Foundations — Chi-squared, t, F distributions — definition, degrees of freedom

This page assumes you know nothing. We will name every squiggle the parent note uses, draw the picture behind it, and say why the topic cannot live without it. Read top to bottom — each item leans on the one above.

1. A random variable, and $\sim$ ("is distributed as")

The wiggle $\sim$ is read "is distributed as". It does not mean "approximately equal". It means "the chance-behaviour of the left side is described by the recipe on the right."

$X\sim N(\mu ,{\sigma }^2)\text{reads}\text{"}X\text{ follows a normal distribution with these settings."}$

WHY the topic needs it: the whole chapter is a catalogue of histogram-shapes (${\chi }^2$, $t$, $F$) and the rule for which shape a computed quantity obeys.

2. The normal distribution $N(\mu ,{\sigma }^2)$ and its two dials

Look at the figure. The peak sits at $\mu$. Move $\mu$ and the whole bell slides sideways. The distance from the centre to the "shoulder" (where the curve bends) is exactly $\sigma$. A bigger $\sigma$ flattens and widens the bell; a smaller $\sigma$ makes it tall and narrow.

WHY the topic needs it: the parent note builds everything from normals. To understand it we must first be fluent in what "centre" and "spread" mean visually.

See Standard Normal Distribution for the fully-detailed treatment.

3. The standard normal $Z$ — the single Lego brick

WHY the topic needs it: if we always convert to $Z$, we only ever need one table / one shape as the raw ingredient. ${\chi }^2$, $t$, $F$ are all "$Z$ cooked three different ways."

4. Expectation $E[\cdot ]$ — the long-run average

Two facts we'll lean on hard:

For the standard normal, $E[Z]=0$ (it's centred at 0) and $E[Z^2]=1$ (proved in the parent note). Keep these two numbers — they seed every mean formula in the chapter.

5. Variance $\mathrm{Var}(\cdot )$ and the independence rule

The second form ("mean of the square minus square of the mean") is the workhorse the parent uses to get $E[Z^2]=\mathrm{Var}(Z)+(E[Z]{)}^2=1$.

WHY the topic needs it: degrees of freedom counts independent pieces. No independence, no clean df.

6. Squaring a normal: $Z^2$ and where ${\chi }^2$ begins

Look at the figure: the symmetric bell of $Z$ (top) becomes the lopsided, always-positive shape of $Z^2$ (bottom). Squaring folds the negative half onto the positive half and stretches the tail. Summing $k$ of these independent squares is the definition of ${\chi }_k^2$.

WHY the topic needs it: ${\chi }^2$ literally is a sum of squared $Z$'s. You cannot read its definition without being comfortable that $Z^2\ge 0$ always and has a skewed shape.

7. Degrees of freedom $k$ — counting free pieces

The rule to memorise: df = (number of data points) − (number of parameters you estimated from them). This is the single most-tested idea in the topic.

See Sample Variance and Bessel's Correction for the $n-1$ story in full.

8. The sample tools: $n$, $\bar{x}$, $s^2$, and $\Sigma$

WHY the topic needs it: $t$ and ${\chi }^2$ in practice are made of $\bar{x}$ and $s^2$; the whole reason $t$ exists is that we must use the random $s$ in place of the unknown true $\sigma$.

9. The three assembled symbols at a glance

Now every ingredient is defined, the parent's headline objects read cleanly:

The fraction bar, the square root  ​, and division here are ordinary school arithmetic — the only new content was steps 1–8. That is the point of this page: once the bricks are named, the buildings are simple.

Prerequisite map

Each foundation flows into the next; the bottom node $K$ is the parent topic Chi-squared, t, F distributions — definition, degrees of freedom. The Central Limit Theorem — Central Limit Theorem — is why $\bar{x}$ is normal in the first place, and the ${\chi }^2$ shape is a special case of the Gamma Distribution. These distributions then power Hypothesis Testing and ANOVA.

Equipment checklist

Cover the right side; can you answer before revealing?