# Notation: indices, bars, hats and Greek letters

> How to read x_i, sorted values, x̄ and ŷ, the Greek letters μ, σ, α, β and λ, Δ for a change, and sample vs population.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: The language of formulas
- Lesson 5/36, ~23 min
- Source: https://productovi.com/courses/math-basics/formula-notation

## A formula is written in shorthand

**Alex Tutor:** Module 2 is about the language of formulas. You're still on Klasna's product team, and from now on you'll read formulas from the statistics and finance courses without stumbling over the symbols.

**Sofia:** In the statistics lessons I keep seeing x with a small i, x with a bar, μ, σ... It looks like a secret code.

**Alex Tutor:** A code with very few rules. Start with the small letter below the line, the **index**. Five learners took 6, 9, 4, 7 and 14 lessons in March. Call the list x. The index says which value you mean: x₁ = 6, x₂ = 9, x₅ = 14.

**Alex Tutor:** xᵢ means "the i-th value, any of them", and **n** is how many values there are: n = 5. So i runs from 1 to n.

**Alex Tutor:** Quick check before the brackets: read the index as a position.

> **Exercise:** Five learners took 6, 9, 4, 7 and 14 lessons in March, in that order. Call the list x. What are n and x₄?

**Sofia:** And when the index is in brackets?

**Alex Tutor:** Then the values are sorted first. x₍₁₎ is the smallest, x₍ₖ₎ the k-th from the bottom. Sorted, our list is 4, 6, 7, 9, 14, so x₍₂₎ = 6, while x₂ = 9. The median of five values is the middle one, x₍₃₎ = 7.

![Five cats sit in a row in the order they arrived; below, the same five cats sit again, lined up from the smallest to the largest.](https://productovi.com/courses/math-basics/inline/m2-l1-i1.webp)

*The same five values in collection order and in sorted order.*

![The list 6, 9, 4, 7, 14 with indices x₁ to x₅; sorted it becomes 4, 6, 7, 9, 14, so x₂ = 9 but x₍₂₎ = 6, and the median is x₍₃₎ = 7.](https://productovi.com/courses/math-basics/diagrams/m2-l1-indices.png)

*An index in brackets counts from the bottom of the sorted list.*

> **Exercise:** Four learners took 3, 8, 5 and 12 lessons, in that order. What are x₃ and x₍₃₎?

## Bars and hats

**Alex Tutor:** A **bar** on top means "the average of". x̄, read "x bar", is the mean of the list: (6 + 9 + 4 + 7 + 14) ÷ 5 = 40 ÷ 5 = **8 lessons**. Later you'll meet p̄, an average share, and c̄m, an average margin.

**Sofia:** And the hat, like on ŷ?

**Alex Tutor:** A **hat** means "estimated" or "predicted by a model". A trend line said May would bring ŷ = 9,800 lessons. The actual number was y = 9,620. The gap y − ŷ = −180 is the model's error for that month, the **residual**.

**Sofia:** So y is what happened, ŷ is the model's guess, and the order y − ŷ tells me the model guessed too high.

**Alex Tutor:** Exactly: a negative residual means the model was too optimistic. Bar: an average of. Hat: a guess about.

> **Exercise:** A model predicted ŷ = 1,250 sign-ups for a week; the week brought y = 1,190. What is the residual y − ŷ?

## Greek for the whole, Latin for the sample

**Sofia:** Why μ and σ, if x̄ is already the mean?

**Alex Tutor:** Because they describe different groups. In March Klasna had **N = 1,120** active learners: that's the **population**. Our five are a **sample**, n = 5. Capital N for the whole, small n for the part you looked at.

**Alex Tutor:** The population's mean is **μ** (mu) and its spread is **σ** (sigma). We almost never see them. We compute x̄ and s from the sample and use them as estimates: x̄ = 8 is our best guess for μ.

![The population of N = 1,120 learners has a true mean μ and spread σ we cannot see; a sample of n = 5 gives x̄ = 8 and s, which estimate them.](https://productovi.com/courses/math-basics/diagrams/m2-l1-sample-population.png)

*Latin letters are computed from the sample; Greek letters describe the whole.*

**Sofia:** And α, β, λ?

**Alex Tutor:** **α** (alpha) is the false-alarm level you choose for a test, usually 0.05. **β** (beta) is the chance of missing a real effect, and in regression the same letter names the coefficients. **λ** (lambda) shows a factor's strength in factor analysis.

**Alex Tutor:** So one letter can mean different things in different lessons. The caption under the formula is the dictionary: always read it.

> **Exercise:** You surveyed 200 of Klasna's learners and got an average of 4.3 out of 5. It is correct to write μ = 4.3.

> **Exercise:** Match each symbol with what it means.

**Alex Tutor:** Back to α, β and λ for a moment. What does this test plan mean?

> **Exercise:** A test plan says: α = 0.05. What does α stand for here?

## Δ: how much it changed

**Alex Tutor:** **Δ** (delta) means "the change in". ΔQ = Q_new − Q_old. Klasna delivered 7,840 lessons in March and 9,620 in May, so ΔQ = 9,620 − 7,840 = **+1,780 lessons**.

**Sofia:** And ΔQ%?

**Alex Tutor:** The same change as a share of the old value, the relative change from module 1: 1,780 ÷ 7,840 = 0.2270, so **+22.70%**. The finance course also writes Δspend for extra marketing money and ΔAR for the change in what customers owe.

**Alex Tutor:** One more mark: a **brace** under part of a formula is just a label for that part.

$$
CM = \underbrace{P - 0.025P}_{\text{Klasna keeps}} - \underbrace{(T + V)}_{\text{costs per lesson}}
$$

*The same contribution margin as in module 1; the braces only name the parts: what is left of the price after the 2.5% fee, and the tutor and video costs.*

**Sofia:** So a brace changes nothing in the arithmetic. It just tells me what each chunk is.

> **Exercise:** Klasna Club, the school's paid monthly membership, had 640 subscribers in April and 712 in May. What is ΔQ%, the relative change, in percent?

**Alex Tutor:** And the braces from the formula above: do they change anything?

> **Exercise:** In CM = P − 0.025P − (T + V), drawing a brace under (T + V) labelled "costs per lesson" changes how the formula is computed.

## Your turn: fix the note

**Alex Tutor:** A colleague posted in the team chat: "μ = 8 lessons per learner, from our 5-learner sample." The number is fine; the symbol isn't.

**Sofia:** Because 8 came from five people, not from all of them.

**Alex Tutor:** Right. Rewrite the note so a statistician wouldn't wince, and say what each symbol stands for.

> **Exercise:** A colleague writes: "μ = 8 lessons per learner, from our 5-learner sample." Klasna had 1,120 active learners that month. Rewrite the note correctly and explain in one or two sentences what x̄, μ, n and N stand for here.

## Where you will meet this

> **In short**
>
> - **xᵢ** is the i-th value as collected; **x₍ₖ₎** is the k-th after sorting from the smallest.
> - A **bar** means an average (x̄, p̄); a **hat** means a model's prediction (ŷ). The residual is y − ŷ.
> - **Greek** letters (μ, σ) describe the whole population; **Latin** ones (x̄, s) are computed from the sample. **N** is the population size, **n** the sample size.
> - α, β and λ are set by the lesson: read the caption.
> - **Δ** is new − old; ΔQ% is that change divided by the old value. A brace only labels a part.

> **These lessons use it**
>
> - [Mean, median and mode](/courses/statistics-for-pm-and-marketers/typical-value): the median through sorted values x₍ₖ₎.
> - [Sample vs population and the normal distribution](/courses/statistics-for-pm-and-marketers/samples-and-normal-distribution): μ and σ against x̄ and s.
> - [p-value and statistical significance](/courses/statistics-for-pm-and-marketers/p-value): α as the level you set before the test.
> - [Simple linear regression](/courses/statistics-for-pm-and-marketers/linear-regression): ŷ and the residuals y − ŷ.
> - [Price vs volume](/courses/finance-for-product/price-vs-volume): ΔQ% for the volume a discount must win.
> - [Case: plan vs actual](/courses/finance-for-product/plan-vs-actual-case): braces that label the volume, mix and price parts.

**Alex Tutor:** Next time: the Σ sign, the shorthand for "add them all up".

- Previous lesson: [Back-of-the-envelope estimates](https://productovi.com/courses/math-basics/back-of-the-envelope-estimates.md)
- Next lesson: [The Σ sign: sums, limits and running totals](https://productovi.com/courses/math-basics/the-sum-sign.md)

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