solar Energy Production (KWh), which the City describes as the monthly output at each of The City of Calgary’s solar photovoltaic projects. Our download on 23 October 2026 held 2,323 rows across 25 facility names, from December 2010 to December 2025. We chose Southland Leisure Centre: 123 monthly readings from September 2015, no month recorded twice, and nine complete calendar years from 2016 to 2024.
Southland Leisure Centre, mean solar output by calendar month over 2016 to 2024 (kWh), our calculation, beside two sinusoidal models we built. The City of Calgary open data, downloaded 23 October 2026| Month | Mean output | Hand model | Least-squares model |
|---|
| January | 2,147.9 | 1,014.8 | 1,805.1 |
| February | 4,278.6 | 2,556.7 | 5,146.9 |
| March | 11,115.2 | 6,769.4 | 10,591.6 |
| April | 18,042.1 | 12,524.0 | 16,680.5 |
| May | 21,350.8 | 18,278.6 | 21,781.8 |
| June | 23,371.2 | 22,491.2 | 24,528.8 |
| July | 24,033.2 | 24,033.2 | 24,185.3 |
| August | 21,458.3 | 22,491.2 | 20,843.5 |
| September | 15,210.1 | 18,278.6 | 15,398.7 |
| October | 10,507.0 | 12,524.0 | 9,309.9 |
| November | 3,413.3 | 6,769.4 | 4,208.6 |
| December | 1,014.8 | 2,556.7 | 1,461.6 |
The hand model is what a Math 30-1 student builds with a pencil. The highest mean, 24,033.2 kWh in July, and the lowest, 1,014.8 in December, give an amplitude of half their difference, 11,509.2, and a midline halfway between them, 12,524.0. A year is 12 months, so b is 2π/12, or π/6, and putting the peak at month 7 gives y = 11,509.2 cos(π/6 (x − 7)) + 12,524.0. It explains 89 percent of the month-to-month variation, but it runs late: spring is too low and autumn too high.
The least-squares model, fitted with a little more mathematics, moves the peak back to month 6.39, between June and July, with amplitude 11,781.4 and midline 12,995.2. It explains 99.1 percent of the variation. A peak near the end of June makes physical sense, since days are longest then. Here is the Python for the hand model, which students write first:import csv, math, statistics
with open("monthly-solar.csv", encoding="utf-8") as f:
site = {r["Period"]: float(r["Solar PV Production (kWh)"])
for r in csv.DictReader(f)
if r["Facility Name"] == "SOUTHLAND LEISURE CENTRE"}
years = range(2016, 2025) # the complete years
means = [statistics.mean(site[f"{y}-{m:02d}"] for y in years) for m in range(1, 13)]
top, bottom = max(means), min(means)
a = (top - bottom) / 2 # amplitude
d = (top + bottom) / 2 # midline
c = means.index(top) + 1 # month of the peak
for m in range(1, 13):
model = a * math.cos(math.pi / 6 * (m - c)) + d
print(m, round(means[m - 1]), round(model))
Amplitude and midline
Amplitude is half the swing between the sunniest and darkest month; the midline is the level the output swings around. Students should say both in kWh, not just as numbers, and explain why neither is an average of the twelve months in the hand method.
Phase shift
The horizontal shift c places the peak. Our first guess, month 7, came from the largest monthly mean; the better fit says 6.39. The difference is a lesson in itself: a model built from two points is only as good as those two points.
Period
The period is fixed by the calendar at 12 months, which is why b is π/6. In a contrasting exercise, students try a period of 11 or 13 and watch the fit fall apart within two years.
A year that does not fit
June output at this site fell from 25,613 kWh in 2016 to 18,379 in 2024, and 2025 readings are roughly half the usual level, with October 2025 missing altogether. The file does not say why: equipment, shading, maintenance or a recording change are all possible. That is why we fitted only the complete years from 2016 to 2024, and why a student should flag 2025 rather than average it in.
Contains information licensed under the Open Government Licence - City of Calgary. The file is used on those terms, which permit commercial use (terms of use, dataset page). The City of Calgary does not endorse this page or our classes.
