Ottawa data lessonA year of Ottawa River temperatures, modelled with a cosine curve
The City of Ottawa publishes the daily temperature of raw river water at the Britannia water purification plant. We built a sinusoid from the 2025 column and compared it with every day of that year. The model and its errors are our calculations.
The City’s description is unusually exact. Each daily value is the average of two readings, recorded 12 hours apart, around 06:00 and 18:00, using a certified probe accurate to plus or minus half a degree. The intake sits approximately 427 m out from the plant, where 12 reinforced fibreglass bells on the last 29 m of pipe rest at an approximate depth of 10 m, with an intake capacity of 364 ML/day. The workbook we downloaded has one row per calendar day and one column for each year from 2005 to 2025; it has no row for 29 February, so leap days are absent.
Our figures from the City of Ottawa workbook, 2025 column, downloaded 22 October 2026 (degrees Celsius)| Measure | 2025 |
|---|
| Warmest day | 26.75 on 13 August (day 225) |
| Coldest day | 0.27 on 17 January (day 17) |
| Mean and median of the 365 days | 11.17 and 9.5 |
| Monthly means, January to March | 0.89, 1.02, 1.15 |
| Monthly means, July and August | 24.87 and 24.76 |
| Days at or above 20 degrees | 102 |
| Days at or below 1 degree | 47 |
From those numbers a Grade 11 student can write a model in four moves. The amplitude is half the range, (26.75 − 0.27) ÷ 2 = 13.24. The midline is the average of the extremes, 13.51. The period is one year, 365 days. And the curve peaks on day 225, so a cosine shifted right by 225 fits: T(d) = 13.24 cos(2π(d − 225) ÷ 365) + 13.51. The program below builds exactly that model from the workbook and measures how far it sits from the real temperatures:import datetime, math
import openpyxl # pip install openpyxl
sheet = openpyxl.load_workbook("2025_Ottawa_River_water_temperature.xlsx", read_only=True).worksheets[0]
rows = list(sheet.iter_rows(max_row=400, values_only=True))
header = next(i for i, row in enumerate(rows) if row[0] == "Date")
col = rows[header].index(2025)
temps = [row[col] for row in rows[header + 1:] if isinstance(row[0], datetime.datetime)]
amp = (max(temps) - min(temps)) / 2
mid = (max(temps) + min(temps)) / 2
peak_day = temps.index(max(temps)) + 1
def model(day):
return amp * math.cos(2 * math.pi * (day - peak_day) / 365) + mid
errors = [abs(temps[d - 1] - model(d)) for d in range(1, 366)]
print("amplitude", amp, "midline", mid, "peak on day", peak_day)
print("mean error, degrees:", round(sum(errors) / 365, 2))
How good is it?
Across all 365 days the model is off by 2.49 degrees on average. For a curve built from only two numbers and a date, that is respectable, and it captures the shape of the year.
Where it fails
In spring the model runs warm: on 1 June it predicts 17.60 degrees against 13.45 measured. The coldest day is 208 days before the warmest, not the 182.5 a symmetric sine wave assumes, so the river warms more slowly than it cools.
The winter floor
From January to March the monthly means stay between 0.89 and 1.15. River water at the intake cannot cool far below freezing, so the bottom of the curve is flat. A sinusoid has no flat parts, which is a limitation worth naming in any report.
Licence and attribution
Contains information licensed under the Open Government Licence - City of Ottawa (licence, Open Ottawa). The model, residuals and summaries are ours. The City of Ottawa does not endorse this page or our classes.
