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Tutoring abilities

Imagine five friends finish a basketball game with scores of 4, 4, 7, 9, and 26 points. If you had to pick just **one number** to describe how well the team played, which number would you choose?

Statisticians use three different tools called **measures of center** — the mean, median, and mode — to summarize an entire group of numbers with a single typical value.

The **mean** is the "fair share" value. It represents what every data point would equal if you combined everything together and split it up evenly.

<!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; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif; }
    body { background: #ffffff; color: #1e2945; display: flex; flex-direction: column; align-items: center; padding: 12px; min-height: 100%; }

    /* Controls */
    .controls { display: flex; gap: 8px; margin-bottom: 12px; width: 100%; max-width: 440px; justify-content: center; }
    .tab-btn {
      flex: 1; padding: 8px 12px; font-size: 13px; font-weight: 700; border-radius: 8px; border: 1.5px solid #e6e6e6;
      background: #f8fafc; color: #64748b; cursor: pointer; transition: all 0.2s ease; text-align: center;
    }
    .tab-btn.active {
      background: #22b7ff; color: #ffffff; border-color: #22b7ff; box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
    }

    /* Stage / Chart Area */
    .stage {
      position: relative; width: 100%; max-width: 440px; height: 210px;
      background: #f8fafc; border: 1px solid #e6e6e6; border-radius: 12px;
      padding: 10px 12px; margin-bottom: 12px; overflow: hidden;
    }

    /* Mean Target Line */
    .mean-line {
      position: absolute; left: 0; right: 0; bottom: 122px;
      border-top: 2px dashed #0284c7; z-index: 5; pointer-events: none;
      display: flex; justify-content: flex-end; padding-right: 8px;
    }
    .mean-tag {
      background: #e0f2fe; color: #0284c7; font-size: 11px; font-weight: 800;
      padding: 1px 6px; border-radius: 4px; transform: translateY(-50%); border: 1px solid #bae6fd;
    }

    /* Column Headers & Footers */
    .columns-grid {
      position: absolute; inset: 0; display: grid; grid-template-columns: repeat(4, 1fr);
      padding: 8px 12px 28px 12px; pointer-events: none;
    }
    .col-track {
      display: flex; flex-direction: column; justify-content: space-between; align-items: center; height: 100%;
    }
    .count-badge {
      font-size: 14px; font-weight: 800; color: #1e2945; transition: color 0.3s;
    }
    .col-label {
      position: absolute; bottom: 6px; font-size: 12px; font-weight: 600; color: #64748b;
    }

    /* Blocks Layer */
    .blocks-container {
      position: absolute; inset: 0; padding: 8px 12px 28px 12px; pointer-events: none;
    }
    .block {
      position: absolute; height: 18px; border-radius: 4px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.12), inset 0 1px 0 rgba(255,255,255,0.4);
      transition: left 0.65s cubic-bezier(0.4, 0, 0.2, 1), bottom 0.65s cubic-bezier(0.4, 0, 0.2, 1);
    }

    /* Vibrant Block Colors by Original Group */
    .c-0 { background: #3b82f6; } /* Blue */
    .c-1 { background: #f59e0b; } /* Amber */
    .c-2 { background: #10b981; } /* Emerald */
    .c-3 { background: #8b5cf6; } /* Purple */

    /* Calculation Summary Card */
    .calc-card {
      width: 100%; max-width: 440px; background: #ffffff; border: 1.5px solid #e6e6e6;
      border-radius: 10px; padding: 10px 14px; display: flex; flex-direction: column; gap: 6px;
    }
    .calc-row {
      display: flex; justify-content: space-between; align-items: center; font-size: 13px; font-weight: 600; color: #475569;
    }
    .calc-row.result {
      font-size: 15px; font-weight: 800; color: #1e2945; border-top: 1px solid #f1f5f9; padding-top: 6px;
    }
    .highlight {
      color: #0284c7; font-weight: 800;
    }
  </style>
</head>
<body>

  
  <div class="controls">
    <button id="btnUneven" class="tab-btn active" onclick="setState('uneven')">1. Uneven Stacks</button>
    <button id="btnFair" class="tab-btn" onclick="setState('fair')">2. Fair Share (Mean)</button>
  </div>

  
  <div class="stage" id="stage">
    <div class="mean-line">
      <span class="mean-tag">Mean = 5</span>
    </div>

    <div class="columns-grid">
      <div class="col-track"><span id="cnt-0" class="count-badge">2</span><span class="col-label" style="left:calc(12.5% - 10px)">#1</span></div>
      <div class="col-track"><span id="cnt-1" class="count-badge">8</span><span class="col-label" style="left:calc(37.5% - 10px)">#2</span></div>
      <div class="col-track"><span id="cnt-2" class="count-badge">3</span><span class="col-label" style="left:calc(62.5% - 10px)">#3</span></div>
      <div class="col-track"><span id="cnt-3" class="count-badge">7</span><span class="col-label" style="left:calc(87.5% - 10px)">#4</span></div>
    </div>

    <div class="blocks-container" id="blocksBox"></div>
  </div>

  
  <div class="calc-card">
    <div class="calc-row">
      <span>Total Sum</span>
      <span>2 + 8 + 3 + 7 = <b>20 blocks</b></span>
    </div>
    <div class="calc-row">
      <span>Total Stacks</span>
      <span><b>4 stacks</b></span>
    </div>
    <div class="calc-row result">
      <span>Fair Share (Mean)</span>
      <span class="highlight">20 ÷ 4 = 5 blocks each</span>
    </div>
  </div>

  <script>
    // 20 blocks with uneven and fair-share targets
    // col 0..3, row 0..7
    const blocksData = [
      // Stack 0 (2 blocks, stays)
      { color: 'c-0', uCol: 0, uRow: 0, fCol: 0, fRow: 0 },
      { color: 'c-0', uCol: 0, uRow: 1, fCol: 0, fRow: 1 },

      // Stack 1 (8 blocks: 5 stay, 3 move to Stack 0)
      { color: 'c-1', uCol: 1, uRow: 0, fCol: 1, fRow: 0 },
      { color: 'c-1', uCol: 1, uRow: 1, fCol: 1, fRow: 1 },
      { color: 'c-1', uCol: 1, uRow: 2, fCol: 1, fRow: 2 },
      { color: 'c-1', uCol: 1, uRow: 3, fCol: 1, fRow: 3 },
      { color: 'c-1', uCol: 1, uRow: 4, fCol: 1, fRow: 4 },
      { color: 'c-1', uCol: 1, uRow: 5, fCol: 0, fRow: 2 }, // Moves to Stack 0
      { color: 'c-1', uCol: 1, uRow: 6, fCol: 0, fRow: 3 }, // Moves to Stack 0
      { color: 'c-1', uCol: 1, uRow: 7, fCol: 0, fRow: 4 }, // Moves to Stack 0

      // Stack 2 (3 blocks, stays)
      { color: 'c-2', uCol: 2, uRow: 0, fCol: 2, fRow: 0 },
      { color: 'c-2', uCol: 2, uRow: 1, fCol: 2, fRow: 1 },
      { color: 'c-2', uCol: 2, uRow: 2, fCol: 2, fRow: 2 },

      // Stack 3 (7 blocks: 5 stay, 2 move to Stack 2)
      { color: 'c-3', uCol: 3, uRow: 0, fCol: 3, fRow: 0 },
      { color: 'c-3', uCol: 3, uRow: 1, fCol: 3, fRow: 1 },
      { color: 'c-3', uCol: 3, uRow: 2, fCol: 3, fRow: 2 },
      { color: 'c-3', uCol: 3, uRow: 3, fCol: 3, fRow: 3 },
      { color: 'c-3', uCol: 3, uRow: 4, fCol: 3, fRow: 4 },
      { color: 'c-3', uCol: 3, uRow: 5, fCol: 2, fRow: 3 }, // Moves to Stack 2
      { color: 'c-3', uCol: 3, uRow: 6, fCol: 2, fRow: 4 }  // Moves to Stack 2
    ];

    const box = document.getElementById('blocksBox');
    const blockElements = [];

    // Create DOM blocks
    blocksData.forEach((b, i) => {
      const el = document.createElement('div');
      el.className = `block ${b.color}`;
      box.appendChild(el);
      blockElements.push(el);
    });

    let currentState = 'uneven';

    function updatePositions() {
      const isFair = currentState === 'fair';

      // Update count badges
      document.getElementById('cnt-0').textContent = isFair ? '5' : '2';
      document.getElementById('cnt-1').textContent = isFair ? '5' : '8';
      document.getElementById('cnt-2').textContent = isFair ? '5' : '3';
      document.getElementById('cnt-3').textContent = isFair ? '5' : '7';

      // Update buttons
      document.getElementById('btnUneven').classList.toggle('active', !isFair);
      document.getElementById('btnFair').classList.toggle('active', isFair);

      blockElements.forEach((el, i) => {
        const d = blocksData[i];
        const col = isFair ? d.fCol : d.uCol;
        const row = isFair ? d.fRow : d.uRow;

        // Calculate CSS placement
        // 4 cols: width ~20% with spacing
        const leftPercent = 3 + col * 24.5;
        const bottomPx = 28 + row * 22;

        el.style.left = `${leftPercent}%`;
        el.style.width = '20%';
        el.style.bottom = `${bottomPx}px`;
      });
    }

    function setState(st) {
      currentState = st;
      updatePositions();
    }

    // Auto-cycle gently every 4 seconds
    let autoTimer = setInterval(() => {
      setState(currentState === 'uneven' ? 'fair' : 'uneven');
    }, 4000);

    // Pause auto-cycle on user interaction
    document.querySelectorAll('.tab-btn').forEach(btn => {
      btn.addEventListener('click', () => {
        clearInterval(autoTimer);
      });
    });

    // Initial render
    updatePositions();
  </script>
</body>
</html>

To calculate the mean, add all the values together and divide that sum by the total number of values.

For the scores $4, 4, 7, 9, 26$, add them to get $50$, then divide by $5$ to find a mean of $10$ points.

Notice how that one huge score of $26$ pulled the mean way up, even though four of the five players scored under $10$. What tool can we use when an extreme number skews the average?

The **median** is the exact middle number when all values are arranged in order from least to greatest.

<!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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .main {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* Panel Styles */
    .panel-section {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
    }

    .panel-title {
      font-size: 13px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }

    .cards-row {
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 8px;
      width: 100%;
    }

    .num-card {
      position: relative;
      width: 42px;
      height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      background: #f8fafc;
      border: 1.5px solid #cbd5e1;
      border-radius: 8px;
      font-size: 17px;
      font-weight: 700;
      color: #1e2945;
      transition: all 0.3s ease;
    }

    /* Cross-out line */
    .num-card::after {
      content: '';
      position: absolute;
      width: 0%;
      height: 2.5px;
      background: #94a3b8;
      border-radius: 2px;
      transform: rotate(-35deg);
      transition: width 0.3s ease;
    }

    .num-card.crossed {
      opacity: 0.35;
      background: #f1f5f9;
      border-color: #cbd5e1;
      color: #94a3b8;
    }

    .num-card.crossed::after {
      width: 110%;
    }

    .num-card.highlight {
      background: #fef08a;
      border-color: #eab308;
      color: #854d0e;
      box-shadow: 0 0 0 3px rgba(234, 179, 8, 0.25);
    }

    .result-row {
      height: 28px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }

    .calc-text {
      font-size: 14px;
      font-weight: 600;
      color: #1e2945;
      opacity: 0;
      transform: translateY(4px);
      transition: all 0.3s ease;
    }

    .calc-text.visible {
      opacity: 1;
      transform: translateY(0);
    }

    .badge {
      display: inline-flex;
      align-items: center;
      padding: 3px 10px;
      border-radius: 999px;
      font-size: 13px;
      font-weight: 700;
      background: #fef08a;
      border: 1px solid #facc15;
      color: #854d0e;
      opacity: 0;
      transform: scale(0.9);
      transition: all 0.3s ease;
    }

    .badge.visible {
      opacity: 1;
      transform: scale(1);
    }

    .divider {
      width: 100%;
      height: 1px;
      background: #e2e8f0;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-top: 10px;
      width: 100%;
      max-width: 480px;
    }

    .play-btn {
      width: 32px;
      height: 32px;
      display: flex;
      align-items: center;
      justify-content: center;
      background: #f8fafc;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      color: #1e2945;
      font-size: 13px;
      cursor: pointer;
      user-select: none;
    }

    .play-btn:active {
      background: #e2e8f0;
    }

    .step-btn {
      padding: 6px 12px;
      border-radius: 6px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #64748b;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      user-select: none;
      transition: all 0.2s ease;
    }

    .step-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
    }
  </style>
</head>
<body>

  <div class="main">
    
    <div class="panel-section">
      <div class="panel-title">Odd Set (5 numbers)</div>
      <div class="cards-row">
        <div class="num-card" id="o0">4</div>
        <div class="num-card" id="o1">4</div>
        <div class="num-card" id="o2">7</div>
        <div class="num-card" id="o3">9</div>
        <div class="num-card" id="o4">26</div>
      </div>
      <div class="result-row">
        <div class="badge" id="odd-badge">Median = 7</div>
      </div>
    </div>

    <div class="divider"></div>

    
    <div class="panel-section">
      <div class="panel-title">Even Set (6 numbers)</div>
      <div class="cards-row">
        <div class="num-card" id="e0">2</div>
        <div class="num-card" id="e1">4</div>
        <div class="num-card" id="e2">6</div>
        <div class="num-card" id="e3">8</div>
        <div class="num-card" id="e4">10</div>
        <div class="num-card" id="e5">12</div>
      </div>
      <div class="result-row">
        <div class="calc-text" id="even-calc">(6 + 8) ÷ 2 =</div>
        <div class="badge" id="even-badge">Median = 7</div>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <button class="play-btn" id="playToggle" aria-label="Play/Pause">⏸</button>
    <button class="step-btn active" id="btn-0" onclick="goToStep(0)">1. Start</button>
    <button class="step-btn" id="btn-1" onclick="goToStep(1)">2. Outer</button>
    <button class="step-btn" id="btn-2" onclick="goToStep(2)">3. Inner</button>
    <button class="step-btn" id="btn-3" onclick="goToStep(3)">4. Median</button>
  </div>

  <script>
    let currentStep = 0;
    let isPlaying = true;
    let timer = null;

    function renderStep(step) {
      currentStep = step;

      // Update button classes
      for (let i = 0; i < 4; i++) {
        const btn = document.getElementById('btn-' + i);
        if (btn) btn.classList.toggle('active', i === step);
      }

      // Card element references
      const o0 = document.getElementById('o0');
      const o1 = document.getElementById('o1');
      const o2 = document.getElementById('o2');
      const o3 = document.getElementById('o3');
      const o4 = document.getElementById('o4');
      const oddBadge = document.getElementById('odd-badge');

      const e0 = document.getElementById('e0');
      const e1 = document.getElementById('e1');
      const e2 = document.getElementById('e2');
      const e3 = document.getElementById('e3');
      const e4 = document.getElementById('e4');
      const e5 = document.getElementById('e5');
      const evenCalc = document.getElementById('even-calc');
      const evenBadge = document.getElementById('even-badge');

      // Reset all classes
      [o0, o1, o2, o3, o4, e0, e1, e2, e3, e4, e5].forEach(el => {
        el.className = 'num-card';
      });
      oddBadge.className = 'badge';
      evenCalc.className = 'calc-text';
      evenBadge.className = 'badge';

      if (step >= 1) {
        // First outer pair crossed
        o0.classList.add('crossed');
        o4.classList.add('crossed');
        e0.classList.add('crossed');
        e5.classList.add('crossed');
      }

      if (step >= 2) {
        // Second outer pair crossed
        o1.classList.add('crossed');
        o3.classList.add('crossed');
        e1.classList.add('crossed');
        e4.classList.add('crossed');
      }

      if (step >= 3) {
        // Medians revealed
        o2.classList.add('highlight');
        oddBadge.classList.add('visible');

        e2.classList.add('highlight');
        e3.classList.add('highlight');
        evenCalc.classList.add('visible');
        evenBadge.classList.add('visible');
      }
    }

    function goToStep(step) {
      renderStep(step);
    }

    function nextStep() {
      const next = (currentStep + 1) % 4;
      renderStep(next);
    }

    function startTimer() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3200);
    }

    function togglePlay() {
      isPlaying = !isPlaying;
      const playBtn = document.getElementById('playToggle');
      if (isPlaying) {
        playBtn.textContent = '⏸';
        startTimer();
      } else {
        playBtn.textContent = '▶';
        clearInterval(timer);
      }
    }

    document.getElementById('playToggle').addEventListener('click', togglePlay);

    // Initialize
    renderStep(0);
    startTimer();
  </script>
</body>
</html>

If a data set has an **even number** of values, there is no single middle number, so you find the mean of the two middle numbers.

For the ordered list $3, 5, 8, 12$, the two middle numbers are $5$ and $8$, giving a median of $(5 + 8) \div 2 = 6.5$.

What if you own an ice cream shop and need to decide which flavor to restock first? Neither the middle flavor nor the fair-share flavor makes sense here!

The **mode** is the value that appears most frequently in a data set.

<!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;
      -webkit-tap-highlight-color: transparent;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      padding: 10px 4px;
      user-select: none;
    }

    .grid {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
      width: 100%;
      max-width: 680px;
      margin: 0 auto;
    }

    .column {
      display: flex;
      flex-direction: column;
      align-items: center;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 6px;
      cursor: pointer;
      transition: background-color 0.2s ease, box-shadow 0.2s ease;
    }

    .column:hover, .column.active {
      background: #ffffff;
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.06);
    }

    /* Column Header */
    .title {
      font-size: 13px;
      font-weight: 700;
      color: #475569;
      margin-bottom: 4px;
      text-align: center;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      padding: 2px 8px;
      border-radius: 12px;
      margin-bottom: 10px;
      display: inline-block;
    }

    .badge-blue { background: #e0e7ff; color: #4338ca; }
    .badge-amber { background: #fef3c7; color: #b45309; }
    .badge-slate { background: #f1f5f9; color: #64748b; }

    /* Number ribbon */
    .token-row {
      display: flex;
      gap: 3px;
      flex-wrap: wrap;
      justify-content: center;
      margin-bottom: 12px;
      min-height: 24px;
      align-items: center;
    }

    .token {
      width: 20px;
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      <div class="title">1 Mode</div>
      <div class="badge badge-blue">Mode: 5</div>
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        <span class="token dim">3</span>
        <span class="token peak-blue">5</span>
        <span class="token peak-blue">5</span>
        <span class="token peak-blue">5</span>
        <span class="token dim">8</span>
        <span class="token dim">9</span>
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          <span class="x-label">3</span>
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          <span class="tag tag-blue">Peak (3×)</span>
          <div class="stack">
            <div class="block peak-blue"></div>
            <div class="block peak-blue"></div>
            <div class="block peak-blue"></div>
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          <span class="x-label">8</span>
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          <span class="x-label">9</span>
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      <div class="badge badge-amber">Modes: 2 & 7</div>
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        <span class="token peak-amber">2</span>
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        <span class="token peak-amber">7</span>
        <span class="token peak-amber">7</span>
        <span class="token dim">9</span>
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            <div class="block peak-amber"></div>
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            <div class="block peak-amber"></div>
            <div class="block peak-amber"></div>
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          <span class="x-label highlight">7</span>
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        <div class="col-group">
          <div class="stack"><div class="block"></div></div>
          <span class="x-label">9</span>
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      </div>
      <div class="caption">Two equal peaks</div>
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    <div class="column" onclick="activate(this, 400)">
      <div class="title">No Mode</div>
      <div class="badge badge-slate">No Mode</div>
      <div class="token-row">
        <span class="token dim">1</span>
        <span class="token dim">4</span>
        <span class="token dim">6</span>
        <span class="token dim">8</span>
        <span class="token dim">11</span>
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      <div class="plot-area">
        <span class="tag tag-slate" style="top: 38px;">All 1×</span>
        <div class="col-group">
          <div class="stack"><div class="block"></div></div>
          <span class="x-label">1</span>
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        <div class="col-group">
          <div class="stack"><div class="block"></div></div>
          <span class="x-label">4</span>
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        <div class="col-group">
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          <span class="x-label">6</span>
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        <div class="col-group">
          <div class="stack"><div class="block"></div></div>
          <span class="x-label">8</span>
        </div>
        <div class="col-group">
          <div class="stack"><div class="block"></div></div>
          <span class="x-label">11</span>
        </div>
      </div>
      <div class="caption">All counts equal</div>
    </div>
  </div>

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A data set can have **one mode**, **more than one mode**, or **no mode** at all if every value appears the same number of times.

A common mistake when finding the median is picking the middle number before putting the list in order. Always sort your data from least to greatest first!

Another common error is writing "$0$" when a set has no mode. Write "no mode" instead, because $0$ is an actual number that could be a mode itself!

math

Finding Mean, Median, and Mode. Calculate the mean, median, and mode of a data set and explain what each measure represents

Statistics Intro: Mean, Median, and Mode

Calculate Mean, Median, Mode, and Range (6th Grade Math Skill)

How to Find the Mean, Median, Mode and Range

To achieve deep conceptual mastery aligned with TEKS 6th grade standards, introduce the mean concretely using physical linking cubes ('fair share' leveling) before teaching the algebraic summation algorithm. Reinforce the median by having students physically line up cards in numerical order and fold them in half or cross out extremes to eliminate the persistent error of taking the middle of unordered data. Contrast all three measures using contextual real-world scenarios to explain why each measure answers a distinct question about the data.

Finding Mean, Median, and Mode

The Mean

The Median