<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Normal Distribution Concavity</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      min-height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 12px 12px 8px;
    }

    .controls {
      display: flex;
      gap: 6px;
      margin-bottom: 8px;
      flex-wrap: wrap;
      justify-content: center;
    }

    .btn {
      appearance: none;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      color: #475569;
      padding: 5px 12px;
      border-radius: 20px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .btn:hover {
      background: #f1f5f9;
      color: #1e2945;
    }
    .btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.25);
    }

    .svg-container {
      width: 100%;
      max-width: 580px;
      position: relative;
      user-select: none;
      touch-action: none;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    .region-down {
      fill: rgba(249, 115, 22, 0.18);
      transition: fill-opacity 0.3s ease;
    }
    .region-up {
      fill: rgba(34, 197, 94, 0.16);
      transition: fill-opacity 0.3s ease;
    }
    .dimmed {
      fill-opacity: 0.04 !important;
      opacity: 0.25 !important;
    }

    .curve-main {
      fill: none;
      stroke: #334155;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-linejoin: round;
    }

    .guide-line {
      stroke: #94a3b8;
      stroke-dasharray: 4 4;
      stroke-width: 1.2;
    }

    .axis-line {
      stroke: #64748b;
      stroke-width: 1.5;
    }

    .tangent-line {
      stroke: #0284c7;
      stroke-width: 2;
      stroke-dasharray: 3 3;
      transition: opacity 0.15s ease;
    }

    .tracker-dot {
      fill: #0284c7;
      stroke: #ffffff;
      stroke-width: 2;
      filter: drop-shadow(0 2px 4px rgba(2, 132, 199, 0.4));
    }

    .status-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      margin-top: 4px;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
      min-height: 24px;
      text-align: center;
    }

    .badge {
      display: inline-flex;
      align-items: center;
      gap: 5px;
      padding: 3px 10px;
      border-radius: 12px;
      font-size: 12px;
      font-weight: 700;
    }
    .badge-orange {
      background: #ffedd5;
      color: #c2410c;
    }
    .badge-green {
      background: #dcfce7;
      color: #15803d;
    }
    .badge-cyan {
      background: #e0f2fe;
      color: #0369a1;
    }

    @keyframes pulseRing {
      0% { r: 6; opacity: 0.9; stroke-width: 2; }
      100% { r: 15; opacity: 0; stroke-width: 0.5; }
    }
    .glow-ring {
      animation: pulseRing 2s infinite ease-out;
      fill: none;
      stroke: #0284c7;
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn active" onclick="setMode('all')">Overview</button>
    <button class="btn" onclick="setMode('down')">Concave Down ⌢</button>
    <button class="btn" onclick="setMode('up')">Concave Up ⌣</button>
    <button class="btn" onclick="setMode('inflection')">Inflection Points ✦</button>
  </div>

  <div class="svg-container">
    <svg id="chart" viewBox="0 0 600 360">
      <defs>
        <filter id="glow" x="-40%" y="-40%" width="180%" height="180%">
          <feGaussianBlur stdDeviation="3" result="blur" />
          <feComposite in="SourceGraphic" in2="blur" operator="over" />
        </filter>
      </defs>

      
      <path id="left-up-fill" class="region-up" d="" />
      <path id="mid-down-fill" class="region-down" d="" />
      <path id="right-up-fill" class="region-up" d="" />

      
      <line x1="20" y1="280" x2="580" y2="280" class="axis-line" />

      
      <line x1="180" y1="280" x2="180" y2="286" class="axis-line" />
      <line x1="300" y1="280" x2="300" y2="286" class="axis-line" />
      <line x1="420" y1="280" x2="420" y2="286" class="axis-line" />

      
      <line id="guide-left" x1="180" y1="280" x2="180" y2="153.6" class="guide-line" />
      <line id="guide-mid" x1="300" y1="280" x2="300" y2="70" class="guide-line" stroke-dasharray="2 4" stroke-opacity="0.6" />
      <line id="guide-right" x1="420" y1="280" x2="420" y2="153.6" class="guide-line" />

      
      <path id="curve" class="curve-main" d="" />

      
      <g id="label-left-up" style="transition: opacity 0.3s ease;">
        <rect x="35" y="195" width="115" height="34" rx="6" fill="#f0fdf4" stroke="#86efac" stroke-width="1" />
        <text x="92.5" y="210" text-anchor="middle" font-size="11" font-weight="700" fill="#15803d">Concave Up ⌣</text>
        <text x="92.5" y="223" text-anchor="middle" font-size="10" font-weight="600" fill="#16a34a">f''(x) &gt; 0</text>
      </g>

      <g id="label-down" style="transition: opacity 0.3s ease;">
        <rect x="235" y="22" width="130" height="36" rx="6" fill="#fff7ed" stroke="#fdba74" stroke-width="1" />
        <text x="300" y="38" text-anchor="middle" font-size="12" font-weight="700" fill="#c2410c">Concave Down ⌢</text>
        <text x="300" y="52" text-anchor="middle" font-size="10" font-weight="600" fill="#ea580c">f''(x) &lt; 0</text>
      </g>

      <g id="label-right-up" style="transition: opacity 0.3s ease;">
        <rect x="450" y="195" width="115" height="34" rx="6" fill="#f0fdf4" stroke="#86efac" stroke-width="1" />
        <text x="507.5" y="210" text-anchor="middle" font-size="11" font-weight="700" fill="#15803d">Concave Up ⌣</text>
        <text x="507.5" y="223" text-anchor="middle" font-size="10" font-weight="600" fill="#16a34a">f''(x) &gt; 0</text>
      </g>

      
      <g id="ip-left" style="transition: opacity 0.3s ease;">
        <circle cx="180" cy="153.6" r="10" class="glow-ring" />
        <circle cx="180" cy="153.6" r="5.5" fill="#0284c7" stroke="#ffffff" stroke-width="2" filter="url(#glow)" />
        <rect x="85" y="105" width="150" height="32" rx="5" fill="#ffffff" stroke="#bae6fd" stroke-width="1" filter="drop-shadow(0 2px 4px rgba(0,0,0,0.06))" />
        <text x="160" y="119" text-anchor="middle" font-size="10.5" font-weight="700" fill="#0369a1">Inflection Point ✦</text>
        <text x="160" y="131" text-anchor="middle" font-size="9.5" font-weight="600" fill="#0284c7">(μ − σ,  f(μ − σ))</text>
        <line x1="160" y1="137" x2="176" y2="149" stroke="#0284c7" stroke-width="1.2" />
      </g>

      
      <g id="ip-right" style="transition: opacity 0.3s ease;">
        <circle cx="420" cy="153.6" r="10" class="glow-ring" />
        <circle cx="420" cy="153.6" r="5.5" fill="#0284c7" stroke="#ffffff" stroke-width="2" filter="url(#glow)" />
        <rect x="365" y="105" width="150" height="32" rx="5" fill="#ffffff" stroke="#bae6fd" stroke-width="1" filter="drop-shadow(0 2px 4px rgba(0,0,0,0.06))" />
        <text x="440" y="119" text-anchor="middle" font-size="10.5" font-weight="700" fill="#0369a1">Inflection Point ✦</text>
        <text x="440" y="131" text-anchor="middle" font-size="9.5" font-weight="600" fill="#0284c7">(μ + σ,  f(μ + σ))</text>
        <line x1="440" y1="137" x2="424" y2="149" stroke="#0284c7" stroke-width="1.2" />
      </g>

      
      <text x="180" y="304" text-anchor="middle" font-size="12" font-weight="700" fill="#1e2945">μ − σ</text>
      <text x="300" y="304" text-anchor="middle" font-size="13" font-weight="700" fill="#1e2945">μ</text>
      <text x="420" y="304" text-anchor="middle" font-size="12" font-weight="700" fill="#1e2945">μ + σ</text>
      <text x="560" y="302" text-anchor="end" font-size="11" font-weight="600" fill="#64748b">x →</text>

      
      <g id="interactive-cursor" style="display: none;">
        <line id="tangent-line" x1="0" y1="0" x2="0" y2="0" class="tangent-line" />
        <circle id="track-dot" cx="0" cy="0" r="4.5" class="tracker-dot" />
      </g>
    </svg>
  </div>

  <div class="status-bar" id="statusBar">
    <span class="badge badge-orange">Center: Concave Down</span>
    <span class="badge badge-cyan">x = μ ± σ: Inflection</span>
    <span class="badge badge-green">Tails: Concave Up</span>
  </div>

  <script>
    // Mathematical constants for Normal curve plotting
    const mu = 300;
    const sigma = 120; // x-distance for 1 standard deviation
    const baselineY = 280;
    const peakHeight = 210; // Peak amplitude in SVG px

    // Normal curve equation: y = baselineY - peakHeight * exp(-0.5 * ((x - mu)/sigma)^2)
    function getY(x) {
      const z = (x - mu) / sigma;
      return baselineY - peakHeight * Math.exp(-0.5 * z * z);
    }

    // Derivative dy/dx for tangent slope calculation
    function getSlope(x) {
      const z = (x - mu) / sigma;
      // dy_math/dx = - (z/sigma) * peakHeight * exp(-0.5 * z^2)
      // Since SVG y goes down:
      return (z / sigma) * peakHeight * Math.exp(-0.5 * z * z);
    }

    // Generate accurate path points
    function makePathString(startX, endX, step = 2) {
      let d = `M ${startX} ${getY(startX).toFixed(2)}`;
      for (let x = startX + step; x <= endX; x += step) {
        d += ` L ${x} ${getY(x).toFixed(2)}`;
      }
      if (endX % step !== 0) {
        d += ` L ${endX} ${getY(endX).toFixed(2)}`;
      }
      return d;
    }

    // Initialize SVG paths
    const curveD = makePathString(20, 580);
    document.getElementById('curve').setAttribute('d', curveD);

    // Left fill (20 to 180, closed along baseline)
    const leftFillD = makePathString(20, 180) + ` L 180 ${baselineY} L 20 ${baselineY} Z`;
    document.getElementById('left-up-fill').setAttribute('d', leftFillD);

    // Mid fill (180 to 420, closed along baseline)
    const midFillD = makePathString(180, 420) + ` L 420 ${baselineY} L 180 ${baselineY} Z`;
    document.getElementById('mid-down-fill').setAttribute('d', midFillD);

    // Right fill (420 to 580, closed along baseline)
    const rightFillD = makePathString(420, 580) + ` L 580 ${baselineY} L 420 ${baselineY} Z`;
    document.getElementById('right-up-fill').setAttribute('d', rightFillD);

    // Mode filter logic
    function setMode(mode) {
      document.querySelectorAll('.btn').forEach(b => b.classList.remove('active'));
      const activeBtn = Array.from(document.querySelectorAll('.btn')).find(b =>
        (mode === 'all' && b.textContent.includes('Overview')) ||
        (mode === 'down' && b.textContent.includes('Down')) ||
        (mode === 'up' && b.textContent.includes('Up')) ||
        (mode === 'inflection' && b.textContent.includes('Inflection'))
      );
      if (activeBtn) activeBtn.classList.add('active');

      const downFill = document.getElementById('mid-down-fill');
      const leftFill = document.getElementById('left-up-fill');
      const rightFill = document.getElementById('right-up-fill');
      const labelDown = document.getElementById('label-down');
      const labelLeft = document.getElementById('label-left-up');
      const labelRight = document.getElementById('label-right-up');
      const ipLeft = document.getElementById('ip-left');
      const ipRight = document.getElementById('ip-right');
      const bar = document.getElementById('statusBar');

      // Reset dimming
      [downFill, leftFill, rightFill, labelDown, labelLeft, labelRight, ipLeft, ipRight].forEach(el => {
        el.classList.remove('dimmed');
      });

      if (mode === 'down') {
        leftFill.classList.add('dimmed');
        rightFill.classList.add('dimmed');
        labelLeft.classList.add('dimmed');
        labelRight.classList.add('dimmed');
        bar.innerHTML = '<span class="badge badge-orange">Peak Region: Concave Down (f\'\'(x) &lt; 0) — Curve bends downward</span>';
      } else if (mode === 'up') {
        downFill.classList.add('dimmed');
        labelDown.classList.add('dimmed');
        bar.innerHTML = '<span class="badge badge-green">Tails: Concave Up (f\'\'(x) &gt; 0) — Curve bends upward</span>';
      } else if (mode === 'inflection') {
        downFill.classList.add('dimmed');
        leftFill.classList.add('dimmed');
        rightFill.classList.add('dimmed');
        labelDown.classList.add('dimmed');
        labelLeft.classList.add('dimmed');
        labelRight.classList.add('dimmed');
        bar.innerHTML = '<span class="badge badge-cyan">Inflection Points: f\'\'(x) = 0 — Exact transition between curvatures</span>';
      } else {
        bar.innerHTML = '<span class="badge badge-orange">Center: Concave Down</span><span class="badge badge-cyan">x = μ ± σ: Inflection</span><span class="badge badge-green">Tails: Concave Up</span>';
      }
    }

    // Touch / Hover Tangent Inspector
    const svg = document.getElementById('chart');
    const cursor = document.getElementById('interactive-cursor');
    const tangent = document.getElementById('tangent-line');
    const dot = document.getElementById('track-dot');

    function updateInspector(clientX) {
      const rect = svg.getBoundingClientRect();
      const scale = 600 / rect.width;
      let svgX = (clientX - rect.left) * scale;
      if (svgX < 30 || svgX > 570) {
        cursor.style.display = 'none';
        return;
      }
      cursor.style.display = 'inline';
      const svgY = getY(svgX);
      const slope = getSlope(svgX);

      dot.setAttribute('cx', svgX);
      dot.setAttribute('cy', svgY);

      // 60px tangent segment
      const dx = 32;
      const dy = slope * dx;
      tangent.setAttribute('x1', svgX - dx);
      tangent.setAttribute('y1', svgY - dy);
      tangent.setAttribute('x2', svgX + dx);
      tangent.setAttribute('y2', svgY + dy);
    }

    svg.addEventListener('pointermove', (e) => updateInspector(e.clientX));
    svg.addEventListener('pointerleave', () => { cursor.style.display = 'none'; });
  </script>
</body>
</html>

Properties of the Continuous Normal Distribution Curve

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      height: 100%;
      background: #f8fafc;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px 16px;
      user-select: none;
    }

    .controls {
      display: flex;
      gap: 8px;
      flex-wrap: wrap;
      justify-content: center;
      margin-bottom: 4px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #475569;
      padding: 6px 12px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
    }

    .svg-container {
      width: 100%;
      max-width: 620px;
      flex-grow: 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    svg {
      width: 100%;
      height: auto;
      overflow: visible;
    }

    .axis-line { stroke: #64748b; stroke-width: 2; }
    .grid-line { stroke: #e2e8f0; stroke-width: 1; stroke-dasharray: 4,4; }
    .mean-line { stroke: #0284c7; stroke-width: 2; stroke-dasharray: 6,4; }
    .curve-line { stroke: #0284c7; stroke-width: 2.5; fill: none; stroke-linecap: round; }
    
    .area-fill {
      fill: #38bdf8;
      fill-opacity: 0.25;
      transition: fill-opacity 0.3s ease, fill 0.3s ease;
    }
    
    .tick-label {
      font-size: 12px;
      font-family: inherit;
      fill: #475569;
      text-anchor: middle;
      font-weight: 500;
    }

    .mean-label {
      font-size: 13px;
      font-weight: 700;
      fill: #0284c7;
      text-anchor: middle;
    }

    .info-badge {
      font-size: 14px;
      font-weight: 700;
      fill: #0369a1;
      text-anchor: middle;
    }

    .asymptote-label {
      font-size: 11px;
      font-weight: 600;
      fill: #94a3b8;
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn active" onclick="setMode('total')">Total Area</button>
    <button class="btn" onclick="setMode('1sd')">±1σ (68.2%)</button>
    <button class="btn" onclick="setMode('2sd')">±2σ (95.4%)</button>
    <button class="btn" onclick="setMode('3sd')">±3σ (99.7%)</button>
  </div>

  <div class="svg-container">
    <svg viewBox="0 0 600 300" id="plotSvg">
      
      <path id="areaPath" class="area-fill" />

      
      <path id="curvePath" class="curve-line" />

      
      <line x1="20" y1="240" x2="580" y2="240" class="axis-line" />

      
      <line x1="300" y1="45" x2="300" y2="240" class="mean-line" />

      
      
      <line x1="75" y1="45" x2="75" y2="240" class="grid-line" />
      <line x1="75" y1="240" x2="75" y2="246" class="axis-line" />
      <text x="75" y="264" class="tick-label">μ − 3σ</text>

      
      <line x1="150" y1="45" x2="150" y2="240" class="grid-line" />
      <line x1="150" y1="240" x2="150" y2="246" class="axis-line" />
      <text x="150" y="264" class="tick-label">μ − 2σ</text>

      
      <line x1="225" y1="45" x2="225" y2="240" class="grid-line" />
      <line x1="225" y1="240" x2="225" y2="246" class="axis-line" />
      <text x="225" y="264" class="tick-label">μ − σ</text>

      
      <line x1="300" y1="240" x2="300" y2="248" class="axis-line" stroke-width="2.5" />
      <text x="300" y="265" class="mean-label">μ</text>

      
      <line x1="375" y1="45" x2="375" y2="240" class="grid-line" />
      <line x1="375" y1="240" x2="375" y2="246" class="axis-line" />
      <text x="375" y="264" class="tick-label">μ + σ</text>

      
      <line x1="450" y1="45" x2="450" y2="240" class="grid-line" />
      <line x1="450" y1="240" x2="450" y2="246" class="axis-line" />
      <text x="450" y="264" class="tick-label">μ + 2σ</text>

      
      <line x1="525" y1="45" x2="525" y2="240" class="grid-line" />
      <line x1="525" y1="240" x2="525" y2="246" class="axis-line" />
      <text x="525" y="264" class="tick-label">μ + 3σ</text>

      
      <text x="300" y="32" class="mean-label" style="font-size:12px;">Line of Symmetry (x = μ)</text>

      
      <text x="30" y="232" class="asymptote-label">Asymptote: y = 0</text>
      <text x="475" y="232" class="asymptote-label">Asymptote: y = 0</text>

      
      <rect id="badgeBg" x="220" y="125" width="160" height="28" rx="14" fill="#ffffff" stroke="#bae6fd" stroke-width="1.5" />
      <text id="areaText" x="300" y="144" class="info-badge">Total Area = 1.0</text>
    </svg>
  </div>

  <script>
    const meanX = 300;
    const sigmaPx = 75;
    const baseY = 240;
    const peakHeight = 185;

    function gaussianY(x) {
      const z = (x - meanX) / sigmaPx;
      return baseY - peakHeight * Math.exp(-0.5 * z * z);
    }

    // Build the smooth full curve path
    let curveD = `M 20 ${gaussianY(20).toFixed(2)}`;
    for (let x = 21; x <= 580; x += 2) {
      curveD += ` L ${x} ${gaussianY(x).toFixed(2)}`;
    }
    document.getElementById('curvePath').setAttribute('d', curveD);

    function updateArea(zMin, zMax, label) {
      const x1 = Math.max(20, meanX + zMin * sigmaPx);
      const x2 = Math.min(580, meanX + zMax * sigmaPx);

      let d = `M ${x1} ${baseY}`;
      for (let x = x1; x <= x2; x += 2) {
        d += ` L ${x} ${gaussianY(x).toFixed(2)}`;
      }
      d += ` L ${x2} ${gaussianY(x2).toFixed(2)} L ${x2} ${baseY} Z`;

      document.getElementById('areaPath').setAttribute('d', d);
      document.getElementById('areaText').textContent = label;
    }

    function setMode(mode) {
      document.querySelectorAll('.btn').forEach(b => b.classList.remove('active'));
      event.target.classList.add('active');

      if (mode === 'total') {
        updateArea(-3.73, 3.73, 'Total Area = 1.0');
      } else if (mode === '1sd') {
        updateArea(-1, 1, 'Area ≈ 68.2%');
      } else if (mode === '2sd') {
        updateArea(-2, 2, 'Area ≈ 95.4%');
      } else if (mode === '3sd') {
        updateArea(-3, 3, 'Area ≈ 99.7%');
      }
    }

    // Initial render
    updateArea(-3.73, 3.73, 'Total Area = 1.0');
  </script>
</body>
</html>

Calculating Z-Scores and Standardizing Normal Variables

Determining Normal Probabilities Using Standard Normal Tables and Technology

Finding Percentiles and Threshold Values with Inverse Normal Calculations

Finding Unknown Mean and Standard Deviation in Normal Models

Assessing Normality Using Normal Probability Plots and Points of Inflection

Evaluating Limitations and Failures of the Normality Assumption

math

Continuous Normal Distributions and Probabilistic Modeling

Introduction to the Normal Distribution

The 68-95-99.7 Empirical Rule

Finding Percentages Using the Empirical Rule

Calculating and Interpreting Z-Scores

Using a Z-Table to Find Probabilities Under the Normal Curve

Using Technology to Estimate Areas Under the Normal Curve

Recognizing When the Normal Distribution Does Not Apply

OGM.13.STA.3

OGM.13.STA.3

Aligned learning resources

Related diagrams