Understanding Axes

"Total runs per player" and "total runs per match" use the same data and the same function — sum — the only difference is the axis. The axis parameter is one of the most confusing parts of NumPy for beginners, so let's anchor it properly.

Definition

An axis is one dimension of an array, numbered from 0 in the same order as the shape tuple. When you pass axis=k to a reduction function such as sum, mean, max or min, NumPy collapses (reduces) that dimension: it combines all values along axis k into one value, and the result's shape is the original shape with entry k removed.

• Axis number = position in the shape. For shape (3, 14): axis 0 has length 3 (players), axis 1 has length 14 (matches).
• Reduction — a function that turns many values into one (sum, mean, max, min, std …).
• Collapse — the chosen axis disappears from the result's shape.
• No axis (axis=None, the default) — reduce everything to a single number.

Syntax

arr.sum(axis=None, keepdims=False)
np.sum(arr, axis=None, keepdims=False)
# same 'axis' parameter in: mean, max, min, std, var, prod, argmax, argmin, cumsum ...

Axes of a 2D array

team_scores.shape = (3, 14)
                     |   |
                 axis 0  axis 1
                (rows /  (columns /
                 players) matches)

axis=0  -> collapse the ROWS: go DOWN each column -> one value per column, shape (14,)

         [72, 45, 88, ...]   <- Virat
         [56,  0, 74, ...]   <- Rohit
         [45, 32,  0, ...]   <- Hardik
           |   |   |
           v   v   v
         [173, 77, 162, ...] <- per-match totals

axis=1  -> collapse the COLUMNS: go ACROSS each row -> one value per row, shape (3,)

         [72, 45, 88, 31, ...] --> 779   (Virat total)
         [56,  0, 74, 88, ...] --> 710   (Rohit total)
         [45, 32,  0, 67, ...] --> 580   (Hardik total)

Memory trick: axis=k makes dimension k disappear. axis=0 removes the first number of the shape (players gone → one value per match); axis=1 removes the second (matches gone → one value per player).

Example — sum(axis=…) on the team table

import numpy as np
team_scores = np.array([
    [72, 45, 88, 31, 64,  0, 103, 52, 19, 77, 41, 93, 28, 66],
    [56,  0, 74, 88, 32, 45,  61, 17, 90, 38, 72,  5, 83, 49],
    [45, 32,  0, 67, 18, 54,  78, 22, 41, 33, 60, 15, 88, 27],
])
print(team_scores.sum(axis=1))
print(team_scores.sum(axis=0))
print(team_scores.sum())
print(team_scores.mean(axis=1).round(1))

[779 710 580]
[173  77 162 186 114  99 242  91 150 148 173 113 199 142]
2069
[55.6 50.7 41.4]

How it works:

• axis=1 → per player totals (779, 710, 580): shape (3, 14) → (3,).
• axis=0 → per match totals (match 1: 72 + 56 + 45 = 173): shape (3, 14) → (14,).
• No axis → grand total of all 42 scores: 2069.
• mean(axis=1) → each player's average.

Example — a small marks table, result shapes, and keepdims

import numpy as np
marks = np.array([
    [80, 90, 70],
    [60, 85, 95],
    [75, 88, 92],
])
print(marks.sum(axis=0))
print(marks.sum(axis=1))
print(marks.sum())
print(marks.max(axis=0), marks.max(axis=1))
print(marks.sum(axis=0).shape, marks.sum(axis=1).shape)
print(marks.sum(axis=1, keepdims=True))
print(marks.sum(axis=1, keepdims=True).shape)
try:
    marks.sum(axis=2)
except Exception as e:
    print(type(e).__name__ + ":", e)

[215 263 257]
[240 240 255]
735
[80 90 95] [90 95 92]
(3,) (3,)
[[240]
 [240]
 [255]]
(3, 1)
AxisError: axis 2 is out of bounds for array of dimension 2

How it works: rows are students (Rahul, Aman, Munna), columns are subjects. axis=0 gives subject totals, axis=1 gives student totals. Both results have shape (3,) here only because the table is square — always reason from which axis disappears. keepdims=True keeps the collapsed axis as length 1, giving a column of shape (3, 1). A 2D array has only axes 0 and 1, so axis=2 raises an AxisError.

Axes of a 3D array

For two_seasons with shape (2, 3, 8) = (seasons, players, matches):

axis 0 = seasons  (2)   -> "across seasons":  combine 2023 + 2024
axis 1 = players  (3)   -> "across players":  combine Virat + Rohit + Hardik
axis 2 = matches  (8)   -> "across matches":  combine M1 + M2 + ... + M8

   sum(axis=0) -> shape (3, 8): each player's match-by-match total over both seasons
   sum(axis=1) -> shape (2, 8): team total per match, for each season
   sum(axis=2) -> shape (2, 3): each player's season total

import numpy as np
two_seasons = np.array([
    [[72, 45, 88, 31, 64,  0, 103, 52],
     [56,  0, 74, 88, 32, 45,  61, 17],
     [45, 32,  0, 67, 18, 54,  78, 22]],
    [[65, 80, 44, 91, 33, 58,  22, 71],
     [39, 55, 82,  0, 77, 21,  64, 48],
     [28, 44, 63, 37, 82,  0,  55, 71]],
])
print("axis0:\n", two_seasons.sum(axis=0))
print("axis1:\n", two_seasons.sum(axis=1))
print("axis2:\n", two_seasons.sum(axis=2))
print(two_seasons.sum(axis=(0, 2)))
print(two_seasons.sum(axis=-1))

axis0:
 [[137 125 132 122  97  58 125 123]
 [ 95  55 156  88 109  66 125  65]
 [ 73  76  63 104 100  54 133  93]]
axis1:
 [[173  77 162 186 114  99 242  91]
 [132 179 189 128 192  79 141 190]]
axis2:
 [[455 373 316]
 [464 386 380]]
[919 759 696]
[[455 373 316]
 [464 386 380]]

How it works:

• axis=0: seasons disappear → (3, 8). Virat's match 1 over both seasons = 72 + 65 = 137.
• axis=1: players disappear → (2, 8). Season 1, match 1 team total = 72 + 56 + 45 = 173.
• axis=2: matches disappear → (2, 3). Virat's 2023 season total = 455.
• axis=(0, 2): collapse seasons and matches → (3,): each player's career total (Virat 455 + 464 = 919).
• axis=-1 means "the last axis" — identical to axis=2 here. Negative axes are handy when you don't care how many dimensions come before.

Quick reference

Common mistakes

• ❌ "axis=0 means rows, so sum(axis=0) gives row totals." ✅ axis=0 collapses the rows, giving one value per column. Think "which dimension disappears?".
• ❌ Trusting the result shape on a square table to tell you which axis you used. ✅ Check with a recognisable value (e.g. Virat's total 779).
• ❌ Using axis=2 on a 2D array. ✅ Valid axes are 0 … ndim-1 (or -1 … -ndim).
• ❌ Forgetting keepdims=True when you want to divide each row by its own total. ✅ marks / marks.sum(axis=1, keepdims=True) — the (3, 1) column broadcasts correctly.

Interview tip

Q: For an array of shape (5, 4, 3), what is the shape of arr.mean(axis=1)? A: (5, 3) — axis 1 (length 4) is collapsed and removed; the other dimensions stay in order.