Describing Data

Info sheet · Statistics for Psychology & Neuroscience

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

Warning✎ Editing notes — to do / to check

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

• Skew demo now marks mean/median/mode and reports SD/IQR/range on one skew slider.
• Added a “which summary fits?” radio quiz with ambiguous cases (income skew, nominal flavour, Likert ordinal).
• Added an “increase n” demo showing the SD holding steady while the SEM shrinks.
• Added self-checks on mean-vs-median / choosing a summary and on two-group boxplots.
• (optional) a dedicated histogram-vs-boxplot-vs-violin interpretation self-check.

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 range=sorted[n-1]-sorted[0];
  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]++;});
  let modeBin=0; c.forEach((k,i)=>{if(k>c[modeBin])modeBin=i;}); const mode=lo+(modeBin+0.5)*bw;
  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(31,78,121,0.28)"/>`).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(mode,"#7c3aed","mode "+mode.toFixed(1),44)}${vline(median,"#1E7A34","median "+median.toFixed(1),12)}${vline(mean,"#C0501E","mean "+mean.toFixed(1),28)}</svg>`+
    `<figcaption style="margin-top:.5rem">centre — mean <b style="color:#C0501E">${mean.toFixed(2)}</b>, median <b style="color:#1E7A34">${median.toFixed(2)}</b>, mode <b style="color:#7c3aed">${mode.toFixed(1)}</b> · spread — SD <b>${sd.toFixed(2)}</b>, IQR <b>${iqr.toFixed(2)}</b>, range <b>${range.toFixed(1)}</b>. ${mean-median>0.3?'Mean &gt; median &gt; mode → right-skew. Median + IQR are the honest summary; the range is inflated by the tail.':'Roughly symmetric — mean, median and mode nearly coincide.'}</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.

More data, steadier estimates

The sample mean and SD are themselves estimates, and they settle down as the sample grows. One number in particular shrinks: the standard error of the mean (SEM = SD⁄√n), which measures how much the mean itself would wobble from sample to sample. Raise n and watch the SD hold roughly steady (it’s estimating the population’s spread) while the SEM shrinks (the mean becomes more precise):

md`**Sample size n** — currently **${+nBig}**`

viewof nBig = html`<input type="range" min="5" max="500" step="5" value="30" style="width:340px">`

nWidget = {
  let s = 21;
  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 n=+nBig, mu=50, sdPop=10, data=[];
  for(let i=0;i<n;i++) data.push(mu+sdPop*nrm());
  const mean=data.reduce((a,b)=>a+b,0)/n;
  const sd=Math.sqrt(data.reduce((a,v)=>a+(v-mean)**2,0)/(n-1));
  const sem=sd/Math.sqrt(n);
  const W=470,H=170,padL=30,padR=14, lo=30, hi=70, base=H-40;
  const X=x=>padL+(x-lo)/(hi-lo)*(W-padL-padR);
  const sdBand=`<rect x="${X(mean-sd)}" y="${base-40}" width="${X(mean+sd)-X(mean-sd)}" height="26" fill="#1F4E79" fill-opacity="0.12"/>`;
  const semBand=`<rect x="${X(mean-sem)}" y="${base-40}" width="${Math.max(1,X(mean+sem)-X(mean-sem))}" height="26" fill="#1E7A34" fill-opacity="0.5"/>`;
  const meanLine=`<line x1="${X(mean)}" y1="${base-46}" x2="${X(mean)}" y2="${base}" stroke="#111" stroke-width="1.5"/>`;
  const axis=`<line x1="${padL}" y1="${base}" x2="${W-padR}" y2="${base}" stroke="#94a3b8"/><text x="${X(mu)}" y="${base+16}" text-anchor="middle" font-size="10" fill="#667">population mean 50</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">${sdBand}${semBand}${meanLine}${axis}<text x="${padL}" y="16" font-size="11" fill="#1F4E79">light band = ±1 SD (spread of the data)</text><text x="${padL}" y="30" font-size="11" fill="#1E7A34">dark band = ±1 SEM (precision of the mean)</text></svg>`+
    `<figcaption style="margin-top:.5rem">n = <b>${n}</b> · mean = <b>${mean.toFixed(2)}</b> · SD = <b>${sd.toFixed(2)}</b> (≈ population 10, roughly fixed) · SEM = SD⁄√n = <b style="color:#1E7A34">${sem.toFixed(2)}</b> (shrinks as n grows).</figcaption>`;
  return fig;
}

This is the crucial distinction the SD/SEM confusion trips over: the SD describes how spread out the data are and doesn’t shrink with more data; the SEM describes how precisely you’ve pinned down the mean, and it does. More data doesn’t make people more alike — it makes your estimate of their average more trustworthy.

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.

Quick quiz: which summary fits?

Pick a scenario, choose the centre you’d report, and check the reasoning — some are deliberately ambiguous:

NoteQuiz

Answer each question — the feedback appears beneath it.

summaryBtn = {
  const scens=["Household incomes in a city (a few very rich)", "Favourite ice-cream flavour of 200 children", "Heights of adult women (roughly symmetric)", "Marathon finishing times (long tail of slow finishers)", "Agreement on a single 1–5 Likert item"];
  const ans=["Median", "Mode", "Mean", "Median", "Median"];
  const why=["Incomes are strongly right-skewed — a few large values drag the mean up, so the median (with IQR) is the honest centre.", "Flavour is nominal (unordered categories): the only meaningful 'centre' is the most common value — the mode.", "Heights are roughly symmetric interval/ratio data, so the mean (with SD) is efficient and appropriate.", "Finishing times have a long right tail, so the median is more representative than the outlier-sensitive mean.", "A Likert item is ordinal, so strictly the median fits; averaging is a common shortcut that assumes equal spacing."];
  const opts=["Mean", "Median", "Mode"];
  const wrap=document.createElement("div");
  scens.forEach((scen,i)=>{
    const block=document.createElement("div"); block.style.cssText="margin:0 0 1rem 0";
    const q=document.createElement("div"); q.style.cssText="font-weight:600;margin-bottom:.35rem"; q.textContent=(i+1)+". "+scen;
    const row=document.createElement("div"); row.style.cssText="display:flex;gap:8px;flex-wrap:wrap;margin-bottom:.3rem";
    const fb=document.createElement("div"); fb.style.cssText="font-size:14px;min-height:1.2rem;color:#94a3b8"; fb.textContent="Click an answer to check.";
    opts.forEach(opt=>{
      const b=document.createElement("button"); b.textContent=opt;
      b.style.cssText="padding:5px 11px;border:1px solid #cbd5e1;border-radius:6px;background:#fff;cursor:pointer;font-size:14px";
      b.onmouseenter=()=>b.style.background="#f1f5f9"; b.onmouseleave=()=>b.style.background="#fff";
      b.onclick=()=>{ const ok=opt===ans[i];
        fb.innerHTML=`<b style="color:${ok?'#1E7A34':'#c0392b'}">${ok?'✓ '+ans[i]:'✗ Not quite — '+ans[i]}.</b> <span style="color:#334">${why[i]}</span>`; };
      row.appendChild(b);
    });
    block.append(q,row,fb); wrap.appendChild(block);
  });
  return wrap;
}

See it in code

R

Python

MATLAB / Octave

rng(1);
rt = 200 + 300*exprnd(1, 200, 1);           % reaction-time-like, right-skewed
[mean(rt), median(rt)]                       % mean > median → right-skew
[std(rt), iqr(rt)]                           % spread: SD vs the robust IQR
histogram(rt); figure; boxplot(rt)           % ALWAYS look at the shape

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

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.

TipCheck your understanding (2)

Two groups have the same mean and the same SD, but their boxplots look very different — one symmetric, the other with a long upper whisker and high outliers. What’s going on, and what should you report?

Equal mean and SD don’t guarantee equal shape. The skewed group has a long upper tail that a single symmetric spread number (the SD) can’t reveal, while the boxplot’s whisker and outlier dots expose it at a glance. For that group, report the median and IQR (and show the boxplots) rather than letting a matching mean/SD imply the groups are alike. Same lesson as always: plot before you summarise.

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.