Ugg say: LOOK at the rock-pile before you count it

Big rule by the fire. Before Ugg make one clever number — plot the data. Draw the dots, see the shape. Numbers lie by hiding; the picture tell the truth.

Four rock-piles with the same footprints

Hunter Francis Anscombe built Anscombe's quartet — four rock-piles that share the SAME summary numbers but look like four different animals. Each has eleven dots at spots $(x,y)$. Same middle of $x$, same middle of $y$, same spread, same correlation (how much two things move together, about $r\approx 0.816$), same best-fit regression line. Twins by every number.

Then Anscombe DRAW them. Pile 1 is a clean straight cloud. Pile 2 is a smooth curve — the link is not straight at all! Pile 3 is a perfect line with ONE crazy dot dragging it off. Pile 4 stacks all dots at one $x$ plus ONE lonely dot that makes the whole correlation by itself.

One line fits all four because it only listen to $\hat{\beta }=\text{cov}(x,y)/\text{Var}(x)$ and $\hat{\alpha }=\bar{y}-\hat{\beta }\bar{x}$ — same middles and spreads, same line. Cave-wall lesson: statistics squash the data flat; the picture shows the true shape. So no pick your counting-test until AFTER you look.

Ugg check two things before drawing

One: what kind of number is it? Name-bucket (nominal), rank-order (ordinal), even-step (interval), or true-zero (ratio)? Count-thing (discrete) or flow-thing (continuous)? The kind decides which summary make sense.

Two: what is one row? One spear-throw, one hunter, or one tribe? Behaviour-data nests: throw inside hunter inside tribe. Squash at the wrong level and you hide a real effect or invent a fake one.

Two more warnings. A mean hides the shape — a mean of 50 could be everybody at 50, OR half at 0 and half at 100. And squashing hunters together hides their differences — the average say "no effect" while half got much better and half got worse.

Why Ugg draw — five reasons

Ugg draw to:(1) check the assumption — bell-shape, lopsided, two-hump;(2) see structure — learning, tiredness, changed strategy;(3) spot bad dots and mistake-dots;(4) understand the wobble — within one hunter versus between hunters;(5) pick the right test — bell or rank, straight or bendy.

Ugg picture-rock catalogue

Histogram — chop a flow-number into buckets (bins), show how many dots land in each. Best for the shape of ONE flow-number. Catch: too few buckets smear the shape away, too many show only noise — try many bucket-widths.

Boxplot (box-and-whisker) — five numbers: the median line, the box edges $Q_1$ and $Q_3$, whiskers to $1.5\times \text{IQR}$, dots past them flagged as bad-dots. Best for comparing piles side by side. Catch: the box CANNOT see two-hump — two very different piles can wear the same box.

Scatter plot — one dot per spot $(x,y)$. Best for how two flow-numbers dance together. Ugg command: before ANY regression or correlation, look at the scatter first.

Bar chart — a bar per name-bucket, length is the value. Best for counts and proportions across categories. Catch (the test LOVE this): a bar for flow-data like reaction-time HIDES the shape — just a mean and an error-stick. Two groups both averaging 600 ms make twin bars, yet one is one-hump and the other two-hump at 400 and 800 (two strategies!). Use a violin.

Pie chart — a circle cut into wedges. Use RARE: the eye reads length well but angle BADLY (Cleveland and McGill ranked it — position beats length beats angle-and-area). Only for 3 to 5 fat, clear slices.

Violin plot — the honest cousin of the box: a mirrored KDE (kernel density estimate — a smooth guess of the shape), fat where dots pile up. Shows lopsided, two-hump, and tails — all the box hides.

Raincloud plot — violin + box + every real dot beside it. The gold standard for behaviour-data: shape, summary, AND every hunter at once.

Mosaic plot — rectangles whose area shows how often two name-buckets happen together. Heat map — a grid of cells where colour is the value; use viridis, never rainbow. Line plot — a line over time for learning-curves and fatigue; draw each hunter's own thread beside the group thread, or big differences hide.

Know the fancy ones on sight: spider/radar, funnel, treemap, waffle, streamgraph, circos.

Ugg hunt the bad dot — the Tukey rule

The IQR is the spread of the middle half: $\text{IQR}=Q_3-Q_1$. A dot is an outlier by the Tukey rule if

With $Q_1=10$ and $Q_3=20$, the IQR is $10$, so the top fence is $20+15=35$. A dot at 38 is a bad-dot; a dot at 50 is a VERY-bad dot (past $3\times \text{IQR}$). Under a bell-shape this flags past about $\pm 2.7\sigma$, needing no bell assumption.

Which way the pile leans — skew

Long tail to the right — reaction-times (floor at zero), riches. The other way:

Balanced pile: mean $=$ median $=$ mode. A quick lean-guess is $\text{skew}\approx \frac{3(\bar{x}-\text{median})}{s}$. A two-hump (bimodal) pile means two kinds of animal mixed together (man and woman heights).

Ugg squash the long tail — transformation

When a pile leans too hard and Ugg needs a bell-shape for a test, use a transformation. Log for right-lean (reaction-times, riches); square-root is milder; 1/x is very strong; Box-Cox tries many powers and picks the best. Ugg warn: after you squash, only speak in the squashed-tongue — say "log-RT differ", NOT "RT differ by X seconds".

Elder Tufte's four rules, and the bad pictures

Elder Tufte carved four rules:(1) data-to-ink ratio — most ink draws data, not decoration;(2) lie factor — picture-change must match number-change;(3) chart junk — throw out shadows, 3D, gradient backgrounds;(4) proper scales and clear labels.

The common sins: a bar for flow-data; a truncated y-axis (sales creeping 100 to 105 drawn as a cliff); a 3D pie; dual-y axes faking co-movement; too many colours; and rainbow or red+green together — about 8% of men are colour-blind! Use viridis, and add symbols or line-styles so the chart still reads in grey.

Last: the initial picture is for YOU (rough check for outliers and shape); the final picture is for the READER (clean, labelled, honest).

Ugg remember

• Anscombe — same numbers, four different shapes. ALWAYS plot before you count.
• Match plot to number: name-bucket to bar, flow-versus-flow to scatter, shape to histogram or violin or raincloud. Never a bar for flow-data.
• Tukey bad-dot: past $Q_1-1.5\text{IQR}$ or $Q_3+1.5\text{IQR}$, where $\text{IQR}=Q_3-Q_1$.
• Lean: mean > median > mode means right-lean; log-squash tames the right tail (but talk in log-tongue after).
• Tufte four: data-to-ink, lie factor, chart junk, proper scales. Use viridis; no rainbow, no red+green, no truncated axis.