Foundations›Core· 30 min read

Vectorised Maths & Handy NumPy Functions

Do maths on a whole array in one line — no loops — and get instant summaries like mean and max.

What you will learn

Add and combine arrays element-by-element
Use summary functions (mean, sum, max)
Understand “vectorised” operations

Vectorised = the whole array at once

Doing maths on an entire array in one line, with no loop, is called a vectorised operation. It is shorter to write and far faster to run. This is NumPy’s superpower.

Think of it like a checkout queue. With a Python list you ring up each item one at a time, in a loop. With a NumPy array it is as if every item is scanned at once — same result, far less waiting.

Two arrays of the same size combine element by element — the first with the first, the second with the second, and so on.

Add two arrays element by element

import numpy as np

prices = np.array([100, 200, 300])     # before tax
tax    = np.array([18,  36,  54])      # 18% tax on each

total = prices + tax        # element-by-element, no loop!
print(total)

Note: Output: [118 236 354] 100+18, 200+36, 300+54 — all done in one line. With plain lists you would need a for-loop to do this.

The same thing with a loop (so you see the difference)

To really feel why vectorising is nicer, here is the loop you would write without NumPy to add the same two lists. It does the identical job — just longer and slower.

The long way with plain Python lists and a for-loop

prices = [100, 200, 300]
tax    = [18,  36,  54]

total = []
for i in range(len(prices)):       # walk every position by hand
    total.append(prices[i] + tax[i])
print(total)

Note: Output: [118, 236, 354] Same answer as prices + tax — but five lines instead of one, and much slower on big data. That one-line NumPy version is what “vectorised” means.

Instant summaries

NumPy gives you one-word answers about an array: its average, total, biggest and smallest. These are the questions you ask of data every single day.

Summary functions on an array

scores = np.array([72, 85, 90, 65, 88])

print('mean :', scores.mean())   # average
print('sum  :', scores.sum())    # total
print('max  :', scores.max())    # highest
print('min  :', scores.min())    # lowest

Note: Output: mean : 80.0 sum : 400 max : 90 min : 65 Five numbers summarised in four lines. The class average is 80, the top score 90, the lowest 65.

A function applied to every number

Vectorising is not just for + and *. NumPy functions like np.sqrt also run on the whole array at once, returning a new array with the function applied to every item.

np.sqrt runs on every element at once

import numpy as np

areas = np.array([4, 9, 16, 25])   # square areas
sides = np.sqrt(areas)             # square root of each, in one call
print(sides)

Note: Output: [2. 3. 4. 5.] √4=2, √9=3, √16=4, √25=5 — all four in a single call, with no loop. Almost every NumPy function works this way.

Function	Answers the question	Example result
`.mean()`	What is the average?	80.0
`.sum()`	What is the total?	400
`.max()` / `.min()`	Highest / lowest?	90 / 65
`.std()`	How spread out is it?	(see Stats unit)
`np.sqrt(a)`	Square root of each item	[…]

Tip: You can compare a whole array at once too: scores > 80 returns [False True True False True]. This trick — a True/False array — is how you will filter data in Pandas.

Watch out: Two arrays must be the same size to add or multiply element-by-element. Adding a 3-item array to a 5-item one raises a shape error.

Q. What does np.array([2, 4, 6]).mean() return?

Answer: mean() is the average: (2 + 4 + 6) / 3 = 4.0.

✍️ Practice

Create an array of daily temperatures and print the mean, max and min.
Make two arrays (prices and discounts) of the same length and subtract them to get final prices.

🏠 Homework

Make an array of your last five quiz scores. Print the average, then print which scores were above the average using a comparison like scores > scores.mean().