Describing Data

Info sheet · Statistics for Psychology & Neuroscience

Info sheet 0.2 (draft) · Prerequisites: levels of measurement · Give feedback ↗

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What you’ll get from this sheet

Summarising a dataset in two numbers — a centre and a spread — and knowing when each is honest. By the end you should be able to:

1. Choose between mean, median, mode.
2. Choose between range, IQR, and SD — and always plot first.

Describe data with a centre and a spread. Centre: mean (efficient, but outlier‑sensitive), median (robust), mode (for categories). Spread: range (fragile), IQR (robust), variance/SD (the workhorse). A gap between mean and median signals skew — so plot before you model.

Central tendency: where is the middle?

Three measures. The mean is the arithmetic average; it uses every value, which makes it efficient but sensitive to outliers. The median is the middle value when the data are ordered; it ignores the size of extremes, which makes it robust — the median income of a room doesn’t change when a billionaire walks in, but the mean lurches up. The mode is the most frequent value, and the only sensible “centre” for categorical data.

Which to use depends on the data: for roughly symmetric interval/ratio data, the mean; for skewed data — reaction times, incomes, firing rates — the median is often more honest. The relationship between them is itself a clue: when the mean sits well above the median, you’re looking at a right‑skewed distribution with a long upper tail.

Watch the mean and median part ways

Drag the skew and see how each summary responds — and why the mean is the first to flinch:

md`**Skew** — currently **${(+skew).toFixed(2)}**`

viewof skew = html`<input type="range" min="0" max="1" step="0.05" value="0.4" style="width:340px">`

descWidget = {
  const n=400; let s=7;
  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 w=+skew, data=[];
  for(let i=0;i<n;i++){ const sym=5+nrm(); const tail=2+(-Math.log(1-rnd()))*3; data.push((1-w)*sym + w*tail); }
  const sorted=[...data].sort((a,b)=>a-b);
  const mean=data.reduce((a,b)=>a+b,0)/n, median=sorted[Math.floor(n/2)];
  const sd=Math.sqrt(data.reduce((a,v)=>a+(v-mean)**2,0)/n);
  const q1=sorted[Math.floor(n*0.25)], q3=sorted[Math.floor(n*0.75)], iqr=q3-q1;
  const lo=0, hi=Math.max(...data)+1, bins=34, bw=(hi-lo)/bins, c=Array(bins).fill(0);
  data.forEach(v=>{const b=Math.floor((v-lo)/bw); if(b>=0&&b<bins)c[b]++;});
  const W=460,H=280,pad=30, plotH=H-2*pad, mx=Math.max(...c);
  const X=v=>pad+(v-lo)/(hi-lo)*(W-2*pad), Y=k=>H-pad-k/mx*plotH;
  let bars=c.map((k,i)=>`<rect x="${X(lo+i*bw)}" y="${Y(k)}" width="${(W-2*pad)/bins-1}" height="${H-pad-Y(k)}" fill="rgba(31,78,121,0.4)"/>`).join("");
  const vline=(x,col,lab,dy)=>`<line x1="${X(x)}" y1="${pad}" x2="${X(x)}" y2="${H-pad}" stroke="${col}" stroke-width="2"/><text x="${X(x)+3}" y="${pad+dy}" font-size="11" fill="${col}">${lab}</text>`;
  const fig=document.createElement("figure"); fig.style.margin="0";
  fig.innerHTML=`<svg viewBox="0 0 ${W} ${H}" width="${Math.min(W,520)}" style="background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px"><line x1="${pad}" y1="${H-pad}" x2="${W-pad}" y2="${H-pad}" stroke="#94a3b8"/>${bars}${vline(median,"#1E7A34","median "+median.toFixed(1),12)}${vline(mean,"#C0501E","mean "+mean.toFixed(1),28)}</svg>`+
    `<figcaption style="margin-top:.5rem">mean = <b style="color:#C0501E">${mean.toFixed(2)}</b>, median = <b style="color:#1E7A34">${median.toFixed(2)}</b>, SD = <b>${sd.toFixed(2)}</b>, IQR = <b>${iqr.toFixed(2)}</b>. ${mean-median>0.3?'Mean &gt; median → right‑skew, long upper tail. The median is the more honest centre.':'Roughly symmetric — mean and median agree.'}</figcaption>`;
  return fig;
}

At zero skew the mean and median sit on top of each other. Add skew and the long right tail drags the mean up while the median barely moves — which is exactly why, for skewed data, the median (and the IQR) tell a more honest story than the mean (and the SD).

Variability: how spread out?

A centre alone is a dangerously incomplete summary — variability is the real subject of statistics. Two classes can share a mean of 60 and have nothing else in common: one clustered near 60, the other split between 30s and 90s. Three measures capture spread. The range (max − min) is simple but hostage to the two most extreme points. The interquartile range (IQR) — the middle 50% — is far more robust and pairs with the median. Most important is the variance (the average squared deviation from the mean) and its square root, the standard deviation (SD), which returns you to the original units. We square the deviations so values above and below the mean don’t cancel — and because squared deviations are exactly the currency ANOVA and regression trade in.

Look before you leap

Numbers summarise; pictures reveal. Three plots earn their keep: a histogram shows the shape of one distribution (symmetric? skewed? bimodal?); a boxplot compresses it to a five‑number summary and is unbeatable for comparing groups; a violin plot draws the full shape on each side of a boxplot (a neuroscience favourite). In R each is a line — hist(x), boxplot(y ~ group), ggplot2::geom_violin(). Build the habit: plot your data before you model it, because a histogram catches the skew, the outlier, or the hidden second peak that a mean and SD quietly conceal.

See it in code

TipCheck your understanding

A dataset has mean 520 and median 430. What does that tell you, and which centre would you report?

The mean sitting well above the median signals a right‑skewed distribution — a long upper tail (a few large values) pulling the mean up while the median stays put. That’s the classic reaction‑time / income shape. For skewed data the median (with the IQR for spread) is the more honest summary; report the mean/SD only if you also show it’s roughly symmetric — which means plotting a histogram first.

Never summarise with a centre alone — two datasets with identical means can be wildly different in spread, so always pair a centre with a measure of variability (and prefer median + IQR when the data are skewed). And match the summary to the level of measurement: a mean/SD is meaningless for nominal data (use counts/mode) and a stretch for ordinal data (use the median). Above all, plot first — a mean and SD can hide skew, outliers, and bimodality completely.

Where this shows up next

These summaries feed everything — the SD underlies the normal distribution, standard errors, and every test that follows. See Chapter (Foundations), and When Data Aren’t Normal for skewed data.

Related topics

• What Kind of Data Is This? (levels of measurement) — which summary each level allows.
• When Data Aren’t Normal — skew, transforms, and robust choices.
• The Normal Distribution & the CLT — where the mean and SD come into their own.
• It All Comes Down to Noise — why variability is the subject.
• One-Way ANOVA & the F-ratio — squared deviations put to work.