Where rock-pile sit, how far rock-pile scatter

Ugg have big pile of number-rocks. Too many to hold in head. So Ugg squish pile down to few smart numbers. Two questions only. Where is MIDDLE of rocks? How far rocks SPREAD? Answer both, Ugg know the whole pile.

Middle of the pile — three ways

Mean — the average. Add up all rocks, split even between all:

$\bar{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$

Add every rock, divide by how many rocks. Mean is the BALANCE POINT — lay rocks on flat log, mean is where log tip-tip balance. Mean use every rock, so mean sharp and exact. But Ugg WARN: one giant mammoth-rock drag the balance far. One rich chief in poor tribe make "average" tribe look rich. Mean easy to fool.

Median — the middle rock. Line up all rocks small-to-big, walk to exact middle, point. That rock is median (also call 50th percentile — half rocks below, half above). If two rocks in middle, average the two. Median STRONG — one giant mammoth-rock at the end no move the middle rock at all. Use median for lopsided piles: gold-hoards, hunt-wait-times, cave prices.

Mode — the most-common rock. Count which rock-value show up most. Mode is only middle that work for name-rocks (nominal — colors, tribe-names, no order). "Average eye-color" is nonsense grunt; most-common eye-color make sense. If pile have TWO tall humps (bimodal), report BOTH modes — hide nothing.

Which middle for which rock?

Rock-type pick the tool. This the NOIR ladder (Nominal, Ordinal, Interval, Ratio):

• Nominal (names, no order) → mode ONLY.
• Ordinal (rank order, gaps not even) → median and mode. Mean not proper — steps between rank not equal size.
• Interval / Ratio (real even-gap numbers) → mean, median, mode all good. Ratio-rocks with true-zero also allow geometric mean ($\sqrt[n]{x_1\cdots x_n}$) — good for growth and ratios.

How far rocks scatter — the spread

Range — biggest rock minus smallest: $\text{range}=\max -\min$. Simplest. But FRAGILE — one bad scratch-mark of 999 blow range to nonsense.

IQR (interquartile range) — $\text{IQR}=Q_3-Q_1$. Chop pile in four. Throw top quarter and bottom quarter away. IQR is width of the middle-half rocks. Strong against giant outlier-rocks. Tribe use IQR for the box-picture and the outlier-rule: any rock past $Q_3+1.5\cdot \text{IQR}$, or below $Q_1-1.5\cdot \text{IQR}$, get flagged weird.

Variance — the mean squared gap from the middle. Sample formula:

$s^2=\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\bar{x}{)}^2$

For each rock, measure gap to the mean, SQUARE the gap, add all, divide. Why square, not just take size of gap? Two reasons. One: squaring stop minus-gaps and plus-gaps canceling to zero. Two: squares carry smooth magic that build the big fire-tests (significance test, ANOVA). Ugg WARN: variance is in SQUARED units — "minutes-squared" — hard for eye to read.

Standard deviation (SD) — $s=\sqrt{s^2}$. Square-root of variance, back in normal units. This the everyday spread-stick. But SD also get dragged by giant rocks — squares make far rocks scream loud.

MAD (median absolute deviation) — $\text{MAD}={\text{median}}_i|x_i-\tilde{x}|$. Gap of each rock from the MIDDLE rock, then median of those gaps. Tough like the median. For bell-pile, $\sigma \approx 1.4826\cdot \text{MAD}$.

Ugg trick: robust trio = median + IQR + MAD (no fear big rocks). Soft trio = mean + SD + range (giant rock hurt them).

The n-minus-one mystery — Bessel

Why divide variance by $n-1$, not $n$? This Bessel's correction. Ugg use $\bar{x}$ (made FROM the rocks) as the middle. Rocks hug their own mean too tight, so gaps come out too small. Dividing by $n$ under-guess true spread. Divide by $n-1$ fix it exact, so $\mathbb{E}[s^2]={\sigma }^2$ — no bias. Grunt-picture: Ugg SPEND one piece of freedom to find the mean, leaving $n-1$ free pieces for spread. Ugg WARN: forget the $-1$ and you always guess spread too small.

Shiny same-stick — z-score

Rocks in different units no compare. Spear-length in hands, hunt-time in suns. Ugg fix with z-score (standardise):

$z=\frac{x-\mu }{\sigma }$

Gap from mean, divided by SD. Answer: how many SD-steps above or below middle. No units left! Now Ugg mix IQ-rock with GPA-rock fair. Example: SAT 1400, mean 1050, SD 200 → $z=1.75$ → about 96 out of 100. Under bell-shape, $|z|>2$ is weird rock, $|z|>3$ very weird. z-score keep the pile's SHAPE — only move and rescale.

CV (coefficient of variation) — $\text{CV}=s/\bar{x}$. Spread as fraction of the middle, no units. Compare wobble across units: weight CV $0.143$ beat height CV $0.041$ → weight wobble more. Only use for ratio-rocks with true zero.

Lopsided piles and skew

When one tail long, pile is skew. Read it WITHOUT a picture by comparing the middle-tools:

• Right skew (long tail to big side): mean > median > mode. Long tail drag mean right. Think gold, wait-times.
• Left skew: mean < median < mode. Think test-scores at a ceiling.
• Symmetric bell: mean = median = mode, all same spot.

Quick skew-guess (Pearson): $\text{skew}\approx 3(\bar{x}-\tilde{x})/s$.

Bell-shape (Normal distribution): smooth hump, mean=median=mode in center. The 68 / 95 / 99.7 rule: 1 SD hold ~68%, 2 SD ~95%, 3 SD ~99.7%. Almost every rock within 3 SD. Many parametric fire-tests lean on this Normal shape (or the CLT rescue when raw rocks not Normal).

For lopsided pile: use median not mean, IQR not SD, or transform the rocks — log (tame right-tail), square-root (gentler), 1/x (strong). Ugg WARN: after transform, talk ONLY in transform-land. Say "log-income differ," NOT "income differ by X gold."

Ugg warn — no do this!

• No report mean ± SD on lopsided pile. Use median + IQR.
• No divide variance by $n$. Divide by $n-1$.
• No take mean of name-rocks or blind rank-rocks. "Average eye-color" nonsense.
• No trust SD as tough — it soft, giant rock fool it. Use MAD or IQR.
• No read transform-rocks in old units.

Ugg remember

• Middle: mean (sharp but soft to giants), median (tough, 50th percentile), mode (name-rocks and two-hump piles). Rock-type pick tool: nominal→mode, ordinal→median, interval/ratio→all.
• Spread: range (fragile), IQR + MAD (tough), variance + SD (standard but soft). Robust trio = median + IQR + MAD.
• Bessel: sample variance divide by $n-1$ — one freedom spent finding the mean, so $\mathbb{E}[s^2]={\sigma }^2$.
• z-score $z=(x-\mu )/\sigma$ — unit-less same-stick to compare across scales; keeps shape. Variance in squared units, SD in real units.
• Skew read: mean > median > mode means right-skew. Bell obeys 68 / 95 / 99.7.