Some basic maths

Core idea

Statistics needs three pieces of mathematical machinery: algebra (letters as placeholders for unknown numbers, with strict rules of precedence); coordinate graphs (positions on a plane encoding pairs of numbers, with linear equations producing straight lines); and a small Greek-letter notation (Σ to sum, X̄ to average). Get these three on solid ground and the rest of the discipline reads as plain English.

Why it matters

Most readers stall on statistics not because the ideas are hard but because the symbols feel hostile. Σ(X − X̄)² looks like noise until you decode it: "sum, over every value, of the squared distance from the mean." Once the symbols are demystified, formulas become recipes. And the same algebra that converts Celsius to Fahrenheit also computes a regression line — same machinery, different application.

Mental model

Algebraic precedence as a tree

When you read F = 1.8C + 32, the order of operations is fixed by convention, not by left-to-right reading. Multiplication binds tighter than addition. Visualise the expression as a tree: multiplications and divisions form the inner branches; additions and subtractions are the outermost trunk.

Copy sourceZoom

Linear equations on a plane

Every equation of the form Y = a + bX plots as a straight line. The number a is where the line crosses the Y axis (its intercept); the number b is how steeply it climbs (its slope). Change a and the line slides up or down; change b and the line tilts.

Copy sourceZoom

Sigma and the mean

Σ is the capital Greek letter sigma. It is shorthand for "add up everything that follows." ΣX means "add up all the X values"; ΣXY means "for each row, multiply X by Y, then add up those products." The mean is then X̄ = ΣX / n, where n is the count of values.

Practical application

Treat any statistical formula as a recipe with three parts: what's being summed, what's being averaged, and what's being squared.

Example

A freelance designer charges a £75 onboarding fee plus £60 per hour of work. Encode the relationship algebraically.

• In words: total bill equals £75 plus £60 per hour.
• As a formula: B = 75 + 60H, where B is the total bill in pounds and H is hours worked.
• As a graph: intercept at B = 75 when H = 0; slope of 60 (the line rises £60 for every additional hour).

For a 4-hour job: substitute H = 4 into the formula. B = 75 + 60 × 4 = 75 + 240 = 315. The bill is £315. Notice precedence — the multiplication runs before the addition, exactly as the formula's structure requires.

Now suppose the designer takes on five clients in a month with the following hour-counts: 3, 5, 2, 8, 4. What's the mean (average) job length?

• ΣH = 3 + 5 + 2 + 8 + 4 = 22
• n = 5 (five jobs)
• H̄ = ΣH / n = 22 / 5 = 4.4 hours

The same machinery — substitution, precedence, sigma sums, division by count — runs every statistical calculation later in the book. The notation is just shorthand for procedures you can do longhand any time.

Related lessons