Foundations — Types of data (numerical, categorical, ordinal, text)

The parent note throws around $\mu$, $\sigma$, ${\mathbb{R}}^k$, ${\mathbf{e}}_i$, summation signs, Euclidean distance, and words like "variance" and "one-hot vector". If you have never met these, the derivations look like spells. This page builds every one of them from nothing, in the order they stack.

0. What is "data" in the first place?

Look at the figure. The whole topic lives inside a single question you ask column by column: what kind of property is this? The red column is the one we are inspecting. There are only three honest answers — number, name, or ranked-name — and the rest of this page defines the tools each answer needs.

1. Numbers, the number line, and $\in$

The picture of a real number is a dot on a horizontal ruler. "Continuous" data (height $175.3$) can land anywhere on the ruler. "Discrete" data (clicks $=5$) can only land on the evenly-spaced integer ticks.

2. The index $i$ and the summation sign $\sum$

The parent writes ${\sum }_{i=1}^n{x}_i$. Three symbols hide here.

The picture of $\sum$ is a conveyor belt: items $x_1,x_2,\dots$ roll past and drop into one bucket that keeps a running total.

3. The mean $\mu$ — the balance point

The picture is a seesaw. Drop each data value as a weight on the number line; $\mu$ is the exact spot where the plank balances. Values far to the right pull the balance point right; that is why one huge house price drags the mean up.

4. Spread, variance, and the standard deviation $\sigma$

Knowing the balance point is not enough — two datasets can share a mean but one is tightly bunched and the other wildly spread. We need a number for spread.

Why not just average the deviations? Because the left pulls (negative) and right pulls (positive) cancel to exactly zero — that is literally what "balance point" means. So we square each deviation first (a negative squared becomes positive), average those, then take the square root to undo the squaring.

The red band in the figure spans $\mu -\sigma$ to $\mu +\sigma$ — the "usual" zone. The square-then-root trick is why $\sigma$ is measured in the same units as the data (rupees, cm), unlike raw variance (rupees-squared).

5. Vectors, ${\mathbb{R}}^k$, and the basis arrow ${\mathbf{e}}_i$

Categorical encoding lands us in the notation ${\mathbf{e}}_i\in {\mathbb{R}}^k$. Let's earn it.

The figure shows ${\mathbf{e}}_1=[1,0]$ and ${\mathbf{e}}_2=[0,1]$. The red arrow is a chosen category living purely along its own axis, touching no other. That is exactly the promise of one-hot encoding: give each category its own private direction so the machine never thinks one category is "between" two others.

6. Distance — why $\sqrt{2}$ keeps appearing

The parent claims every two one-hot vectors sit $\sqrt{2}$ apart. Here is the tool behind that.

7. Order vs no-order: the $<$ symbol

The three data types differ exactly in whether $<$ is meaningful:

Prerequisite map

Equipment checklist

Cover the right side and check you can say each aloud.

Return to the parent topic once every checklist line answers itself instantly.