Python math and statistics

3Math and Statistics: explained step by step

Python can do arithmetic the moment you open it, but real programs soon need more: the long side of a triangle, the number of pages for 47 items, the average mark of a class, an invoice that adds up to the exact cent. This lesson covers the tools for each, and the places where plain decimal numbers (floats) let you down.

Built-in helpers: no import needed

Python already gives you abs() to drop the sign, round() to round, and divmod(a, b), which returns the quotient and the remainder in one call. That turns 135 minutes into hours and minutes.

print(abs(-7.5))
print(round(3.14159, 2))
hours, minutes = divmod(135, 60)
print(hours, "h", minutes, "min")

The math module

import math unlocks the functions of a scientific calculator. You call each one as math.name(...), so a reader can see where it came from.

import math

print(math.sqrt(144))
print(math.pi, math.e)
print(math.hypot(3, 4))
print(math.log10(1000), math.log2(8), math.log(math.e))
print(math.factorial(5), math.comb(5, 2), math.perm(5, 2))
print(math.gcd(12, 18), math.lcm(4, 6))

• sqrt(x) is the square root. It always returns a float.
• hypot(a, b) is the long side of a right-angled triangle, sqrt(a*a + b*b).
• log(x) is the natural logarithm, log10 and log2 use base 10 and 2, and log(x, base) takes any base.
• factorial(n) is n!. comb(n, k) counts the ways to choose k things out of n when order does not matter, and perm(n, k) when it does.
• gcd is the greatest common divisor and lcm the lowest common multiple.

For powers, write 2 ** 3; math.pow(2, 3) always returns a float.

Angles are in radians

sin, cos and tan expect radians, not degrees. Convert with math.radians() on the way in and math.degrees() on the way out.

import math

print(math.sin(90))
print(math.sin(math.radians(90)))
print(round(math.cos(math.radians(60)), 2))
print(math.degrees(math.pi))

The first line treats 90 as 90 radians, a different angle altogether.

Four ways to get a whole number

They treat the decimal part differently, and negative numbers show it best:

• math.floor(x) always goes down, toward negative infinity.
• math.ceil(x) always goes up. Use it for "how many boxes or pages do I need?".
• math.trunc(x) (the same as int(x)) cuts the decimals off, which means toward zero.
• round(x) picks the nearest whole number. An exact half goes to the even neighbour, so round(2.5) is 2 and round(3.5) is 4. This is banker's rounding: always rounding halves up would push long columns of figures upward, and going to the even side balances that out. A negative second argument rounds to tens or hundreds, so round(1234, -2) is 1200.

import math

for x in (2.5, 3.5, -2.5):
    print(x, math.floor(x), math.ceil(x), math.trunc(x), round(x))
print(round(3.14159, 2), round(1234, -2))
print(math.ceil(47 / 10))

Floats are close, not exact

Computers store a float in binary, and a value such as 0.1 has no exact binary form. Tiny errors appear, and they surface when you compare: 0.1 + 0.2 == 0.3 is False. Use math.isclose() for floats.

import math

total = 0.1 + 0.2
print(total == 0.3, total)
print(math.isclose(total, 0.3))
print(math.isclose(0, 1e-12))
print(math.isclose(0, 1e-12, abs_tol=1e-9))

By default isclose allows a small difference relative to the size of the numbers. Next to zero any relative difference is huge, so when one side may be 0 add abs_tol.

The statistics module

To summarise a list of numbers, use statistics instead of writing the formulas yourself.

import statistics

marks = [62, 70, 70, 75, 81, 90, 98]
print(statistics.mean(marks))
print(statistics.median(marks), statistics.mode(marks))
print(statistics.multimode([1, 1, 2, 2, 3]))
print(round(statistics.stdev(marks), 2), round(statistics.pstdev(marks), 2))
print(statistics.quantiles(marks, n=4))

• mean is the total divided by the count. median is the middle value once sorted, or the average of the two middle values when the count is even.
• mode returns the most common value. multimode returns a list of all values that tie.
• stdev measures how spread out the data is. Use it when your list is a sample of something bigger, and pstdev when it is the whole population. variance is the standard deviation squared.
• quantiles(data, n=4) returns the three cut points that split the data into quarters.

The mean is easy to drag around, and the median is not: one huge value changes the mean a lot and the median hardly at all.

import statistics

salaries = [30, 32, 35, 36, 400]
print(statistics.mean(salaries), statistics.median(salaries))

Exact numbers: Decimal and Fraction

For money, binary floats are the wrong tool. Decimal stores decimal digits exactly, so build it from a string. quantize() rounds to a chosen number of places, and you choose the rounding rule. Fraction keeps a ratio exact, so one third plus one sixth really is one half.

from decimal import Decimal, ROUND_HALF_UP
from fractions import Fraction

print(Decimal("0.1") + Decimal("0.2") == Decimal("0.3"))
print(Decimal(0.1))
print(round(2.675, 2), Decimal("2.675").quantize(Decimal("0.01"), rounding=ROUND_HALF_UP))
print(Fraction(1, 3) + Fraction(1, 6))

Decimal(0.1) copies the float's error along with it. And round(2.675, 2) gives 2.67 because the float behind 2.675 sits a hair below it.

When the input is bad

A math function raises ValueError: math domain error when the maths has no answer, such as the square root of a negative number. The statistics functions raise StatisticsError (a kind of ValueError) when there is not enough data: mean([]), or stdev of a single value. Check the data first, or catch the error.

import math
import statistics

for value in (16, -4):
    try:
        print(math.sqrt(value))
    except ValueError as error:
        print("Cannot take the root of", value, "->", error)

try:
    statistics.mean([])
except statistics.StatisticsError as error:
    print("No data:", error)

Which tool, when?

Operators and built-ins cover everyday sums. Use math for roots, logarithms, angles and rounding direction, statistics to summarise a list, Decimal for money and Fraction for exact ratios. For millions of numbers, switch to NumPy.