Signal and Error

Info sheet · Statistics for Psychology & Neuroscience

Author

Andrew Bell

Published

September 22, 2026

Info sheet 0.2 (draft) · Prerequisites: it all comes down to noise; the mean and variance · Give feedback ↗

✎ Editing notes — to do / to check

Working notes for the author — not shown to students once collapsed; remove before publishing.

Added framing on the page’s purpose: it formalises the noise intuition into Data = Signal + Error and shows F, R², and t are one ratio — the bridge from the previous sheet into every later chapter.

What you’ll get from this sheet

The “noise” idea, written as an equation. By the end you should be able to:

Decompose data into Signal + Error.
See why the F‑statistic, R², and t‑statistic are all the same ratio in disguise.

Every model writes the data as Data = Signal + Error: the part the model explains, plus the leftover it can’t. Almost every statistic is a ratio of those two — F = explained ÷ unexplained variance, R² = explained ÷ total, t = difference ÷ its standard error.

Why this page

The previous sheet gave the intuition — significance is signal relative to noise. This page turns that intuition into a piece of arithmetic you can carry into every later chapter. Writing Data = Signal + Error is the move that makes “noise” something a model can actually compute, and it’s the reason the F‑ratio, R², and t‑statistic all turn out to be the same thing wearing different clothes. Think of it as the Rosetta Stone for the rest of the book: once you see that every test is just explained ÷ unexplained, they stop being separate recipes to memorise and become one idea applied over and over.

Data = Signal + Error

In nearly every model we build, the data split into two parts:

\[\text{Data} = \text{Signal} + \text{Error}\]

The signal (the systematic or explained part) is what our model accounts for — the effect of decade on danceability, of surgery on quality of life, of study time on recall. The error (the residual or unexplained part) is everything left over after the model has done its best. Crucially, error isn’t mistake: it’s the part of the world the model doesn’t capture, either because it’s genuinely random or because it comes from variables we didn’t (or couldn’t) measure. Every model is an admission of partial ignorance, and the error term is where that ignorance lives.

One ratio, many names

Almost every statistic we compute is a ratio of these two quantities — explained versus unexplained. Slide the signal strength and watch the total variance split into an explained part (the model) and a residual (the error), with R² and the F‑ratio reading straight off that split:

md`**Signal strength** — currently **${(+sig).toFixed(1)}**`

viewof sig = html`<input type="range" min="0" max="5" step="0.25" value="2" style="width:340px">`

seWidget = {
  const n=40; let s=5;
  const rnd=()=>{s|=0;s=s+0x6D2B79F5|0;let t=Math.imul(s^s>>>15,1|s);t=t+Math.imul(t^t>>>7,61|t)^t;return((t^t>>>14)>>>0)/4294967296;};
  const nrm=()=>{const u=1-rnd(),v=rnd();return Math.sqrt(-2*Math.log(u))*Math.cos(2*Math.PI*v);};
  const xs=[],ys=[]; for(let i=0;i<n;i++){const x=rnd()*10; xs.push(x); ys.push(20+(+sig)*x+6*nrm());}
  const mx=xs.reduce((a,b)=>a+b,0)/n, my=ys.reduce((a,b)=>a+b,0)/n;
  let Sxy=0,Sxx=0; for(let i=0;i<n;i++){Sxy+=(xs[i]-mx)*(ys[i]-my);Sxx+=(xs[i]-mx)**2;}
  const b1=Sxy/Sxx, b0=my-b1*mx;
  let SStot=0,SSres=0; for(let i=0;i<n;i++){SStot+=(ys[i]-my)**2; SSres+=(ys[i]-(b0+b1*xs[i]))**2;}
  const SSexp=SStot-SSres, R2=SSexp/SStot, df1=1, df2=n-2, F=(SSexp/df1)/(SSres/df2);

  const W=460,H=300,pad=34,xmin=0,xmax=10,ymin=Math.min(...ys)-3,ymax=Math.max(...ys)+3;
  const X=x=>pad+(x-xmin)/(xmax-xmin)*(W*0.62-2*pad), Y=y=>H-pad-(y-ymin)/(ymax-ymin)*(H-2*pad);
  let seg="",dots=""; xs.forEach((x,i)=>{const yh=b0+b1*x; seg+=`<line x1="${X(x)}" y1="${Y(ys[i])}" x2="${X(x)}" y2="${Y(yh)}" stroke="#C0501E" stroke-width="1" opacity="0.5"/>`; dots+=`<circle cx="${X(x)}" cy="${Y(ys[i])}" r="3" fill="#1f2937"/>`;});
  const line=`<line x1="${X(xmin)}" y1="${Y(b0+b1*xmin)}" x2="${X(xmax)}" y2="${Y(b0+b1*xmax)}" stroke="#1F4E79" stroke-width="2.5"/>`;
  // variance bar
  const bx=W*0.72, bw=46, bTop=pad, bH=H-2*pad;
  const expH=bH*SSexp/SStot;
  const bar=`<rect x="${bx}" y="${bTop}" width="${bw}" height="${expH}" fill="#1E7A34"/><rect x="${bx}" y="${bTop+expH}" width="${bw}" height="${bH-expH}" fill="#cbd5e1"/>`+
    `<text x="${bx+bw+8}" y="${bTop+expH/2+4}" font-size="11" fill="#1E7A34">signal ${(100*R2).toFixed(0)}%</text>`+
    `<text x="${bx+bw+8}" y="${bTop+expH+(bH-expH)/2+4}" font-size="11" fill="#667">error ${(100*(1-R2)).toFixed(0)}%</text>`+
    `<text x="${bx+bw/2}" y="${bTop-6}" text-anchor="middle" font-size="10" fill="#334">total variance</text>`;
  const fig=document.createElement("figure"); fig.style.margin="0";
  fig.innerHTML=`<svg viewBox="0 0 ${W} ${H}" width="${Math.min(W,540)}" style="background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px">${seg}${line}${dots}${bar}</svg>`+
    `<figcaption style="margin-top:.5rem">R² = signal ÷ total = <b style="color:#1E7A34">${R2.toFixed(2)}</b> · F(${df1},${df2}) = signal ÷ error = <b>${F.toFixed(1)}</b>. The <span style="color:#C0501E">orange stubs</span> are the residuals (error); the line is the signal.</figcaption>`;
  return fig;
}

Turn the signal up and the green (explained) slice of the total‑variance bar grows while the residual stubs on the scatter shrink — R² climbs toward 1 and F climbs with it. That’s the whole game in one picture: the F‑statistic in an ANOVA is literally explained ÷ unexplained variance; R² in regression is the proportion of total variance explained; a t‑statistic is a difference (signal) ÷ its standard error (noise). Once you see the pattern, the alphabet soup of tests stops looking like unrelated recipes and starts looking like variations on one theme.

The recurring three-step

So throughout the book we do the same three things: (1) build a model that proposes a signal, (2) measure how much variation that signal explains, and (3) ask whether that explained variation is large enough, relative to the leftover error, to take seriously. Hold onto that framing — it’s the thread tying every chapter together.

See it in code

rng(1);
x = 10*rand(40,1);
y = 20 + 2*x + 6*randn(40,1);
m = fitlm(x, y);
anova(m, 'summary')     % SS explained vs residual → F = signal / error
m.Rsquared.Ordinary     % R² = explained / total variance

The R and Python tabs run live; MATLAB is a static reference.

Check your understanding

What do R² and the F‑statistic have in common, and how do they differ?

Both are built from the same split — explained variance versus unexplained (residual) variance. R² is a proportion: explained ÷ total variance (bounded 0–1, “how much of the wobble did we account for?”). The F‑statistic is a ratio: explained ÷ unexplained variance (adjusted for degrees of freedom), used to ask whether the explained part is big enough to be more than chance. Same ingredients, arranged for two different questions.

“Error” doesn’t mean you made a mistake — it’s just the variation your model didn’t capture, some of it genuinely random, some of it from variables you never measured. So a large residual isn’t necessarily a bad model; it may be an honest one facing a noisy world. And a high R² isn’t automatically good (it can signal overfitting), just as a low one isn’t automatically bad — always read the explained and the unexplained together.

Where this shows up next

This ratio is the backbone of ANOVA (the F‑ratio), regression (R²), and the t‑test. See those sheets — you’ll now recognise them all as the same move.

What you’ll get from this sheet

Why this page

Data = Signal + Error

One ratio, many names

The recurring three-step

See it in code

Where this shows up next

Related topics