Area-weighting (cos-latitude) · Learn

In one lineOn a normal lat/lon grid, cells near the poles cover far less ground than cells at the equator. A plain average treats them as equal — so weight each cell by cos(latitude), its true share of area.

The trap

Climate grids are usually a regular lat/lon mesh — say 0.25° × 0.25°. That sounds uniform, but the Earth is a sphere: meridians (lines of longitude) converge toward the poles. A 0.25° cell at the equator is a ~28 × 28 km square; the same 0.25° cell at 60° latitude is only ~14 km wide — half the ground. At 80° it's a sliver.

So if you just average every cell equally (field.mean()), you give a tiny patch of Arctic the same vote as a big patch of tropics. With thousands of skinny polar cells, that bias is real — it can shift a "global mean" by a noticeable amount and exaggerate polar swings.

The fix

Weight each cell by how much area it actually covers. That area is proportional to the cosine of its latitude (cos 0° = 1 at the equator, cos 90° = 0 at the pole):

weighted mean = Σ(value × cos(lat)) / Σ(cos(lat))

That's it. Every honest spatial mean on this site — including the ERA5 trends on /verify — uses it.

Play with it

Each row is a latitude band; its drawn width is its true ground area (cos lat) — see them shrink toward the pole. Add a polar anomaly and watch the naïve average chase it while the area-weighted average barely moves.

Polar anomaly

naïve mean (every cell equal): area-weighted mean: error from skipping it:

Do it yourself

editable · runs in your browser

import numpy as np
# a tiny global grid; a uniform field of 1.0 so the TRUE area mean is exactly 1.0
lat = np.arange(-85, 86, 10.0)
field = np.ones((lat.size, 36))
w = np.cos(np.deg2rad(lat))[:, None]            # a cell's area is proportional to cos(latitude)
def wmean(f): return float((f * w).sum() / (w * np.ones_like(f)).sum())
print("uniform field -> the true mean is 1.0")
print("  naive .mean() =", round(float(field.mean()), 4))
print("  area-weighted =", round(wmean(field), 4))
# now make the poles hot and watch the naive mean inflate:
field[np.abs(lat) > 60] = 5.0
print("with hot poles:")
print("  naive    =", round(float(field.mean()), 3), " <- over-counts the tiny polar cells")
print("  weighted =", round(wmean(field), 3), " <- honest")

Two footnotes for later: cos-latitude is the right weight for a regular lat/lon grid; other projections (equal-area grids, EASE-Grid) bake the area in differently. And for a small region the effect is tiny — it matters most for wide or high-latitude domains.