Unit 3 • Lesson 5

Array Maths and Broadcasting

Overview

Arrays of different shapes can still meet, under one clear rule.

Intermediate 20–25 min

← Previous Next: Summing and Axes

What You Will Learn in This Lesson

By the end of this lesson, you will be able to:

Combine two arrays of the same shape without a loop or a zip.
State the broadcasting rule and apply it to a scalar and a row.
Predict which pairs of shapes cannot be combined.
Centre and scale a column in one expression.

Why This Matters

Broadcasting is why + 5 worked in the first lesson, and it is the reason a per-column adjustment does not need a loop. It is also the source of the shape errors that stop a script dead — and those errors are readable once you know the rule they are enforcing.

Step 1: Elementwise by default

Two arrays of the same shape combine position by position. No loop, no zip.

Same shape, straightforward

Every arithmetic operator works this way, and so do the comparisons you met in the last lesson.

import numpy as np

marks = np.array([45, 30, 50])
bonus = np.array([5, 10, 0])

print(marks + bonus)
print(marks * 2)
print(marks / 50)

The methods that reduce

Alongside the operators, an array carries its own summaries. Each collapses the array to one number, unless you say otherwise — which is the next lesson.

import numpy as np

marks = np.array([45, 30, 50])
print(marks.sum(), marks.mean(), marks.min(), marks.max())
print(round(marks.std(), 3))

Mini Practice #1: Weighted total

Try It Yourself

Two tests carry different weights. Multiply each column by its weight and total each student, in two expressions and no loop.

import numpy as np

marks = np.array([[80, 60],
                  [50, 90]])
weights = np.array([0.7, 0.3])

weighted = marks * weights
print(weighted)
print(None)   # one total per student

Press Run to see output

Step 2: Broadcasting stretches the smaller one

A single number is treated as if repeated to fit. That is why + 5 worked, and the same rule extends to whole rows and columns.

The rule

Line the shapes up from the right. Two dimensions are compatible if they are equal, or if one of them is 1. Anything else has no defined meaning and numpy says so rather than guessing.

import numpy as np

table = np.array([[1, 2, 3],
                  [4, 5, 6]])      # shape (2, 3)
per_column = np.array([10, 20, 30]) # shape (3,)

# (2, 3) and (3,) line up on the right: 3 == 3, and the missing
# dimension is treated as 1. The row is applied to every row.
print(table + per_column)

When it refuses

A shape mismatch is a real error, not a numpy quirk — there is no single sensible answer for adding three numbers to rows of four.

import numpy as np

try:
    np.array([1, 2, 3]) + np.array([1, 2, 3, 4])
except ValueError as exc:
    print("ValueError:", exc)

Centring, the everyday use

Subtracting the mean is broadcasting a single number across the whole array. Dividing by the standard deviation as well gives you a standardised column, which is the form most models want.

import numpy as np

values = np.array([2.0, 4.0, 6.0])
centred = values - values.mean()
print(centred)
print(centred / values.std())

Mini Practice #2: Read the shape error

Try It Yourself

This raises. Read the message, then change the second array so the shapes line up.

import numpy as np

table = np.array([[1, 2, 3],
                  [4, 5, 6]])
adjust = np.array([1, 2])

try:
    print(table + adjust)
except ValueError as exc:
    print("ValueError:", exc)

Press Run to see output

Step 3: When to Use It

Applying one value to everything

A scalar broadcasts across the whole array: offsets, scaling, unit conversions.

celsius = (fahrenheit - 32) * 5 / 9

A row or column against a table

Subtract a per-column mean from every row, where the shapes line up from the right.

When Not to Use It

Do not rely on broadcasting when you are not sure of the shapes. Print .shape first — a silent broadcast between the wrong axes gives a result of the wrong size, not an error.

End-of-Lesson Exercises

Exercise 1: Centre on the mean

Write centred(values) that returns a list of each value minus the mean of them all, rounded to two decimal places.

values - values.mean() does the whole thing. np.round then rounds the array, and tolist() turns it back into plain numbers.

import numpy as np

def centred(values):
    # Each value minus the average, rounded to 2dp.
    return []

print(centred(np.array([1.0, 2.0, 3.0])))

Press Run to try it, then Check my work to mark it.

Exercise 2: Apply a weight to every column

Write weighted_totals(table, weights) that multiplies each column of a 2-D array by the matching weight and returns one total per row, as a list. Let broadcasting apply the weights; do not loop over the columns.

(table * weights) scales every row by the weights, because the shapes line up on the right. Then sum along axis=1 for one total per row.

import numpy as np

def weighted_totals(table, weights):
    # Each column scaled by its weight, then one total per row.
    return []

print(weighted_totals(np.array([[80, 60], [50, 90]]), np.array([0.5, 0.5])))

Press Run to try it, then Check my work to mark it.

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