# The Σ sign: sums, limits and running totals

> Read Σ with its limits, see why deviations from the mean add up to zero, use sums of products, running totals and double indices, and two shortcuts: n(n + 1)/2 and k(k − 1)/2.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: The language of formulas
- Lesson 6/36, ~26 min
- Source: https://productovi.com/courses/math-basics/the-sum-sign

## Σ means "add them up"

**Alex Tutor:** Still on Klasna's product team, still learning to read formulas. Today's sign is the one statistics uses most.

**Sofia:** Almost every statistics formula has a big Σ in it. What exactly does it ask me to do?

**Alex Tutor:** Add things up. Σ is the Greek capital sigma, for "sum". The part after it says **what** to add, and the small marks below and above say **which ones**: start at i = 1 and stop at n.

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

*i = 1 below: the first term; n above: the last; xᵢ: what is added each time. The mean is that sum divided by n.*

**Alex Tutor:** Our five learners from last time took 6, 9, 4, 7 and 14 lessons. Σxᵢ = 40, and x̄ = 40 ÷ 5 = 8. The mean formula is just "add them all, divide by how many".

**Sofia:** And if the limits are different, say from 2 to 4?

**Alex Tutor:** Then add only x₂, x₃ and x₄: 9 + 4 + 7 = 20. The limits are a filter on the index. When a formula drops them and writes just Σx, it means "all of them".

> **Exercise:** The values are 3, 8, 5, 12 and 10, in that order. Compute the sum from i = 2 to 4: x₂ + x₃ + x₄.

## Deviations always add up to zero

**Alex Tutor:** Now a sum you'll see in every spread formula: Σ(xᵢ − x̄). Subtract the mean from each value first, then add. For our list: −2, +1, −4, −1 and +6.

**Sofia:** That's... zero.

**Alex Tutor:** Always, for any list. The mean is the balance point: what sits below it exactly cancels what sits above. That's why you can't measure spread as an average deviation, and why statistics squares them first.

**Alex Tutor:** Check that claim in general, not just for our list.

> **Exercise:** For any list of numbers, the deviations from the mean add up to zero: Σ(xᵢ − x̄) = 0.

![On a number line the values 4, 6, 7, 9 and 14 sit around the mean 8: deviations below add to −7, above to +7, so their sum is 0.](https://productovi.com/courses/math-basics/diagrams/m2-l2-deviations.png)

*The mean is the balance point of the list.*

**Alex Tutor:** One more thing about order. Σxᵢ² means square each value, then add: 36 + 81 + 16 + 49 + 196 = 378. (Σxᵢ)² means add first, then square: 40² = 1,600. The brackets decide.

> **Exercise:** For the values 2, 3 and 5, what is Σxᵢ²?

## Sums of products

**Alex Tutor:** Often the thing being added is a product. In March Klasna sold English lessons in three packages: 1,200 lessons at 650 UAH, 3,400 at 620 and 2,000 at 580. Revenue is Σ qᵢ · pᵢ, quantity times price, package by package.

**Alex Tutor:** 780,000 + 2,108,000 + 1,160,000 = **4,048,000 UAH**. Multiply inside each term first, then add the terms, as in the order of operations from module 1.

**Sofia:** And divided by Σqᵢ = 6,600 lessons, that's an average price of about 613.33 UAH.

**Alex Tutor:** Yes, and that average is weighted by quantity. Module 3 is built on it. Correlation in statistics is a sum of products too: Σ(xᵢ − x̄)(yᵢ − ȳ).

> **Exercise:** In one week three tutor rates applied: 40 lessons at 390 UAH, 25 at 430 and 10 at 500. Compute the total payout Σ qᵢ · pᵢ, in UAH.

## Running totals and two indices

**Alex Tutor:** When the upper limit moves, you get a **running total**: Sₖ = x₁ + … + xₖ. A cohort of learners brings 120, 100, 90 and 80 UAH of margin per learner in months 1 to 4. The running totals are 120, 220, 310 and 390.

**Sofia:** That's what an LTV curve plots, right?

**Alex Tutor:** Right. And payback is the first month where the running total reaches what a learner cost. Find it.

> **Exercise:** A cohort brings 120, 100, 90 and 80 UAH of margin per learner in months 1 to 4. Winning a learner cost 300 UAH. In which month does the running total first reach that cost?

**Alex Tutor:** Now two indices. To forecast active users, product analytics adds up cohorts: MAUₜ = Σ N_c · r₍ₜ₋c₎.

**Alex Tutor:** N_c is how many joined in month c; r₍ₜ₋c₎ is the share still active t − c months later. Cohorts of 400, 500 and 600 joined in January to March; retention is 100%, 50% and 40% at ages 0, 1 and 2. March: 400 × 0.4 + 500 × 0.5 + 600 × 1 = **1,010**.

![A cohort grid: January 400 → 200 → 160, February 500 → 250, March 600. Adding the March column gives 160 + 250 + 600 = 1,010 active learners.](https://productovi.com/courses/math-basics/diagrams/m2-l2-cohort-sum.png)

*One index walks down the cohorts, the other across the months.*

> **Exercise:** Cohorts of 300, 450 and 500 learners joined in April, May and June. Retention is 100% at age 0, 60% at age 1 and 45% at age 2. Using MAUₜ = Σ N_c · r₍ₜ₋c₎, how many are active in June?

## Two shortcuts

**Alex Tutor:** First: 1 + 2 + … + n = n(n + 1)/2. Pair the first with the last: 1 + 10, 2 + 9 and so on, each pair is n + 1, and there are n/2 pairs. So 1 + … + 10 = 10 × 11 ÷ 2 = 55. It's the total of ranks in the Mann–Whitney test.

**Alex Tutor:** Use the pairing trick on a longer list.

> **Exercise:** Twenty test users are ranked from 1 to 20. What is the total of all the ranks, 1 + 2 + … + 20?

**Alex Tutor:** Second: how many pairs can you make from k things? Each of the k can pair with k − 1 others, but that counts every pair twice, so k(k − 1)/2. Four test variants give 4 × 3 ÷ 2 = 6 pairwise comparisons.

**Sofia:** So pairs grow much faster than variants.

> **Exercise:** An A/B/n test has 6 variants. How many different pairs of variants can be compared?

## Your turn: a spread that is always zero

**Alex Tutor:** A colleague proposes a simple measure of how much lesson counts vary: "the average of xᵢ − x̄ across learners". Explain what will happen.

**Sofia:** I can already guess the answer for every list.

> **Exercise:** A colleague wants to measure how much lesson counts vary with (1/n) Σ (xᵢ − x̄). What result will this give for any list, why, and what do statistics formulas do instead? Use the list 6, 9, 4, 7, 14 to show it.

## Where you will meet this

> **In short**
>
> - **Σ** adds up what follows it; the limits below and above say which terms. No limits means all of them.
> - **Σ(xᵢ − x̄) = 0** for any list, so spread formulas square the deviations.
> - **Σxᵢ²** squares then adds; **(Σxᵢ)²** adds then squares.
> - **Σ qᵢ · pᵢ** multiplies inside each term, then adds. A moving upper limit gives a **running total**.
> - Two indices, as in **Σ N_c · r₍ₜ₋c₎**, walk over cohorts and ages at once.
> - **1 + … + n = n(n + 1)/2**; **k things make k(k − 1)/2 pairs**.

> **These lessons use it**
>
> - [Measures of spread](/courses/statistics-for-pm-and-marketers/spread): Σ(x − x̄)² inside the variance.
> - [Mann–Whitney](/courses/statistics-for-pm-and-marketers/mann-whitney): U = R − n(n + 1)/2.
> - [One-way ANOVA](/courses/statistics-for-pm-and-marketers/anova): sums of squares and the 6 pairs among 4 variants.
> - [Correlation coefficient](/courses/statistics-for-pm-and-marketers/correlation): a sum of products of deviations.
> - [The LTV curve](/courses/product-analytics/ltv-curve): a running total divided by cohort users.
> - [Forecasting active users](/courses/product-analytics/forecasting): MAUₜ = Σ N_c × r₍ₜ₋c₎.
> - [A twelve-month cash plan](/courses/finance-for-product/twelve-month-cash-plan): a month-by-month running balance.

**Alex Tutor:** Next time: rearranging a formula, so you can solve for the number you actually need.

- Previous lesson: [Notation: indices, bars, hats and Greek letters](https://productovi.com/courses/math-basics/formula-notation.md)
- Next lesson: [Rearranging a formula: solve for what you need](https://productovi.com/courses/math-basics/rearranging-a-formula.md)

---
Productovi Learning. Free, no sign-up needed to read.