Interactive explanations

Law of Large Numbers

Observe how the sample mean stabilizes as the sample size increases.

Learning objective

The Law of Large Numbers explains why averages calculated from increasingly large random samples become reliable estimates of the population mean.

If X₁, X₂, … are i.i.d. with finite mean μ, then
X̄_n = (1/n) Σ X_i → μ in probability as n → ∞.

How to explore

1. Select a distribution.
2. Change the sample size.
3. Generate several trajectories.
4. Compare the final error.

What does choosing n actually mean?

If you choose n = 20, the lab generates one sequence of 20 observations: X₁, X₂, …, X₂₀. It then recalculates the cumulative mean every time a new observation is added.

Step 1X̄1 = X1

Step 2X̄2 = (X1+X2)/2

Step 3X̄3 = (X1+X2+X3)/3

Continue to nX̄n = ΣXi/n

The important idea: the graph shows X̄₁, X̄₂, …, X̄₂₀. These are not 20 unrelated samples. They are nested prefixes of the same sequence: {X₁} ⊂ {X₁,X₂} ⊂ … ⊂ {X₁,…,X₂₀}. Therefore, the slider selects the maximum sample size displayed.

Running Sample Mean

Running sample mean X̄ₙTheoretical mean μ

Theoretical mean μ—

Final sample mean—

Absolute error—

Sample size—

What to notice: for small n, the sample mean can fluctuate strongly. As n grows, the trajectory usually settles near μ. Convergence does not mean that every trajectory is monotonic.

Discussion question: Which distribution appears to require more observations before the running mean becomes stable?

Experiment settings

Population distributionBernoulli(p = 0.30)Uniform U(0,1)Exponential(λ = 1)

Distribution

—

Mean E[X]—

Variance Var(X)—

—

Sample size n =

800

Animated Example — Rolling a Fair Die

Each roll is an independent observation X_i. The running mean uses every result obtained so far and should progressively approach the theoretical mean 3.5.

⚀

Fair die distribution
P(X = x) = 1/6,  x ∈ {1,2,3,4,5,6}
E[X] = 3.5
Var(X) = 35/12 ≈ 2.9167

Last roll—

Number of rolls0

Running mean—

|X̄ₙ − 3.5|—

Running mean after each roll

Running meanTheoretical mean 3.5

Early averages can move sharply because each roll has a large influence. Later, one additional roll has much less effect on the cumulative mean.

// The original interactive controller is executed by the Observable JS runtime.
(function(){
const $=id=>document.getElementById(id);
function info(type){
  if(type==='bernoulli')return{name:'Bernoulli distribution',equation:'P(X = x) = pˣ(1 − p)¹⁻ˣ, &nbsp;x ∈ {0,1}, &nbsp;p = 0.30',support:'Support: {0, 1}',mu:.3,variance:.3*.7,meanText:'p = 0.3000',varianceText:'p(1 − p) = 0.2100',sample:()=>Math.random()<.3?1:0};
  if(type==='uniform')return{name:'Continuous uniform distribution',equation:'f(x) = 1/(b − a) = 1, &nbsp;0 ≤ x ≤ 1',support:'Support: [0, 1]; f(x) = 0 outside this interval',mu:.5,variance:1/12,meanText:'(a + b)/2 = 0.5000',varianceText:'(b − a)²/12 = 0.0833',sample:()=>Math.random()};
  return{name:'Exponential distribution',equation:'f(x) = λe<sup>−λx</sup> = e<sup>−x</sup>, &nbsp;x ≥ 0, &nbsp;λ = 1',support:'Support: [0, ∞); f(x) = 0 for x < 0',mu:1,variance:1,meanText:'1/λ = 1.0000',varianceText:'1/λ² = 1.0000',sample:()=>-Math.log(1-Math.random())};
}
function updateDistributionCard(){const d=info($('dist').value);$('distName').textContent=d.name;$('distEquation').innerHTML=d.equation;$('distMean').textContent=d.meanText;$('distVariance').textContent=d.varianceText;$('distSupport').textContent=d.support}
function setup(c){const x=c.getContext('2d');x.clearRect(0,0,c.width,c.height);x.fillStyle='#ffffff';x.fillRect(0,0,c.width,c.height);return x}
function label(ctx,t,x,y,align='left'){ctx.fillStyle='#5f6b7f';ctx.font='13px system-ui';ctx.textAlign=align;ctx.fillText(t,x,y)}
let values=[],timer=null;
function generate(){if(timer){clearInterval(timer);timer=null}updateDistributionCard();const n=+$('n').value,d=info($('dist').value);values=[];let total=0;for(let i=1;i<=n;i++){total+=d.sample();values.push(total/i)}draw(n)}
function draw(showN=values.length){const c=$('chart'),ctx=setup(c),d=info($('dist').value),n=values.length,L=62,R=748,T=35,B=390,shown=values.slice(0,Math.max(1,showN));let lo=Math.min(d.mu,...shown),hi=Math.max(d.mu,...shown),pad=(hi-lo)*.18||.2;lo-=pad;hi+=pad;const X=i=>L+(i-1)/(Math.max(2,n)-1)*(R-L),Y=v=>B-(v-lo)/(hi-lo)*(B-T);ctx.strokeStyle='#dfe5ef';ctx.lineWidth=1;ctx.beginPath();ctx.moveTo(L,T);ctx.lineTo(L,B);ctx.lineTo(R,B);ctx.stroke();ctx.strokeStyle='#177e68';ctx.lineWidth=2;ctx.setLineDash([7,6]);ctx.beginPath();ctx.moveTo(L,Y(d.mu));ctx.lineTo(R,Y(d.mu));ctx.stroke();ctx.setLineDash([]);ctx.strokeStyle='#315efb';ctx.lineWidth=2.6;ctx.beginPath();shown.forEach((v,i)=>i?ctx.lineTo(X(i+1),Y(v)):ctx.moveTo(X(i+1),Y(v)));ctx.stroke();label(ctx,'Observation number n',R,B+28,'right');label(ctx,'Running mean',L,T-12);label(ctx,'μ = '+d.mu.toFixed(3),R-5,Y(d.mu)-9,'right');$('nOut').value=$('n').value;$('nStat').textContent=$('n').value;$('mu').textContent=d.mu.toFixed(4);const last=values[values.length-1];$('mean').textContent=last.toFixed(4);$('error').textContent=Math.abs(last-d.mu).toFixed(4)}
$('newRun').onclick=generate;$('dist').onchange=generate;$('n').oninput=generate;$('animate').onclick=()=>{generate();let k=5;timer=setInterval(()=>{draw(k);k+=Math.max(3,Math.floor(values.length/80));if(k>=values.length){draw(values.length);clearInterval(timer);timer=null}},35)};

const dieFaces=['⚀','⚁','⚂','⚃','⚄','⚅'];
let diceValues=[],diceTimer=null,diceSpin=null;
function randomDie(){return Math.floor(Math.random()*6)+1}
function showDie(value){$('die').textContent=dieFaces[value-1];$('die').setAttribute('aria-label','Die showing '+value)}
function recordRoll(value){diceValues.push(value);showDie(value);const avg=diceValues.reduce((s,x)=>s+x,0)/diceValues.length;$('lastRoll').textContent=value;$('rollCount').textContent=diceValues.length;$('diceMean').textContent=avg.toFixed(4);$('diceError').textContent=Math.abs(avg-3.5).toFixed(4);drawDiceChart()}
function drawDiceChart(){const c=$('diceChart'),ctx=setup(c),L=60,R=748,T=30,B=370,n=Math.max(20,diceValues.length),X=i=>L+(i-1)/(n-1)*(R-L),Y=v=>B-(v-1)/5*(B-T);ctx.strokeStyle='#dfe5ef';ctx.lineWidth=1;ctx.beginPath();ctx.moveTo(L,T);ctx.lineTo(L,B);ctx.lineTo(R,B);ctx.stroke();[1,2,3,4,5,6].forEach(v=>{label(ctx,String(v),L-10,Y(v)+4,'right');ctx.strokeStyle='#f3f6fb';ctx.beginPath();ctx.moveTo(L,Y(v));ctx.lineTo(R,Y(v));ctx.stroke()});ctx.strokeStyle='#177e68';ctx.lineWidth=2;ctx.setLineDash([7,6]);ctx.beginPath();ctx.moveTo(L,Y(3.5));ctx.lineTo(R,Y(3.5));ctx.stroke();ctx.setLineDash([]);label(ctx,'μ = 3.5',R-5,Y(3.5)-8,'right');if(diceValues.length){let total=0;ctx.strokeStyle='#315efb';ctx.lineWidth=2.8;ctx.beginPath();diceValues.forEach((v,i)=>{total+=v;const avg=total/(i+1);i?ctx.lineTo(X(i+1),Y(avg)):ctx.moveTo(X(i+1),Y(avg))});ctx.stroke()}label(ctx,'Number of rolls',R,B+28,'right');label(ctx,'Running mean',L,T-12)}
function setDiceButtons(running){$('rollOnce').disabled=running;$('roll100').disabled=running}
function animateOneRoll(){if(diceTimer||diceSpin)return;setDiceButtons(true);$('die').classList.add('rolling');let frames=0;diceSpin=setInterval(()=>{showDie(randomDie());frames++;if(frames>=10){clearInterval(diceSpin);diceSpin=null;$('die').classList.remove('rolling');recordRoll(randomDie());setDiceButtons(false)}},55)}
function animateHundred(){if(diceTimer||diceSpin)return;resetDice();setDiceButtons(true);$('die').classList.add('rolling');let count=0;diceTimer=setInterval(()=>{recordRoll(randomDie());count++;if(count>=100){clearInterval(diceTimer);diceTimer=null;$('die').classList.remove('rolling');setDiceButtons(false)}},65)}
function resetDice(){if(diceTimer){clearInterval(diceTimer);diceTimer=null}if(diceSpin){clearInterval(diceSpin);diceSpin=null}$('die').classList.remove('rolling');diceValues=[];showDie(1);$('lastRoll').textContent='—';$('rollCount').textContent='0';$('diceMean').textContent='—';$('diceError').textContent='—';setDiceButtons(false);drawDiceChart()}
$('rollOnce').onclick=animateOneRoll;$('roll100').onclick=animateHundred;$('resetDice').onclick=resetDice;
generate();resetDice();
})();

Central Limit Theorem

The Central Limit Theorem explains why the sample mean often behaves like a normal random variable, even when the original data do not.

1. The main idea

Original variable

X

May be discrete, skewed, or non-normal.

→

Take a sample

X₁,…,Xₙ

Use n independent observations.

→

Compute the mean

X̄ = (1/n) ΣXᵢ

Repeat this experiment many times.

CLT statement: if the observations are independent and identically distributed, with finite mean μ and variance σ², then for sufficiently large n,

X̄ ≈ N( μ , σ² / n )    and    (X̄ − μ)/(σ/√n) ≈ N(0,1).

2. What becomes normal?

A common mistake is to think that the original observations become normal. They do not.

One observation

X

Still follows the original distribution.

One sample

X₁,…,Xₙ

A collection of original observations.

Many sample means

X̄₁,X̄₂,…

Their distribution becomes approximately normal.

3. LLN vs CLT

Law of Large Numbers:

X̄ₙ → μ

tells us where the sample mean goes.

Central Limit Theorem:

X̄ₙ ≈ N(μ,σ²/n)

tells us how the sample mean fluctuates around μ.

4. Why do we repeat the sampling experiment M times?

The simulation uses two numbers, n and M. They have completely different roles.

n

Sample size

n is the number of observations inside one experiment / one sample.

X₁, X₂, …, Xₙ → one X̄

Increasing n changes the sampling distribution itself:

SE(X̄) = σ / √n

Therefore, larger n makes the distribution of X̄ narrower and, under the CLT, generally more bell-shaped.

M

Number of repeated experiments

M is the number of times we repeat the whole sampling procedure.

X̄₁, X̄₂, …, X̄_M

Each experiment gives one sample mean. Increasing M gives us more sample means, so the histogram becomes smoother and represents the theoretical sampling distribution more accurately.

Example: n = 30 and M = 1000
Perform 1000 experiments. In each experiment, roll the die 30 times and calculate one mean. At the end:

X̄₁, X̄₂, …, X̄₁₀₀₀

These 1000 means are the values used to construct the histogram.

Experiment 1: n observations

153 62

→

Compute one mean

X̄₁ = 3.40

→

Repeat M times

X̄₁ = 3.40 X̄₂ = 3.67 X̄₃ = 3.27 …

Important: increasing M does not make the CLT stronger, and it does not reduce SE = σ/√n. M is mainly used here so that we can see the sampling distribution.

If n increases

The distribution of X̄ changes.

n ↑ → σ/√n ↓

The histogram becomes narrower around μ.

If M increases

The theoretical distribution of X̄ does not change.

M ↑ → more X̄ values

The histogram simply becomes smoother and more stable.

Quick check

Choose A or B.

5. Interactive experiment

Choose a distribution, select the sample size n, and choose the number of experiments M. In each experiment, the computer generates n observations and computes one sample mean. After M repetitions, we obtain X̄₁, X̄₂, …, X̄_M. The histogram below is therefore a histogram of sample means, not of the original observations.

Original distribution Fair die: X ∈ {1,2,3,4,5,6} Bernoulli: X ∈ {0,1}, p = 0.30

Sample size n: 5

Number of experiments M: 1000

Theoretical μ3.500

Theoretical σ1.708

SE = σ/√n0.764

Observed mean of X̄—

Click Run simulation, or use Animate to increase n automatically from 1 to 100, one value at a time. Watch the distribution of sample means become smoother and narrower around μ.

6. Numerical example: fair die

For one fair die roll,

μ = E[X] = (1+2+3+4+5+6)/6 = 3.5

σ² = 35/12 ≈ 2.917

σ ≈ 1.708

If we roll the die 30 times and compute the average, then

X̄ ≈ N(3.5, (1.708/√30)²)

SE = 1.708/√30 ≈ 0.312.

So the sample means are centered near 3.5 and most of them stay relatively close to 3.5.

7. Why the curve gets narrower

The standard deviation of the sampling distribution is

SE(X̄) = σ / √n.

Therefore increasing n reduces the variability of the sample mean.

n = 4 → SE = σ/2
n = 25 → SE = σ/5
n = 100 → SE = σ/10

This is why larger samples give more stable estimates of the population mean.

8. Final intuition

Imagine repeating the same sampling procedure again and again. Every repetition gives one value of X̄. The CLT says that the cloud of these sample means approaches a bell-shaped distribution centered at μ, with spread σ/√n.

This is why the CLT is fundamental for confidence intervals, hypothesis tests, estimation, and statistical inference.

// The original interactive controller is executed by the Observable JS runtime.
(function(){
  const dist = document.getElementById('dist');
  const nSlider = document.getElementById('n');
  const repsSlider = document.getElementById('reps');
  const nVal = document.getElementById('nVal');
  const repsVal = document.getElementById('repsVal');
  const muEl = document.getElementById('mu');
  const sigmaEl = document.getElementById('sigma');
  const seEl = document.getElementById('se');
  const obsMeanEl = document.getElementById('obsMean');
  const runBtn = document.getElementById('run');
  const animateBtn = document.getElementById('animate');
  const canvas = document.getElementById('hist');
  const ctx = canvas.getContext('2d');
  const interp = document.getElementById('interpretation');
  let animationToken = 0;

  let processToken = 0;
  const procSample = document.getElementById('procSample');
  const procMean = document.getElementById('procMean');
  const procMeans = document.getElementById('procMeans');
  const procTitle = document.getElementById('procTitle');
  const oneExpBtn = document.getElementById('showOneExperiment');
  const manyExpBtn = document.getElementById('showManyExperiments');
  const quizButtons = Array.from(document.querySelectorAll('[data-answer]'));
  const quizFeedback = document.getElementById('quizFeedback');


  function params(){
    if(dist.value === 'die'){
      return {mu:3.5, sigma:Math.sqrt(35/12), min:1, max:6, name:'Fair die'};
    }
    const p = 0.30;
    return {mu:p, sigma:Math.sqrt(p*(1-p)), min:0, max:1, name:'Bernoulli(0.30)'};
  }

  function sampleOne(){
    if(dist.value === 'die') return 1 + Math.floor(Math.random()*6);
    return Math.random() < 0.30 ? 1 : 0;
  }


  function renderOneExperiment(index=1, maxShow=12){
    const n = +nSlider.value;
    const values = [];
    let total = 0;

    for(let i=0;i<n;i++){
      const x = sampleOne();
      total += x;
      if(i < Math.min(n,maxShow)) values.push(x);
    }

    const mean = total/n;
    procTitle.textContent = `Experiment ${index}: n = ${n} observations`;
    procSample.innerHTML =
      values.map(v => `<span class="chip">${v}</span>`).join('') +
      (n > maxShow ? '<span class="chip">…</span>' : '');
    procMean.textContent = `X̄${index} = ${mean.toFixed(3)}`;
    return mean;
  }

  async function animateRepeatedExperiments(){
    processToken++;
    const token = processToken;
    oneExpBtn.disabled = true;
    manyExpBtn.disabled = true;
    procMeans.innerHTML = '';

    const demoM = 20;
    const means = [];

    for(let m=1; m<=demoM; m++){
      if(token !== processToken) break;
      const mean = renderOneExperiment(m);
      means.push(mean);

      const start = Math.max(0, means.length-8);
      procMeans.innerHTML = means.slice(start).map((v,j) => {
        const idx = start + j + 1;
        return `<span class="mini-mean">X̄${idx} = ${v.toFixed(2)}</span>`;
      }).join('');

      await new Promise(resolve => setTimeout(resolve, 140));
    }

    if(token === processToken){
      procMeans.innerHTML += '<span class="mini-mean">… up to M</span>';
    }

    oneExpBtn.disabled = false;
    manyExpBtn.disabled = false;
  }

  function updateLabels(){
    nVal.textContent = nSlider.value;
    repsVal.textContent = repsSlider.value;
    const p = params();
    muEl.textContent = p.mu.toFixed(3);
    sigmaEl.textContent = p.sigma.toFixed(3);
    seEl.textContent = (p.sigma/Math.sqrt(+nSlider.value)).toFixed(3);
  }

  function makeMeans(reps){
    const n = +nSlider.value;
    const means = new Array(reps);
    for(let r=0;r<reps;r++){
      let s=0;
      for(let i=0;i<n;i++) s += sampleOne();
      means[r]=s/n;
    }
    return means;
  }

  function normalPdf(x,mu,sd){
    const z=(x-mu)/sd;
    return Math.exp(-0.5*z*z)/(sd*Math.sqrt(2*Math.PI));
  }

  function draw(means){
    const W=canvas.width,H=canvas.height;
    ctx.clearRect(0,0,W,H);
    ctx.fillStyle='#ffffff';
    ctx.fillRect(0,0,W,H);

    const p=params();
    const n=+nSlider.value;
    const se=p.sigma/Math.sqrt(n);

    let xmin=Math.min(...means), xmax=Math.max(...means);
    const pad=Math.max((xmax-xmin)*0.12, se*0.8, 0.02);
    xmin=Math.min(xmin-pad, p.mu-4*se);
    xmax=Math.max(xmax+pad, p.mu+4*se);
    if(xmax<=xmin){xmin-=1;xmax+=1}

    const bins=Math.max(12, Math.min(42, Math.round(Math.sqrt(means.length)/1.7)));
    const counts=new Array(bins).fill(0);
    for(const x of means){
      let b=Math.floor((x-xmin)/(xmax-xmin)*bins);
      if(b<0)b=0;if(b>=bins)b=bins-1;
      counts[b]++;
    }
    const maxC=Math.max(...counts,1);

    const L=70,R=25,T=25,B=55;
    const pw=W-L-R, ph=H-T-B;

    ctx.strokeStyle='#dfe5ef';ctx.lineWidth=1;
    ctx.fillStyle='#5f6b7f';ctx.font='14px system-ui';
    for(let k=0;k<=5;k++){
      const y=T+ph*k/5;
      ctx.beginPath();ctx.moveTo(L,y);ctx.lineTo(W-R,y);ctx.stroke();
      const val=Math.round(maxC*(1-k/5));
      ctx.fillText(val,12,y+5);
    }
    ctx.beginPath();ctx.moveTo(L,T);ctx.lineTo(L,H-B);ctx.lineTo(W-R,H-B);ctx.strokeStyle='#8f9bb0';ctx.stroke();

    const bw=pw/bins;
    for(let i=0;i<bins;i++){
      const h=counts[i]/maxC*ph;
      ctx.fillStyle='rgba(49,94,251,.72)';
      ctx.fillRect(L+i*bw+1,H-B-h,Math.max(1,bw-2),h);
    }

    // Normal curve scaled to histogram count scale
    const binWidth=(xmax-xmin)/bins;
    let maxPdf=0;
    for(let i=0;i<=300;i++){
      const x=xmin+(xmax-xmin)*i/300;
      maxPdf=Math.max(maxPdf,normalPdf(x,p.mu,se)*means.length*binWidth);
    }
    ctx.beginPath();
    for(let i=0;i<=300;i++){
      const x=xmin+(xmax-xmin)*i/300;
      const expected=normalPdf(x,p.mu,se)*means.length*binWidth;
      const px=L+(x-xmin)/(xmax-xmin)*pw;
      const py=H-B-(expected/maxC)*ph;
      if(i===0)ctx.moveTo(px,py);else ctx.lineTo(px,py);
    }
    ctx.strokeStyle='#177e68';ctx.lineWidth=3;ctx.stroke();

    // mu line
    const mux=L+(p.mu-xmin)/(xmax-xmin)*pw;
    ctx.setLineDash([7,6]);ctx.strokeStyle='#b7791f';ctx.lineWidth=2;
    ctx.beginPath();ctx.moveTo(mux,T);ctx.lineTo(mux,H-B);ctx.stroke();ctx.setLineDash([]);

    ctx.fillStyle='#182033';ctx.font='15px system-ui';
    for(let k=0;k<=5;k++){
      const x=xmin+(xmax-xmin)*k/5;
      const px=L+pw*k/5;
      ctx.fillText(x.toFixed(2),px-18,H-20);
    }
    ctx.fillStyle='#177e68';ctx.fillText('Normal approximation',W-205,23);
    ctx.fillStyle='#b7791f';ctx.fillText('μ',mux+6,T+16);
    ctx.fillStyle='#5f6b7f';ctx.fillText('sample mean X̄',W/2-48,H-5);

    const observed=means.reduce((a,b)=>a+b,0)/means.length;
    obsMeanEl.textContent=observed.toFixed(3);

    let msg='';
    if(n===1){
      msg='With n = 1, the histogram mostly reflects the original distribution. The CLT effect is not yet visible.';
    } else if(n<10){
      msg='The sampling distribution is beginning to concentrate around μ. Increase n further to make the normal shape clearer.';
    } else if(n<30){
      msg='The bell shape is becoming more visible, and the sample means are less spread out because SE = σ/√n is smaller.';
    } else {
      msg='Now the CLT is clearly visible: the sample means are approximately normal, centered near μ, with a much smaller spread.';
    }
    interp.innerHTML='<strong>Interpretation:</strong> '+msg;
  }

  function run(){
    animationToken++;
    updateLabels();
    draw(makeMeans(+repsSlider.value));
  }

  async function animate(){
    animationToken++;
    const token=animationToken;
    const originalN=+nSlider.value;
    animateBtn.disabled=true;
    runBtn.disabled=true;
    const seq = Array.from({length:100}, (_,i) => i + 1);
    for(const n of seq){
      if(token!==animationToken) break;
      nSlider.value=n;
      updateLabels();
      draw(makeMeans(Math.min(+repsSlider.value,1200)));
      await new Promise(r=>setTimeout(r,110));
    }
    if(token===animationToken){
      interp.innerHTML='<strong>Animation complete:</strong> n increased step by step from 1 to 100. Notice how the histogram becomes progressively more bell-shaped and narrower around μ.';
    }
    animateBtn.disabled=false;
    runBtn.disabled=false;
  }

  nSlider.addEventListener('input',updateLabels);
  repsSlider.addEventListener('input',updateLabels);
  dist.addEventListener('change',()=>{updateLabels();run();});
  runBtn.addEventListener('click',run);
  animateBtn.addEventListener('click',animate);

  oneExpBtn.addEventListener('click', () => {
    processToken++;
    const mean = renderOneExperiment(1);
    procMeans.innerHTML =
      `<span class="mini-mean">X̄₁ = ${mean.toFixed(2)}</span>` +
      '<span class="mini-mean">Repeat → X̄₂, X̄₃, …, X̄_M</span>';
  });

  manyExpBtn.addEventListener('click', animateRepeatedExperiments);

  quizButtons.forEach(btn => {
    btn.addEventListener('click', () => {
      quizButtons.forEach(b => b.classList.remove('correct','wrong'));
      if(btn.dataset.answer === 'correct'){
        btn.classList.add('correct');
        quizFeedback.innerHTML =
          '<strong>Correct.</strong> n appears in SE = σ/√n. Increasing n reduces the standard error. Increasing M only gives us more sample means to visualize the same distribution.';
      } else {
        btn.classList.add('wrong');
        quizFeedback.innerHTML =
          '<strong>Not quite.</strong> M does not appear in SE = σ/√n. Increasing M makes the simulated histogram smoother; increasing n changes the sampling distribution itself.';
      }
    });
  });


  updateLabels();
  run();
})();

Hypothesis Testing

Hypothesis testing is a formal way to decide whether the evidence in a sample is strong enough to question a claim about a population.

1. The idea

We begin with a claim called the null hypothesis, denoted by H₀. We then ask:

If H₀ were true, would our observed sample result be reasonably common, or surprisingly extreme?

Null hypothesis

H₀: μ = μ₀

The reference claim. We assume it is true while calculating the probability of observing results as extreme as ours.

Alternative hypothesis

H₁: μ ≠ μ₀

The competing claim. Depending on the question, it may also be μ > μ₀ or μ < μ₀.

2. The five-step logic

1

State H₀ and H₁

Define the population claim being tested.

2

Choose α

Typical value: α = 0.05.

3

Compute a statistic

Measure how far the sample is from what H₀ predicts.

4

Find the p-value

Quantify how extreme the result is under H₀.

5

Decision

Compare the p-value with α.

3. Which test statistic?

Case 1 — Population variance known

Use the population standard deviation σ and the standard normal distribution.

z = (X̄ − μ₀) / (σ/√n)

Case 2 — Population variance unknown

Replace σ by the sample standard deviation s and use Student's t distribution.

t = (X̄ − μ₀) / (s/√n),    df = n − 1

In both cases, the statistic measures how many estimated standard errors the observed sample mean is away from the value assumed by H₀.

4. The p-value

The p-value is the probability, assuming H₀ is true, of obtaining a test statistic at least as extreme as the one observed.

Important: the p-value is not the probability that H₀ is true.

p ≤ α → reject H₀ p > α → fail to reject H₀

5. Choose the case and study the corresponding example

First choose whether the population variance is known or unknown. The worked example below changes automatically according to your choice. The same choice is then used in the interactive demonstration in Part 6.

Choose the variance case Case 1 — Population variance known → z-test Case 2 — Population variance unknown → t-test

Case 1 — Population variance known → z-test

A manufacturer claims that the mean lifetime of a battery is μ = 100 hours. The population standard deviation is known: σ = 12 hours.

A sample of n = 36 batteries gives X̄ = 104 hours. We use a two-tailed test with α = 0.05.

H₀: μ = 100

H₁: μ ≠ 100

SE = σ/√n = 12/√36 = 2 z = (104 − 100)/2 = 2 p ≈ 0.0455

Since p ≈ 0.0455 < 0.05, Reject H₀.

Case 2 — Population variance unknown → Student's t-test

The population standard deviation is unknown. A sample of n = 16 observations gives X̄ = 54 and sample standard deviation s = 8. We use a two-tailed test with α = 0.05.

H₀: μ = 50

H₁: μ ≠ 50

SE = s/√n = 8/√16 = 2 t = (54 − 50)/2 = 2 df = n − 1 = 15

For df = 15 and a two-tailed test with α = 0.05, the critical values are approximately ±2.131.

Since |t| = 2 < 2.131, Fail to reject H₀.

6. Interactive hypothesis test for a population mean

This demonstration uses the variance case selected in Part 5. Change the sample information below and observe the test statistic, critical region, p-value, and decision update automatically.

Selected in Part 5: Case 1 — z-test, population standard deviation σ is known.

Null mean μ₀

Observed sample mean X̄

Population SD σ

Sample size n

Significance α 0.10 0.05 0.01

Alternative H₁ μ ≠ μ₀ — two-tailed μ > μ₀ — right-tailed μ < μ₀ — left-tailed

Standard Error2.000

z statistic2.000

p-value0.0455

Critical value(s)±1.960

Degrees of freedom35

DecisionReject H₀

// The original interactive controller is executed by the Observable JS runtime.
(function(){
  const varianceModeEl = document.getElementById('varianceMode');
  const mu0El = document.getElementById('mu0');
  const xbarEl = document.getElementById('xbar');
  const sigmaEl = document.getElementById('sigma');
  const nEl = document.getElementById('n');
  const alphaEl = document.getElementById('alpha');
  const testTypeEl = document.getElementById('testType');
  const calcBtn = document.getElementById('calculate');

  const seOut = document.getElementById('se');
  const zOut = document.getElementById('zval');
  const pOut = document.getElementById('pval');
  const critOut = document.getElementById('crit');
  const decisionMini = document.getElementById('decisionMini');
  const statLabel = document.getElementById('statLabel');
  const spreadLabel = document.getElementById('spreadLabel');
  const methodBanner = document.getElementById('methodBanner');
  const dfKpi = document.getElementById('dfKpi');
  const dfOut = document.getElementById('dfval');
  const zExampleBlock = document.getElementById('zExampleBlock');
  const tExampleBlock = document.getElementById('tExampleBlock');
  const decisionBox = document.getElementById('decisionBox');
  const canvas = document.getElementById('plot');
  const ctx = canvas.getContext('2d');

  function erf(x){
    const sign = x < 0 ? -1 : 1;
    x = Math.abs(x);
    const a1=0.254829592,a2=-0.284496736,a3=1.421413741,a4=-1.453152027,a5=1.061405429,p=0.3275911;
    const t=1/(1+p*x);
    const y=1-(((((a5*t+a4)*t)+a3)*t+a2)*t+a1)*t*Math.exp(-x*x);
    return sign*y;
  }

  function cdf(z){
    return 0.5*(1+erf(z/Math.sqrt(2)));
  }

  function invNorm(p){
    if(p<=0 || p>=1) return NaN;
    const a=[-39.69683028665376,220.9460984245205,-275.9285104469687,138.3577518672690,-30.66479806614716,2.506628277459239];
    const b=[-54.47609879822406,161.5858368580409,-155.6989798598866,66.80131188771972,-13.28068155288572];
    const c=[-0.007784894002430293,-0.3223964580411365,-2.400758277161838,-2.549732539343734,4.374664141464968,2.938163982698783];
    const d=[0.007784695709041462,0.3224671290700398,2.445134137142996,3.754408661907416];
    const plow=0.02425, phigh=1-plow;
    let q,r;
    if(p<plow){
      q=Math.sqrt(-2*Math.log(p));
      return (((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) /
             ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1);
    }
    if(p>phigh){
      q=Math.sqrt(-2*Math.log(1-p));
      return -(((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) /
              ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1);
    }
    q=p-0.5;r=q*q;
    return (((((a[0]*r+a[1])*r+a[2])*r+a[3])*r+a[4])*r+a[5])*q /
           (((((b[0]*r+b[1])*r+b[2])*r+b[3])*r+b[4])*r+1);
  }

  function pdf(z){
    return Math.exp(-0.5*z*z)/Math.sqrt(2*Math.PI);
  }

  function logGamma(z){
    const c=[676.5203681218851,-1259.1392167224028,771.32342877765313,
      -176.61502916214059,12.507343278686905,-0.13857109526572012,
      9.9843695780195716e-6,1.5056327351493116e-7];
    if(z < 0.5) return Math.log(Math.PI)-Math.log(Math.sin(Math.PI*z))-logGamma(1-z);
    z -= 1;
    let x=0.99999999999980993;
    for(let i=0;i<c.length;i++) x += c[i]/(z+i+1);
    const t=z+c.length-0.5;
    return 0.5*Math.log(2*Math.PI)+(z+0.5)*Math.log(t)-t+Math.log(x);
  }

  function betacf(a,b,x){
    const MAXIT=120, EPS=3e-12, FPMIN=1e-30;
    const qab=a+b, qap=a+1, qam=a-1;
    let c=1, d=1-qab*x/qap;
    if(Math.abs(d)<FPMIN) d=FPMIN;
    d=1/d;
    let h=d;
    for(let m=1;m<=MAXIT;m++){
      const m2=2*m;
      let aa=m*(b-m)*x/((qam+m2)*(a+m2));
      d=1+aa*d; if(Math.abs(d)<FPMIN)d=FPMIN;
      c=1+aa/c; if(Math.abs(c)<FPMIN)c=FPMIN;
      d=1/d; h*=d*c;
      aa=-(a+m)*(qab+m)*x/((a+m2)*(qap+m2));
      d=1+aa*d; if(Math.abs(d)<FPMIN)d=FPMIN;
      c=1+aa/c; if(Math.abs(c)<FPMIN)c=FPMIN;
      d=1/d;
      const del=d*c; h*=del;
      if(Math.abs(del-1)<EPS) break;
    }
    return h;
  }

  function regIncompleteBeta(x,a,b){
    if(x<=0) return 0;
    if(x>=1) return 1;
    const bt=Math.exp(logGamma(a+b)-logGamma(a)-logGamma(b)+a*Math.log(x)+b*Math.log(1-x));
    if(x < (a+1)/(a+b+2)) return bt*betacf(a,b,x)/a;
    return 1-bt*betacf(b,a,1-x)/b;
  }

  function tCDF(t,df){
    if(t===0) return 0.5;
    const x=df/(df+t*t);
    const ib=regIncompleteBeta(x,df/2,0.5);
    return t>0 ? 1-0.5*ib : 0.5*ib;
  }

  function tPdf(t,df){
    const logC=logGamma((df+1)/2)-logGamma(df/2)-0.5*Math.log(df*Math.PI);
    return Math.exp(logC-((df+1)/2)*Math.log(1+t*t/df));
  }

  function tInv(p,df){
    if(p<=0 || p>=1) return NaN;
    let lo=-20, hi=20;
    for(let i=0;i<90;i++){
      const mid=(lo+hi)/2;
      if(tCDF(mid,df)<p) lo=mid; else hi=mid;
    }
    return (lo+hi)/2;
  }

  function readValues(){
    let mu0 = parseFloat(mu0El.value);
    let xbar = parseFloat(xbarEl.value);
    let sigma = parseFloat(sigmaEl.value);
    let n = parseInt(nEl.value,10);
    let alpha = parseFloat(alphaEl.value);
    const mode = varianceModeEl.value;
    if(!Number.isFinite(mu0)) mu0=0;
    if(!Number.isFinite(xbar)) xbar=mu0;
    if(!Number.isFinite(sigma) || sigma<=0){sigma=1;sigmaEl.value='1'}
    const minN = mode==='unknown' ? 2 : 1;
    if(!Number.isFinite(n) || n<minN){n=minN;nEl.value=String(minN)}
    nEl.min=String(minN);
    return {mu0,xbar,sigma,n,alpha,type:testTypeEl.value,mode};
  }

  function compute(){
    const v=readValues();
    const se=v.sigma/Math.sqrt(v.n);
    const stat=(v.xbar-v.mu0)/se;
    const isT=v.mode==='unknown';
    const df=Math.max(1,v.n-1);
    const CDF=isT ? (x=>tCDF(x,df)) : cdf;
    const INV=isT ? (p=>tInv(p,df)) : invNorm;

    let p,critText,critical=[];
    if(v.type==='two'){
      p=2*(1-CDF(Math.abs(stat)));
      const c=INV(1-v.alpha/2);
      critical=[-c,c];
      critText='±'+c.toFixed(3);
    } else if(v.type==='right'){
      p=1-CDF(stat);
      const c=INV(1-v.alpha);
      critical=[c];
      critText=c.toFixed(3);
    } else {
      p=CDF(stat);
      const c=INV(v.alpha);
      critical=[c];
      critText=c.toFixed(3);
    }

    p=Math.max(0,Math.min(1,p));
    const reject=p<=v.alpha;

    if(isT){
      zExampleBlock.style.display='none';
      tExampleBlock.style.display='block';
    }else{
      zExampleBlock.style.display='block';
      tExampleBlock.style.display='none';
    }

    if(isT){
      spreadLabel.innerHTML='Sample SD s';
      statLabel.textContent='t statistic';
      methodBanner.innerHTML='<strong>Selected in Part 5:</strong> Case 2 — t-test. Population variance is unknown, so σ is replaced by the sample standard deviation s.';
      dfKpi.style.display='';
      dfOut.textContent=String(df);
    }else{
      spreadLabel.innerHTML='Population SD σ';
      statLabel.textContent='z statistic';
      methodBanner.innerHTML='<strong>Selected in Part 5:</strong> Case 1 — z-test. Population standard deviation σ is known.';
      dfKpi.style.display='none';
    }

    seOut.textContent=se.toFixed(3);
    zOut.textContent=stat.toFixed(3);
    pOut.textContent=p<0.0001?'< 0.0001':p.toFixed(4);
    critOut.textContent=critText;
    decisionMini.textContent=reject?'Reject H₀':'Fail to reject H₀';
    decisionMini.style.color=reject?'#c62828':'#17743b';

    let h1='';
    if(v.type==='two') h1=`μ ≠ ${v.mu0}`;
    if(v.type==='right') h1=`μ > ${v.mu0}`;
    if(v.type==='left') h1=`μ < ${v.mu0}`;

    decisionBox.classList.remove('reject','keep');
    if(reject){
      decisionBox.classList.add('reject');
      decisionBox.innerHTML =
        `<strong>Decision: Reject H₀.</strong> Since p = ${p.toFixed(4)} ≤ α = ${v.alpha.toFixed(2)}, `+
        `the observed result is sufficiently unusual under H₀. The data provide evidence in favor of H₁: ${h1}.`;
    } else {
      decisionBox.classList.add('keep');
      decisionBox.innerHTML =
        `<strong>Decision: Fail to reject H₀.</strong> Since p = ${p.toFixed(4)} > α = ${v.alpha.toFixed(2)}, `+
        `the observed result is not sufficiently unusual under H₀. This does not prove H₀ is true.`;
    }

    drawPlot(stat,v.alpha,v.type,critical,isT,df);
  }

  function drawPlot(z,alpha,type,critical,isT,df){
    const W=canvas.width,H=canvas.height;
    const L=65,R=25,T=25,B=60;
    const pw=W-L-R, ph=H-T-B;
    const xmin=-4.2,xmax=4.2;
    const ymax=0.42;

    ctx.clearRect(0,0,W,H);
    ctx.fillStyle='#ffffff';ctx.fillRect(0,0,W,H);

    function xpix(x){return L+(x-xmin)/(xmax-xmin)*pw}
    function ypix(y){return H-B-y/ymax*ph}

    ctx.strokeStyle='#dfe5ef';ctx.lineWidth=1;
    for(let x=-4;x<=4;x++){
      const px=xpix(x);
      ctx.beginPath();ctx.moveTo(px,T);ctx.lineTo(px,H-B);ctx.stroke();
      ctx.fillStyle='#5f6b7f';ctx.font='14px system-ui';
      ctx.fillText(String(x),px-4,H-28);
    }
    ctx.beginPath();ctx.moveTo(L,H-B);ctx.lineTo(W-R,H-B);ctx.strokeStyle='#8f9bb0';ctx.stroke();

    // Rejection regions
    function shade(a,b,color){
      ctx.beginPath();
      ctx.moveTo(xpix(a),H-B);
      for(let i=0;i<=220;i++){
        const x=a+(b-a)*i/220;
        ctx.lineTo(xpix(x),ypix(isT ? tPdf(x,df) : pdf(x)));
      }
      ctx.lineTo(xpix(b),H-B);
      ctx.closePath();
      ctx.fillStyle=color;ctx.fill();
    }

    if(type==='two'){
      shade(xmin,critical[0],'rgba(255,142,155,.28)');
      shade(critical[1],xmax,'rgba(255,142,155,.28)');
    } else if(type==='right'){
      shade(critical[0],xmax,'rgba(255,142,155,.28)');
    } else {
      shade(xmin,critical[0],'rgba(255,142,155,.28)');
    }

    // Curve
    ctx.beginPath();
    for(let i=0;i<=500;i++){
      const x=xmin+(xmax-xmin)*i/500;
      const px=xpix(x),py=ypix(isT ? tPdf(x,df) : pdf(x));
      if(i===0)ctx.moveTo(px,py);else ctx.lineTo(px,py);
    }
    ctx.strokeStyle='#177e68';ctx.lineWidth=3;ctx.stroke();

    // Critical lines
    ctx.setLineDash([6,6]);
    ctx.strokeStyle='#c62828';ctx.lineWidth=2;
    critical.forEach(c=>{
      const px=xpix(c);
      ctx.beginPath();ctx.moveTo(px,T+35);ctx.lineTo(px,H-B);ctx.stroke();
    });

    // Observed z
    ctx.setLineDash([]);
    const zx=Math.max(xmin,Math.min(xmax,z));
    const pz=xpix(zx);
    ctx.strokeStyle='#9a6700';ctx.lineWidth=3;
    ctx.beginPath();ctx.moveTo(pz,T);ctx.lineTo(pz,H-B);ctx.stroke();
    ctx.fillStyle='#9a6700';ctx.font='bold 15px system-ui';
    ctx.fillText(`${isT ? 'observed t' : 'observed z'} = ${z.toFixed(2)}`,Math.min(W-180,Math.max(75,pz+8)),T+18);

    ctx.fillStyle='#b4232f';ctx.font='14px system-ui';
    ctx.fillText('rejection region',W-185,T+42);
    ctx.fillStyle='#5f6b7f';
    ctx.fillText(isT ? `Student t distribution under H₀ (df = ${df})` : 'Standard normal distribution under H₀',L,T+18);
    ctx.fillText('z',W/2,H-8);
  }

  calcBtn.addEventListener('click',compute);
  [mu0El,xbarEl,sigmaEl,nEl,alphaEl,testTypeEl].forEach(el=>{
    el.addEventListener('input',compute);
    el.addEventListener('change',compute);
  });

  varianceModeEl.addEventListener('change',()=>{
    if(varianceModeEl.value==='known'){
      mu0El.value='100';
      xbarEl.value='104';
      sigmaEl.value='12';
      nEl.value='36';
      alphaEl.value='0.05';
      testTypeEl.value='two';
    }else{
      mu0El.value='50';
      xbarEl.value='54';
      sigmaEl.value='8';
      nEl.value='16';
      alphaEl.value='0.05';
      testTypeEl.value='two';
    }
    compute();
  });





  compute();
})();