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System default

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Tutoring abilities

Imagine two gaming companies both advertise that their "average" player spends 5 hours a day playing.

At Company A, almost everyone plays around 5 hours; at Company B, most people play for 1 hour while a few streamers play for 20 hours straight.

Single numbers can hide the truth, so how do we inspect the full picture of our data?

The **shape** of a distribution tells us how data points are spread out across a number line.

<!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>Data Shapes and Outliers</title>
  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    }

    body {
      background: transparent;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      user-select: none;
    }

    .nav-tabs {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      width: 100%;
      max-width: 540px;
      margin-bottom: 8px;
    }

    .tab-btn {
      background: #ffffff;
      border: 1.5px solid #e6e6e6;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #6b7280;
      border-radius: 20px;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.04);
      transition: all 0.2s ease;
    }

    .tab-btn:hover {
      background: #f8f9fa;
      color: #1e2945;
    }

    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }

    .chart-area {
      position: relative;
      width: 100%;
      max-width: 540px;
      height: 250px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e6e6e6;
      display: flex;
      flex-direction: column;
      justify-content: flex-end;
      padding: 20px 16px 36px 16px;
    }

    .svg-layer {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      pointer-events: none;
    }

    .curve-path {
      fill: rgba(34, 183, 255, 0.12);
      stroke: #22b7ff;
      stroke-width: 2.5;
      stroke-linecap: round;
      stroke-linejoin: round;
      transition: d 0.4s ease;
    }

    .dots-grid {
      position: relative;
      width: 100%;
      height: 160px;
      display: flex;
      justify-content: space-between;
      align-items: flex-end;
      z-index: 1;
    }

    .dot-col {
      display: flex;
      flex-direction: column-reverse;
      align-items: center;
      gap: 3px;
      width: 8%;
      height: 100%;
      justify-content: flex-start;
    }

    .dot {
      width: 14px;
      height: 14px;
      border-radius: 50%;
      background: #22b7ff;
      transition: all 0.35s ease;
      opacity: 0;
      transform: translateY(15px) scale(0.5);
    }

    .dot.show {
      opacity: 1;
      transform: translateY(0) scale(1);
    }

    .dot.outlier-dot {
      background: #ff5964;
      box-shadow: 0 0 0 3px rgba(255, 89, 100, 0.25);
    }

    .axis {
      position: absolute;
      bottom: 28px;
      left: 16px;
      right: 16px;
      height: 2px;
      background: #cbd5e1;
      display: flex;
      justify-content: space-between;
    }

    .axis-labels {
      position: absolute;
      bottom: 10px;
      left: 16px;
      right: 16px;
      display: flex;
      justify-content: space-between;
    }

    .axis-label {
      font-size: 11px;
      font-weight: 600;
      color: #94a3b8;
      width: 8%;
      text-align: center;
    }

    .indicator-badge {
      position: absolute;
      padding: 4px 10px;
      font-size: 12px;
      font-weight: 700;
      border-radius: 12px;
      transition: all 0.35s ease;
      pointer-events: none;
      z-index: 2;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .badge-peak {
      background: #e0f2fe;
      color: #0284c7;
      border: 1px solid #bae6fd;
    }

    .badge-tail {
      background: #fef3c7;
      color: #d97706;
      border: 1px solid #fde68a;
    }

    .badge-outlier {
      background: #fee2e2;
      color: #dc2626;
      border: 1px solid #fecaca;
    }

    .footer-note {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-top: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
    }
  </style>
</head>
<body>

  <div class="nav-tabs">
    <button class="tab-btn active" onclick="setShape(0)">Symmetric</button>
    <button class="tab-btn" onclick="setShape(1)">Skewed Right</button>
    <button class="tab-btn" onclick="setShape(2)">Skewed Left</button>
    <button class="tab-btn" onclick="setShape(3)">Outlier</button>
  </div>

  <div class="chart-area" id="chart">
    <svg class="svg-layer" viewBox="0 0 500 250" preserveAspectRatio="none">
      <path id="curve" class="curve-path" d="" />
    </svg>

    <div class="dots-grid" id="grid"></div>

    <div class="axis"></div>
    <div class="axis-labels" id="axisLabels"></div>

    <div id="badge1" class="indicator-badge"></div>
    <div id="badge2" class="indicator-badge"></div>
  </div>

  <div class="footer-note" id="footerText">
    <span>Balanced</span>
  </div>

  <script>
    const dataSets = [
      {
        name: "Symmetric",
        counts: [1, 2, 4, 7, 9, 7, 4, 2, 1],
        curvePoints: [[0, 210], [55, 200], [110, 185], [165, 150], [220, 95], [250, 70], [280, 95], [335, 150], [390, 185], [445, 200], [500, 210]],
        badge1: { text: "Peak", type: "badge-peak", left: "50%", top: "15%" },
        badge2: { text: "", type: "", left: "0%", top: "0%" },
        note: "Even on both sides"
      },
      {
        name: "Skewed Right",
        counts: [9, 8, 6, 4, 2, 1, 1, 0, 0],
        curvePoints: [[0, 100], [45, 68], [100, 80], [155, 115], [210, 150], [265, 180], [320, 195], [375, 200], [430, 208], [500, 212]],
        badge1: { text: "Peak", type: "badge-peak", left: "18%", top: "16%" },
        badge2: { text: "Long Tail →", type: "badge-tail", left: "75%", top: "62%" },
        note: "Tail points right"
      },
      {
        name: "Skewed Left",
        counts: [0, 0, 1, 1, 2, 4, 6, 8, 9],
        curvePoints: [[0, 212], [70, 208], [125, 200], [180, 195], [235, 180], [290, 150], [345, 115], [400, 80], [455, 68], [500, 100]],
        badge1: { text: "Peak", type: "badge-peak", left: "82%", top: "16%" },
        badge2: { text: "← Long Tail", type: "badge-tail", left: "25%", top: "62%" },
        note: "Tail points left"
      },
      {
        name: "Outlier",
        counts: [1, 3, 6, 8, 6, 3, 1, 0, 1],
        curvePoints: [[0, 210], [55, 195], [110, 160], [165, 115], [200, 85], [235, 115], [290, 160], [345, 195], [400, 210], [500, 212]],
        badge1: { text: "Main Cluster", type: "badge-peak", left: "40%", top: "20%" },
        badge2: { text: "★ Outlier", type: "badge-outlier", left: "90%", top: "60%" },
        note: "Isolated value far away"
      }
    ];

    const grid = document.getElementById('grid');
    const labelsEl = document.getElementById('axisLabels');
    const curve = document.getElementById('curve');
    const badge1 = document.getElementById('badge1');
    const badge2 = document.getElementById('badge2');
    const footerText = document.getElementById('footerText');
    const buttons = document.querySelectorAll('.tab-btn');

    // Build 9 columns with up to 10 dots each
    for (let c = 0; c < 9; c++) {
      const col = document.createElement('div');
      col.className = 'dot-col';
      col.id = `col-${c}`;
      for (let r = 0; r < 10; r++) {
        const dot = document.createElement('div');
        dot.className = 'dot';
        dot.id = `dot-${c}-${r}`;
        col.appendChild(dot);
      }
      grid.appendChild(col);

      const lbl = document.createElement('div');
      lbl.className = 'axis-label';
      lbl.textContent = (c + 1) * 10;
      labelsEl.appendChild(lbl);
    }

    function createSmoothPath(points) {
      let d = `M ${points[0][0]} ${points[0][1]}`;
      for (let i = 0; i < points.length - 1; i++) {
        const p0 = points[i];
        const p1 = points[i + 1];
        const mx = (p0[0] + p1[0]) / 2;
        d += ` Q ${p0[0]} ${p0[1]}, ${mx} ${(p0[1] + p1[1]) / 2}`;
      }
      const last = points[points.length - 1];
      d += ` L ${last[0]} ${last[1]} L 500 215 L 0 215 Z`;
      return d;
    }

    function setShape(idx) {
      const data = dataSets[idx];

      buttons.forEach((b, i) => {
        b.classList.toggle('active', i === idx);
      });

      // Update dots
      for (let c = 0; c < 9; c++) {
        const targetCount = data.counts[c];
        const isOutlierCol = (idx === 3 && c === 8);
        for (let r = 0; r < 10; r++) {
          const dot = document.getElementById(`dot-${c}-${r}`);
          if (r < targetCount) {
            dot.classList.add('show');
            dot.classList.toggle('outlier-dot', isOutlierCol);
          } else {
            dot.classList.remove('show');
            dot.classList.remove('outlier-dot');
          }
        }
      }

      // Update curve
      curve.setAttribute('d', createSmoothPath(data.curvePoints));

      // Update badge 1
      if (data.badge1.text) {
        badge1.textContent = data.badge1.text;
        badge1.className = `indicator-badge ${data.badge1.type}`;
        badge1.style.left = data.badge1.left;
        badge1.style.top = data.badge1.top;
        badge1.style.transform = 'translate(-50%, -50%)';
        badge1.style.opacity = '1';
      } else {
        badge1.style.opacity = '0';
      }

      // Update badge 2
      if (data.badge2.text) {
        badge2.textContent = data.badge2.text;
        badge2.className = `indicator-badge ${data.badge2.type}`;
        badge2.style.left = data.badge2.left;
        badge2.style.top = data.badge2.top;
        badge2.style.transform = 'translate(-50%, -50%)';
        badge2.style.opacity = '1';
      } else {
        badge2.style.opacity = '0';
      }

      // Update footer
      footerText.textContent = data.note;
    }

    // Init first state
    setShape(0);
  </script>
</body>
</html>

A distribution is **skewed** in the direction of its long tail, not where its tall peak sits.

In a **symmetrical** distribution (such as bell-shaped or uniform), data is balanced evenly on both sides, meaning $\text{mean} \approx \text{median}$.

A **positively skewed** (skewed right) distribution has a long tail stretching to the right, which pulls the $\text{mean} > \text{median}$.

A **negatively skewed** (skewed left) distribution has a long tail stretching to the left, dragging the $\text{mean} < \text{median}$.

In a **symmetrical** distribution (such as bell-shaped or uniform data), both halves mirror each other, making $\text{mean} \approx \text{median}$.

A **positively skewed** (skewed right) distribution has a long tail to the right, pulling the average upward so that $\text{mean} > \text{median}$.

A **negatively skewed** (skewed left) distribution has a long tail to the left, dragging the average downward so that $\text{mean} < \text{median}$.

Why does this tail change which summary numbers we should trust?

The **mean** adds up every single value and divides by the total count, which means extreme values pull it strongly toward them.

The **median** is just the physical middle value when numbers are ordered from least to greatest, so extreme values do not move it.

<!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 {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    padding: 16px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
  }

  .main {
    width: 100%;
    max-width: 580px;
    display: flex;
    flex-direction: column;
    gap: 16px;
  }

  /* Preset buttons */
  .btn-group {
    display: flex;
    justify-content: center;
    gap: 8px;
  }
  .toggle-btn {
    padding: 8px 16px;
    font-size: 14px;
    font-weight: 600;
    border-radius: 20px;
    border: 1px solid #d1d5db;
    background: #ffffff;
    color: #4b5563;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .toggle-btn.active {
    background: #1e2945;
    color: #ffffff;
    border-color: #1e2945;
  }

  /* Plot stage */
  .plot-card {
    background: #ffffff;
    border-radius: 12px;
    padding: 24px 16px 16px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.06);
    position: relative;
  }

  /* Metric cards row */
  .metrics {
    display: grid;
    grid-template-columns: 1fr 1fr;
    gap: 12px;
    margin-bottom: 24px;
  }
  .metric-badge {
    border-radius: 8px;
    padding: 10px 12px;
    text-align: center;
  }
  .metric-badge.mean-badge {
    background: #e0f2fe;
    border: 1px solid #7dd3fc;
  }
  .metric-badge.median-badge {
    background: #ede9fe;
    border: 1px solid #c4b5fd;
  }
  .metric-title {
    font-size: 12px;
    font-weight: 700;
    text-transform: uppercase;
    letter-spacing: 0.5px;
    margin-bottom: 2px;
  }
  .mean-badge .metric-title { color: #0369a1; }
  .median-badge .metric-title { color: #6d28d9; }
  .metric-val {
    font-size: 22px;
    font-weight: 800;
    line-height: 1;
  }
  .mean-badge .metric-val { color: #0284c7; }
  .median-badge .metric-val { color: #7c3aed; }

  /* SVG container */
  .svg-wrap {
    width: 100%;
    position: relative;
  }
  svg {
    width: 100%;
    height: auto;
    display: block;
    overflow: visible;
  }

  /* Slider controls */
  .slider-box {
    display: flex;
    flex-direction: column;
    gap: 6px;
    background: #ffffff;
    border-radius: 12px;
    padding: 12px 16px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.06);
  }
  .slider-label {
    display: flex;
    justify-content: space-between;
    font-size: 13px;
    font-weight: 600;
    color: #4b5563;
  }
  input[type=range] {
    -webkit-appearance: none;
    width: 100%;
    height: 6px;
    border-radius: 3px;
    background: #e5e7eb;
    outline: none;
    cursor: pointer;
  }
  input[type=range]::-webkit-slider-thumb {
    -webkit-appearance: none;
    width: 20px;
    height: 20px;
    border-radius: 50%;
    background: #f97316;
    border: 2px solid #ffffff;
    box-shadow: 0 1px 4px rgba(0,0,0,0.25);
    cursor: pointer;
  }
  input[type=range]::-moz-range-thumb {
    width: 20px;
    height: 20px;
    border-radius: 50%;
    background: #f97316;
    border: 2px solid #ffffff;
    box-shadow: 0 1px 4px rgba(0,0,0,0.25);
    cursor: pointer;
  }
</style>
</head>
<body>

<div class="main">
  <div class="btn-group">
    <button class="toggle-btn active" id="btnSym" onclick="setPreset(7)">Symmetric</button>
    <button class="toggle-btn" id="btnOutlier" onclick="setPreset(19)">With Outlier</button>
  </div>

  <div class="plot-card">
    <div class="metrics">
      <div class="metric-badge median-badge">
        <div class="metric-title">Median 🔒</div>
        <div class="metric-val" id="valMedian">5.0</div>
      </div>
      <div class="metric-badge mean-badge">
        <div class="metric-title">Mean ↗</div>
        <div class="metric-val" id="valMean">5.0</div>
      </div>
    </div>

    <div class="svg-wrap">
      <svg viewBox="0 0 500 130">
        
        <line x1="25" y1="100" x2="475" y2="100" stroke="#cbd5e1" stroke-width="2" stroke-linecap="round"/>
        
        
        <g id="ticks" font-size="11" fill="#64748b" text-anchor="middle" font-family="sans-serif">
          
        </g>

        
        <g id="medianMarker">
          <line x1="0" y1="18" x2="0" y2="98" stroke="#7c3aed" stroke-width="2.5" stroke-dasharray="3,3"/>
          <circle cx="0" cy="18" r="4" fill="#7c3aed"/>
        </g>

        
        <g id="meanMarker">
          <line x1="0" y1="18" x2="0" y2="98" stroke="#0284c7" stroke-width="2.5"/>
          <polygon points="0,23 -5,14 5,14" fill="#0284c7"/>
        </g>

        
        <g id="dotsGroup"></g>
      </svg>
    </div>
  </div>

  <div class="slider-box">
    <div class="slider-label">
      <span>Drag outlier point:</span>
      <span id="dragVal" style="color: #f97316;">7</span>
    </div>
    <input type="range" id="outlierSlider" min="7" max="20" step="0.5" value="7" oninput="onSlide(this.value)">
  </div>
</div>

<script>
  const svgWidth = 500;
  const padLeft = 30;
  const padRight = 30;
  const plotW = svgWidth - padLeft - padRight;
  const maxVal = 20;

  function toX(val) {
    return padLeft + (val / maxVal) * plotW;
  }

  // Draw ticks
  const ticksG = document.getElementById('ticks');
  let ticksHTML = '';
  for (let i = 0; i <= maxVal; i += 2) {
    const x = toX(i);
    ticksHTML += `<line x1="${x}" y1="100" x2="${x}" y2="106" stroke="#94a3b8" stroke-width="1.5"/>`;
    ticksHTML += `<text x="${x}" y="120">${i}</text>`;
  }
  ticksG.innerHTML = ticksHTML;

  // Base data points: 3, 4, 5, 5, 5, 6
  const basePoints = [3, 4, 5, 5, 5, 6];

  function update(xVal) {
    const all = [...basePoints, xVal].sort((a, b) => a - b);
    const sum = all.reduce((acc, v) => acc + v, 0);
    const mean = sum / all.length;
    const median = all[3]; // 4th item out of 7

    document.getElementById('valMean').textContent = mean.toFixed(1);
    document.getElementById('valMedian').textContent = median.toFixed(1);
    document.getElementById('dragVal').textContent = xVal;

    // Update Indicators
    const medianX = toX(median);
    const meanX = toX(mean);

    const medianMarker = document.getElementById('medianMarker');
    medianMarker.setAttribute('transform', `translate(${medianX}, 0)`);

    const meanMarker = document.getElementById('meanMarker');
    meanMarker.setAttribute('transform', `translate(${meanX}, 0)`);

    // Render Data Dots (stack identical values)
    const counts = {};
    const dotsG = document.getElementById('dotsGroup');
    let dotsHTML = '';

    // Render base dots
    basePoints.forEach(pt => {
      counts[pt] = (counts[pt] || 0) + 1;
      const cy = 90 - (counts[pt] - 1) * 14;
      dotsHTML += `<circle cx="${toX(pt)}" cy="${cy}" r="6" fill="#22b7ff" stroke="#0284c7" stroke-width="1.5"/>`;
    });

    // Render outlier dot
    counts[xVal] = (counts[xVal] || 0) + 1;
    const cy = 90 - (counts[xVal] - 1) * 14;
    const isOutlier = xVal > 7;
    const color = isOutlier ? '#f97316' : '#22b7ff';
    const stroke = isOutlier ? '#ea580c' : '#0284c7';
    dotsHTML += `<circle cx="${toX(xVal)}" cy="${cy}" r="6.5" fill="${color}" stroke="${stroke}" stroke-width="2"/>`;

    dotsG.innerHTML = dotsHTML;

    // Toggle active buttons
    document.getElementById('btnSym').classList.toggle('active', xVal === 7);
    document.getElementById('btnOutlier').classList.toggle('active', xVal === 19);
  }

  function setPreset(val) {
    document.getElementById('outlierSlider').value = val;
    update(val);
  }

  function onSlide(val) {
    update(parseFloat(val));
  }

  // Initial call
  update(7);
</script>

</body>
</html>

For symmetric data without outliers, we use the **mean** and standard deviation; for skewed data or data with outliers, we use the **median** and the **interquartile range (IQR)**.

When does an unusual data point officially become an outlier?

In 1977, statistician John Tukey developed a standard boundary rule to detect extreme values called **outliers** without guessing.

An **outlier** is any data value that sits more than $1.5 \times \text{IQR}$ beyond the lower quartile ($Q_1$) or upper quartile ($Q_3$).

<!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>1.5 x IQR Outlier Rule</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }
    :root {
      --primary: #22b7ff;
      --text: #1e2945;
      --muted: #6b7280;
      --outlier: #f43f5e;
      --bg-fence: rgba(244, 63, 94, 0.08);
      --box-bg: rgba(34, 183, 255, 0.12);
    }
    body {
      background: transparent;
      color: var(--text);
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      gap: 12px;
    }
    .viz-container {
      width: 100%;
      max-width: 540px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .controls {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }
    .slider-row {
      width: 100%;
      display: flex;
      align-items: center;
      gap: 12px;
    }
    input[type="range"] {
      flex: 1;
      height: 8px;
      border-radius: 4px;
      background: #e2e8f0;
      outline: none;
      -webkit-appearance: none;
      cursor: pointer;
    }
    input[type="range"]::-webkit-slider-thumb {
      -webkit-appearance: none;
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: var(--primary);
      box-shadow: 0 2px 6px rgba(0,0,0,0.2);
      cursor: grab;
      transition: background 0.2s;
    }
    input[type="range"].is-outlier::-webkit-slider-thumb {
      background: var(--outlier);
    }
    .btn-group {
      display: flex;
      gap: 8px;
    }
    .btn {
      padding: 6px 14px;
      border-radius: 8px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: var(--text);
      font-size: 14px;
      font-weight: 500;
      cursor: pointer;
      transition: background 0.15s, border-color 0.15s;
    }
    .btn.active {
      background: #f1f5f9;
      border-color: var(--text);
    }
  </style>
</head>
<body>

  <div class="viz-container">
    <svg viewBox="0 0 500 200" id="plot">
      
      <rect x="397" y="20" width="83" height="140" fill="var(--bg-fence)" rx="6" />

      
      <line x1="397" y1="20" x2="397" y2="160" stroke="#f43f5e" stroke-width="2" stroke-dasharray="4 4" />
      <text x="397" y="16" fill="#f43f5e" font-size="12" font-weight="600" text-anchor="middle">Fence (85)</text>

      
      <line x1="30" y1="130" x2="470" y2="130" stroke="#94a3b8" stroke-width="2" stroke-linecap="round" />
      
      <g fill="#94a3b8" font-size="11" text-anchor="middle">
        <text x="72" y="148">10</text>
        <text x="156" y="148">30</text>
        <text x="240" y="148">50</text>
        <text x="324" y="148">70</text>
        <text x="408" y="148">90</text>
      </g>

      
      
      <line x1="114" y1="80" x2="177" y2="80" stroke="#1e2945" stroke-width="2" />
      <line x1="114" y1="72" x2="114" y2="88" stroke="#1e2945" stroke-width="2" />

      
      <rect x="177" y="60" width="84" height="40" fill="var(--box-bg)" stroke="#1e2945" stroke-width="2" rx="4" />
      
      <line x1="219" y1="60" x2="219" y2="100" stroke="#1e2945" stroke-width="2.5" />

      
      <line id="whiskerRight" x1="261" y1="80" x2="324" y2="80" stroke="#1e2945" stroke-width="2" />
      <line id="whiskerEnd" x1="324" y1="72" x2="324" y2="88" stroke="#1e2945" stroke-width="2" />

      
      <text x="177" y="116" fill="#1e2945" font-size="12" font-weight="600" text-anchor="middle">Q₁ (35)</text>
      <text x="261" y="116" fill="#1e2945" font-size="12" font-weight="600" text-anchor="middle">Q₃ (55)</text>

      
      
      <path d="M 177,52 L 177,46 L 261,46 L 261,52" fill="none" stroke="#22b7ff" stroke-width="1.5" />
      <text x="219" y="41" fill="#22b7ff" font-size="11" font-weight="600" text-anchor="middle">IQR = 20</text>

      
      <path d="M 261,52 L 261,46 L 397,46 L 397,52" fill="none" stroke="#f43f5e" stroke-width="1.5" />
      <text x="329" y="41" fill="#f43f5e" font-size="11" font-weight="600" text-anchor="middle">1.5 × IQR = 30</text>

      
      <g id="pointGroup" transform="translate(324, 80)">
        <circle id="pointHalo" r="14" fill="rgba(34, 183, 255, 0.2)" />
        <circle id="pointDot" r="7" fill="var(--primary)" stroke="#ffffff" stroke-width="2" />
        <text id="pointLabel" y="-14" fill="var(--primary)" font-size="13" font-weight="700" text-anchor="middle">70</text>
        <text id="outlierTag" y="24" fill="var(--outlier)" font-size="11" font-weight="700" text-anchor="middle" opacity="0">✦ OUTLIER</text>
      </g>
    </svg>
  </div>

  <div class="controls">
    <div class="slider-row">
      <input type="range" id="valSlider" min="55" max="98" value="70">
    </div>
    <div class="btn-group">
      <button class="btn active" id="btnNormal" onclick="setValue(70)">Normal (70)</button>
      <button class="btn" id="btnFence" onclick="setValue(85)">Fence (85)</button>
      <button class="btn" id="btnOutlier" onclick="setValue(95)">Outlier (95)</button>
    </div>
  </div>

  <script>
    const slider = document.getElementById('valSlider');
    const pointGroup = document.getElementById('pointGroup');
    const pointDot = document.getElementById('pointDot');
    const pointHalo = document.getElementById('pointHalo');
    const pointLabel = document.getElementById('pointLabel');
    const outlierTag = document.getElementById('outlierTag');
    const whiskerRight = document.getElementById('whiskerRight');
    const whiskerEnd = document.getElementById('whiskerEnd');

    // 0 = 30px, 100 = 450px  => x = 30 + val * 4.2
    function toSvgX(val) {
      return 30 + val * 4.2;
    }

    function update(val) {
      const x = toSvgX(val);
      const isOutlier = val > 85;

      pointGroup.setAttribute('transform', `translate(${x}, 80)`);
      pointLabel.textContent = val;

      if (isOutlier) {
        slider.classList.add('is-outlier');
        pointDot.setAttribute('fill', 'var(--outlier)');
        pointHalo.setAttribute('fill', 'rgba(244, 63, 94, 0.25)');
        pointLabel.setAttribute('fill', 'var(--outlier)');
        outlierTag.setAttribute('opacity', '1');
        
        // Whisker stops at non-outlier max (70)
        whiskerRight.setAttribute('x2', toSvgX(70));
        whiskerEnd.setAttribute('x1', toSvgX(70));
        whiskerEnd.setAttribute('x2', toSvgX(70));
        whiskerEnd.setAttribute('opacity', '1');
      } else {
        slider.classList.remove('is-outlier');
        pointDot.setAttribute('fill', 'var(--primary)');
        pointHalo.setAttribute('fill', 'rgba(34, 183, 255, 0.2)');
        pointLabel.setAttribute('fill', 'var(--primary)');
        outlierTag.setAttribute('opacity', '0');

        // Whisker connects to point
        whiskerRight.setAttribute('x2', x);
        whiskerEnd.setAttribute('x1', x);
        whiskerEnd.setAttribute('x2', x);
        whiskerEnd.setAttribute('opacity', '1');
      }

      // Update button styles
      document.querySelectorAll('.btn').forEach(b => b.classList.remove('active'));
      if (val === 70) document.getElementById('btnNormal').classList.add('active');
      else if (val === 85) document.getElementById('btnFence').classList.add('active');
      else if (val === 95) document.getElementById('btnOutlier').classList.add('active');
    }

    function setValue(v) {
      slider.value = v;
      update(v);
    }

    slider.addEventListener('input', (e) => {
      update(parseInt(e.target.value, 10));
    });

    update(70);
  </script>
</body>
</html>

Let's see how all four features—Center, Spread, Shape, and Outliers—work together in a real comparison.

## Worked Example: Comparing Class Scores

Two math classes took the same test out of 100 points, and their results are displayed on parallel box plots below.

<!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 {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      min-height: 100%;
      user-select: none;
    }
    .toolbar {
      display: flex;
      gap: 8px;
      margin-bottom: 8px;
    }
    .btn {
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 20px;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      color: #64748b;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .btn:hover {
      background: #f1f5f9;
      color: #1e2945;
    }
    .btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
    }
    .svg-container {
      width: 100%;
      max-width: 540px;
      flex-grow: 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: auto;
      overflow: visible;
    }
    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
    }
    .axis-line {
      stroke: #cbd5e1;
      stroke-width: 1.5;
    }
    .tick-label {
      font-size: 11px;
      fill: #64748b;
      font-weight: 500;
      text-anchor: middle;
    }
    .row-label {
      font-size: 13px;
      font-weight: 700;
      fill: #1e2945;
    }
    .whisker {
      stroke: #1e2945;
      stroke-width: 2;
    }
    .box-a {
      fill: #e0f2fe;
      stroke: #0284c7;
      stroke-width: 2;
      rx: 4;
    }
    .box-b {
      fill: #f3e8ff;
      stroke: #9333ea;
      stroke-width: 2;
      rx: 4;
    }
    .median-a {
      stroke: #0369a1;
      stroke-width: 3;
    }
    .median-b {
      stroke: #7e22ce;
      stroke-width: 3;
    }
    .outlier-dot {
      fill: #ef4444;
      stroke: #ffffff;
      stroke-width: 2;
      cursor: pointer;
      transition: r 0.2s ease;
    }
    .fence-line {
      stroke: #ef4444;
      stroke-dasharray: 4 4;
      stroke-width: 2;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .fence-zone {
      fill: rgba(239, 68, 68, 0.08);
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .iqr-bracket {
      stroke: #9333ea;
      stroke-width: 1.5;
      fill: none;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .callout-text {
      font-size: 11px;
      font-weight: 700;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .visible {
      opacity: 1 !important;
    }
    .pulse {
      animation: pulse-ring 1.8s cubic-bezier(0.215, 0.61, 0.355, 1) infinite;
    }
    @keyframes pulse-ring {
      0% { r: 6px; stroke-width: 2px; }
      50% { r: 10px; stroke-width: 4px; }
      100% { r: 6px; stroke-width: 2px; }
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="btn active" id="btn-all" onclick="setMode('all')">Compare</button>
    <button class="btn" id="btn-fence" onclick="setMode('fence')">IQR Fence</button>
    <button class="btn" id="btn-outlier" onclick="setMode('outlier')">Outlier</button>
  </div>

  <div class="svg-container">
    <svg viewBox="0 0 460 210">
      
      
      <line x1="50" y1="20" x2="50" y2="165" class="grid-line" />
      <line x1="126" y1="20" x2="126" y2="165" class="grid-line" />
      <line x1="202" y1="20" x2="202" y2="165" class="grid-line" />
      <line x1="278" y1="20" x2="278" y2="165" class="grid-line" />
      <line x1="354" y1="20" x2="354" y2="165" class="grid-line" />
      <line x1="430" y1="20" x2="430" y2="165" class="grid-line" />

      
      <rect id="fence-zone" x="50" y="88" width="123.5" height="66" class="fence-zone" rx="4" />

      
      <text x="50" y="36" class="row-label">Class A</text>
      
      <line x1="240" y1="52" x2="297" y2="52" class="whisker" />
      <line x1="240" y1="44" x2="240" y2="60" class="whisker" />
      <line x1="373" y1="52" x2="411" y2="52" class="whisker" />
      <line x1="411" y1="44" x2="411" y2="60" class="whisker" />
      
      <rect x="297" y="38" width="76" height="28" class="box-a" />
      
      <line x1="335" y1="38" x2="335" y2="66" class="median-a" />

      
      <text x="50" y="104" class="row-label">Class B</text>
      
      <line x1="259" y1="120" x2="316" y2="120" class="whisker" />
      <line x1="259" y1="112" x2="259" y2="128" class="whisker" />
      <line x1="411" y1="120" x2="430" y2="120" class="whisker" />
      <line x1="430" y1="112" x2="430" y2="128" class="whisker" />
      
      <rect x="316" y="106" width="95" height="28" class="box-b" />
      
      <line x1="373" y1="106" x2="373" y2="134" class="median-b" />

      
      <line id="fence-line" x1="173.5" y1="88" x2="173.5" y2="155" class="fence-line" />
      <text id="fence-text" x="173.5" y="82" fill="#ef4444" text-anchor="middle" class="callout-text">Fence (32.5)</text>

      
      <path id="iqr-bracket" d="M 316 142 L 316 148 L 411 148 L 411 142" class="iqr-bracket" />
      <text id="iqr-text" x="363.5" y="159" fill="#9333ea" text-anchor="middle" class="callout-text">IQR = 25</text>

      
      <circle id="outlier-dot" cx="126" cy="120" r="6" class="outlier-dot" onclick="setMode('outlier')" />
      <text id="outlier-text" x="126" y="104" fill="#ef4444" text-anchor="middle" class="callout-text">20 (Outlier)</text>

      
      <line x1="50" y1="170" x2="430" y2="170" class="axis-line" />
      
      <line x1="50" y1="170" x2="50" y2="176" class="axis-line" />
      <text x="50" y="190" class="tick-label">0</text>

      <line x1="126" y1="170" x2="126" y2="176" class="axis-line" />
      <text x="126" y="190" class="tick-label">20</text>

      <line x1="202" y1="170" x2="202" y2="176" class="axis-line" />
      <text x="202" y="190" class="tick-label">40</text>

      <line x1="278" y1="170" x2="278" y2="176" class="axis-line" />
      <text x="278" y="190" class="tick-label">60</text>

      <line x1="354" y1="170" x2="354" y2="176" class="axis-line" />
      <text x="354" y="190" class="tick-label">80</text>

      <line x1="430" y1="170" x2="430" y2="176" class="axis-line" />
      <text x="430" y="190" class="tick-label">100</text>
    </svg>
  </div>

  <script>
    function setMode(mode) {
      const btnAll = document.getElementById('btn-all');
      const btnFence = document.getElementById('btn-fence');
      const btnOutlier = document.getElementById('btn-outlier');

      const fenceZone = document.getElementById('fence-zone');
      const fenceLine = document.getElementById('fence-line');
      const fenceText = document.getElementById('fence-text');
      const iqrBracket = document.getElementById('iqr-bracket');
      const iqrText = document.getElementById('iqr-text');
      const outlierDot = document.getElementById('outlier-dot');
      const outlierText = document.getElementById('outlier-text');

      [btnAll, btnFence, btnOutlier].forEach(b => b.classList.remove('active'));
      [fenceZone, fenceLine, fenceText, iqrBracket, iqrText, outlierText].forEach(el => el.classList.remove('visible'));
      outlierDot.classList.remove('pulse');

      if (mode === 'all') {
        btnAll.classList.add('active');
      } else if (mode === 'fence') {
        btnFence.classList.add('active');
        fenceZone.classList.add('visible');
        fenceLine.classList.add('visible');
        fenceText.classList.add('visible');
        iqrBracket.classList.add('visible');
        iqrText.classList.add('visible');
      } else if (mode === 'outlier') {
        btnOutlier.classList.add('active');
        fenceZone.classList.add('visible');
        fenceLine.classList.add('visible');
        fenceText.classList.add('visible');
        outlierText.classList.add('visible');
        outlierDot.classList.add('pulse');
      }
    }
  </script>
</body>
</html>

For Class B, the IQR is $95 - 70 = 25$, so the lower boundary is $70 - 1.5(25) = 32.5$, making the score of $20$ a confirmed **outlier**.

When comparing them, Class B had a higher typical performance (median of $85$ vs. $75$), but also slightly more spread in the middle 50% (IQR of $25$ vs. $20$) and one extreme low score.

A common mistake is claiming that Class A performed better overall just because Class B's calculated mean was dragged down by that single score of $20$.

Whenever a distribution is skewed or has outliers, always use the **median** and **IQR** to make fair, accurate comparisons.

math

Comparing Data Sets with Different Shapes and Outliers. Compare data sets that have different distribution shapes and identify the effect of outliers.

Comparing Distributions Using Shape, Center, Spread, and Outliers

Comparing Box Plots and Interpreting Data Distributions

Compare Populations Using Measures of Centre and Spread

To achieve mastery under AC9M9ST03, guide students through the structured C-U-S-S protocol (Centre, Unusual points/Outliers, Spread, Shape) whenever comparing distributions. Emphasize why specific summary measures are chosen based on data topology: explicitly pairing Mean with Standard Deviation/Range for symmetric data, and Median with IQR for skewed data or data containing outliers. Always encourage students to interpret comparisons in real-world contexts rather than merely listing numerical values.

Comparing Data Sets with Different Shapes and Outliers

Distribution Shapes