Foundations — z-test, t-test, chi-squared goodness of fit, F-test

Before you can read the parent note comfortably, you need to own every symbol it throws at you. This page builds each one from nothing: what it means in plain words, what picture it draws, and why the topic can't live without it. Read top to bottom — each block uses only symbols defined above it.

1. Data, samples, and the humble subscript

Picture a row of dots on a number line — each dot is one $x_i$, and there are $n$ of them.

2. The average $\bar{x}$ and the true mean $\mu$

The bent-E symbol $\sum$ (capital Greek sigma) just says "add them all up." The $i=1$ underneath and $n$ on top mean "let $i$ run from $1$ to $n$." So ${\sum }_{i=1}^n{x}_i$ is shorthand for $x_1+x_2+\cdots +x_n$ — nothing more.

3. Spread: variance ${\sigma }^2$, standard deviation $\sigma$, and their sample cousins $s^2,s$

A middle isn't enough — we need to know how wobbly the data is around that middle.

Why squared distances and not plain distances? Plain distances $x_i-\bar{x}$ cancel out — the ones above the mean exactly kill the ones below (their sum is always zero). Squaring stops that cancellation, and it happens to be the spread measure that all the reference distributions are built around.

Why divide by $n-1$ and not $n$? This is Bessel's Correction. We measure distances from $\bar{x}$, but $\bar{x}$ was pulled toward our own data, so those distances come out a touch too small. Dividing by the smaller number $n-1$ nudges the answer up to compensate. The parent note needs this the moment $\sigma$ is unknown.

4. Standard error $\mathrm{SE}=\sigma /\sqrt{n}$ — the wobble of the average

Here is the pivot that makes the z-test and t-test work.

Why $\sqrt{n}$ and not $n$? Averaging lets errors partly cancel, but randomly, so the shrinkage grows like the square root of how many you averaged, not linearly. This is the single most important number-fact the topic uses: quadruple your data, halve the wobble of the average.

5. The bell curve: $\mathcal{N}(0,1)$ and the family of reference shapes

The topic then meets three other reference shapes, each a distortion of the bell for a specific job:

• Student's t-distribution — a bell with fatter tails, used when we had to estimate the spread.
• Chi-squared Distribution — a right-skewed curve for adding up squared surprises of counts.
• F-distribution — a curve for a ratio of two spreads.

You don't need their formulas yet — just know each is a "what does luck alone do?" curve.

6. The standardizing move: $Z$

This ratio is the template the parent note repeats four times. Everything else is choosing the right denominator and the right reference curve.

7. Greek letters and Yes/No machinery

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.