sum sign (Σ)

Also called: sigma notation, summation, Σ sign

The Greek capital sigma, meaning "add up": Σ with i = 1 below and n above adds x₁ + x₂ + … + xₙ. The term after it says what to add; the limits say which terms.

The Σ sign is shorthand for a sum. Read it in three parts: the expression after Σ is what you add each time, the line below (i = 1) is where the index starts, and the number above (n) is where it stops:

$\sum_{i=1}^{n} x_i = x_1 + x_2 + \dots + x_n$

When a formula drops the limits and writes just Σx, it means "all of them". The mean is the simplest use: $\bar{x} = \frac{1}{n}\sum x_i$ , add everything and divide by how many.

The expression inside can be anything: squares (Σxᵢ², square each, then add), products (Σqᵢpᵢ, multiply in each term, then add) or deviations (Σ(xᵢ − x̄), which always equals zero). Brackets decide the order: (Σxᵢ)² adds first and squares the total.

Example

Five learners took 6, 9, 4, 7 and 14 lessons.

Σxᵢ = 40, so x̄ = 40 ÷ 5 = 8.
From i = 2 to 4: 9 + 4 + 7 = 20.
Σxᵢ² = 36 + 81 + 16 + 49 + 196 = 378, while (Σxᵢ)² = 40² = 1,600.
Σ(xᵢ − x̄) = −2 + 1 − 4 − 1 + 6 = 0.

Common mistakes

Mixing up Σx² and (Σx)². One squares then adds, the other adds then squares.
Ignoring the limits. Σ from 2 to 4 is not the whole list.
Measuring spread with Σ(x − x̄). It is always zero; spread formulas square the deviations first.