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If you ask five of your friends whether school starts too early, you will probably get a unanimous "yes." But does that actually prove that the entire student body agrees?

To answer real-world questions with real evidence, you cannot rely on casual chats—you need a structured statistical investigation.

How do professional researchers take a simple question and turn it into reliable science?

**The Statistical Enquiry Cycle** (the **PPDAC framework**) guides an investigation through five stages: **Problem** (formulating the question), **Plan** (sampling and methodology), **Data** (collection and management), **Analysis** (graphs and statistics), and **Conclusion** (answering the question).

In 1999, statisticians Chris Wild and Maxine Pfannkuch developed this cycle to show that statistics is an ongoing investigation, not just calculating numbers.

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Everything starts with the first two stages: formulating your problem and building your collection plan.

What makes an investigation question clear enough to actually test?

## Formulating Statistical Investigative Questions

An **investigative question** is a question that addresses a whole group and expects variability in the data, rather than asking for a single factual answer.

For example, asking **'Do Year 9 students at our school spend more time on screens on weekends than on weekdays?'** is measurable and anticipates variability, whereas 'Is screen time bad?' is subjective.

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There are three main question types: **summary questions** (describing one group), **comparison questions** (comparing two groups), and **relationship questions** (checking if two variables change together).

Once your question specifies what to study, what kinds of information will you need to record?

A **variable** is any characteristic, number, or quantity that can vary between individuals in a dataset.

We organize variables into two main families: **categorical variables**, which describe qualitative labels or categories, and **numerical variables**, which represent countable or measurable numbers.

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    border: 2px solid #cbd5e1;
    border-radius: 12px;
    padding: 8px 24px;
    font-weight: 700;
    font-size: 16px;
    color: #0f172a;
    box-shadow: 0 2px 4px rgba(0,0,0,0.04);
  }

  .connector-svg {
    width: 100%;
    height: 24px;
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    grid-template-columns: 1fr 1fr;
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    border-radius: 12px;
    border: 2px solid #e2e8f0;
    padding: 12px;
    display: flex;
    flex-direction: column;
    gap: 10px;
    transition: border-color 0.2s ease, box-shadow 0.2s ease;
  }

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    border-color: #2563eb;
    box-shadow: 0 4px 12px rgba(37,99,235,0.12);
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  .branch-title {
    font-size: 15px;
    font-weight: 700;
    text-align: center;
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 6px;
  }

  .cat-color { color: #d97706; }
  .num-color { color: #2563eb; }

  .sub-grid {
    display: flex;
    flex-direction: column;
    gap: 8px;
  }

  .sub-btn {
    background: #f8fafc;
    border: 1.5px solid #e2e8f0;
    border-radius: 8px;
    padding: 8px 10px;
    cursor: pointer;
    text-align: left;
    transition: all 0.2s ease;
  }

  .sub-btn:hover {
    background: #f1f5f9;
  }

  .sub-btn.selected {
    background: #eff6ff;
    border-color: #2563eb;
    box-shadow: 0 0 0 1px #2563eb;
  }

  .sub-card-header {
    display: flex;
    justify-content: space-between;
    align-items: center;
    font-weight: 600;
    font-size: 13px;
    margin-bottom: 4px;
  }

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    font-weight: 700;
    padding: 2px 6px;
    border-radius: 4px;
    text-transform: uppercase;
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  .tag-label { background: #fef3c7; color: #b45309; }
  .tag-order { background: #ffedd5; color: #c2410c; }
  .tag-count { background: #dbeafe; color: #1d4ed8; }
  .tag-measure { background: #e0e7ff; color: #4338ca; }

  .examples {
    font-size: 13px;
    color: #475569;
    display: flex;
    align-items: center;
    gap: 6px;
    font-weight: 500;
  }

  .inspect-box {
    background: #ffffff;
    border: 1.5px solid #e2e8f0;
    border-radius: 10px;
    padding: 10px 14px;
    display: flex;
    align-items: center;
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    font-size: 13px;
    font-weight: 600;
  }

  .preview-pills {
    display: flex;
    gap: 6px;
    flex-wrap: wrap;
  }

  .pill {
    padding: 4px 8px;
    background: #f1f5f9;
    border-radius: 6px;
    font-size: 12px;
    font-weight: 600;
    color: #1e2945;
  }
</style>
</head>
<body>

<div class="main-wrap">
  <div class="tree-root">Variables</div>

  <svg class="connector-svg" viewBox="0 0 400 24" preserveAspectRatio="none">
    <path d="M 200 0 L 200 12 L 100 12 L 100 24 M 200 12 L 300 12 L 300 24" fill="none" stroke="#cbd5e1" stroke-width="2" />
  </svg>

  <div class="branches-grid">
    
    <div class="branch-card" id="card-cat">
      <div class="branch-title cat-color">
        <span>🏷️</span> Categorical
      </div>
      <div class="sub-grid">
        <div class="sub-btn selected" onclick="selectType(0)">
          <div class="sub-card-header">
            <span>Nominal</span>
            <span class="sub-tag tag-label">Labels</span>
          </div>
          <div class="examples">🐕 🐈 🦜 • Red, Blue</div>
        </div>

        <div class="sub-btn" onclick="selectType(1)">
          <div class="sub-card-header">
            <span>Ordinal</span>
            <span class="sub-tag tag-order">Ordered</span>
          </div>
          <div class="examples">🥇 🥈 🥉 • S, M, L</div>
        </div>
      </div>
    </div>

    
    <div class="branch-card" id="card-num">
      <div class="branch-title num-color">
        <span>🔢</span> Numerical
      </div>
      <div class="sub-grid">
        <div class="sub-btn" onclick="selectType(2)">
          <div class="sub-card-header">
            <span>Discrete</span>
            <span class="sub-tag tag-count">Counted</span>
          </div>
          <div class="examples">👥 1, 2, 3 • Whole</div>
        </div>

        <div class="sub-btn" onclick="selectType(3)">
          <div class="sub-card-header">
            <span>Continuous</span>
            <span class="sub-tag tag-measure">Measured</span>
          </div>
          <div class="examples">⏱️ 14.8s • 172.5 cm</div>
        </div>
      </div>
    </div>
  </div>

  <div class="inspect-box">
    <span id="inspect-label" style="color: #64748b;">Sample values:</span>
    <div class="preview-pills" id="inspect-pills">
      <span class="pill">🐕 Dog</span>
      <span class="pill">🐈 Cat</span>
      <span class="pill">🦜 Bird</span>
    </div>
  </div>
</div>

<script>
  const data = [
    {
      catIndex: 0,
      pills: ['🐕 Dog', '🐈 Cat', '🦜 Bird']
    },
    {
      catIndex: 0,
      pills: ['🥉 Bronze', '🥈 Silver', '🥇 Gold']
    },
    {
      catIndex: 1,
      pills: ['0 cars', '1 car', '2 cars', '3 cars']
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      catIndex: 1,
      pills: ['1.75 m', '23.4 °C', '45.8 kg']
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    document.getElementById('card-num').classList.toggle('active', item.catIndex === 1);

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    pillsContainer.innerHTML = item.pills.map(p => `<span class="pill">${p}</span>`).join('');
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</script>

</body>
</html>

When investigating a relationship, we label the factor we believe influences a change as the **explanatory variable** (or independent variable).

The outcome we observe and measure in response is called the **response variable** (or dependent variable).

For example, if you investigate whether study time impacts test scores, weekly study hours is the explanatory variable, and the percentage score is the response variable.

Now that you know what variables you are measuring, whom do you actually collect them from?

A **census** collects data from the entire population ($N$), while a **sample** gathers data from a representative subset ($n$) of that population.

A census provides complete certainty, but surveying everyone is often impossible because of limited time, budget, or access.

<!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, sans-serif; }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 12px;
      user-select: none;
    }

    /* Switcher controls */
    .controls {
      display: flex;
      gap: 8px;
      background: #e2e8f0;
      padding: 4px;
      border-radius: 24px;
      width: 100%;
      max-width: 320px;
    }
    .toggle-btn {
      flex: 1;
      padding: 8px 12px;
      border: none;
      border-radius: 20px;
      background: transparent;
      color: #64748b;
      font-weight: 600;
      font-size: 15px;
      cursor: pointer;
      transition: all 0.25s ease;
      text-align: center;
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    .toggle-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 2px 6px rgba(0,0,0,0.08);
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    /* Main Display Stage */
    .stage {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      max-width: 420px;
      flex: 1;
      gap: 12px;
    }

    /* Population Box */
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      background: #ffffff;
      border-radius: 16px;
      padding: 14px;
      width: 100%;
      box-shadow: 0 4px 12px rgba(0,0,0,0.04);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
      position: relative;
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    .pop-header {
      font-size: 13px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      color: #64748b;
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    .grid {
      display: grid;
      grid-template-columns: repeat(6, 1fr);
      gap: 6px;
      width: 100%;
      max-width: 280px;
      justify-items: center;
    }

    .unit {
      width: 32px;
      height: 32px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 17px;
      transition: all 0.35s ease;
      opacity: 0.25;
      background: #f1f5f9;
      transform: scale(0.9);
    }

    /* Selected state */
    .unit.selected {
      opacity: 1;
      transform: scale(1.05);
      box-shadow: 0 0 0 2px #0284c7;
      background: #e0f2fe;
    }

    /* Meters Section */
    .metrics {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
      width: 100%;
    }

    .metric-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 6px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.03);
    }

    .metric-label {
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
      display: flex;
      justify-content: space-between;
    }

    .bar-bg {
      height: 8px;
      background: #f1f5f9;
      border-radius: 4px;
      overflow: hidden;
    }

    .bar-fill {
      height: 100%;
      border-radius: 4px;
      transition: width 0.4s ease, background-color 0.4s ease;
    }

    .fill-time { background: #f97316; }
    .fill-coverage { background: #0284c7; }
  </style>
</head>
<body>

  
  <div class="controls">
    <button class="toggle-btn active" id="btn-census" onclick="setMode('census')">Census</button>
    <button class="toggle-btn" id="btn-sample" onclick="setMode('sample')">Sample</button>
  </div>

  
  <div class="stage">
    <div class="population-card">
      <div class="pop-header" id="pop-title">Population (100%)</div>
      <div class="grid" id="unit-grid"></div>
    </div>

    
    <div class="metrics">
      <div class="metric-card">
        <div class="metric-label">
          <span>⏱ Time & Cost</span>
          <span id="cost-val">High</span>
        </div>
        <div class="bar-bg">
          <div class="bar-fill fill-time" id="cost-bar" style="width: 100%;"></div>
        </div>
      </div>

      <div class="metric-card">
        <div class="metric-label">
          <span>🎯 Surveyed</span>
          <span id="coverage-val">All</span>
        </div>
        <div class="bar-bg">
          <div class="bar-fill fill-coverage" id="coverage-bar" style="width: 100%;"></div>
        </div>
      </div>
    </div>
  </div>

  <script>
    // 24 Population units with balanced characteristics
    const population = [
      { icon: '👩', sample: true },
      { icon: '👨', sample: false },
      { icon: '🧑', sample: false },
      { icon: '👵', sample: true },
      { icon: '👦', sample: false },
      { icon: '👧', sample: false },
      { icon: '👨', sample: false },
      { icon: '👩', sample: false },
      { icon: '🧓', sample: true },
      { icon: '👧', sample: false },
      { icon: '👦', sample: false },
      { icon: '👩', sample: false },
      { icon: '🧑', sample: true },
      { icon: '👨', sample: false },
      { icon: '👵', sample: false },
      { icon: '👩', sample: false },
      { icon: '👧', sample: true },
      { icon: '👦', sample: false },
      { icon: '🧓', sample: false },
      { icon: '🧑', sample: false },
      { icon: '👨', sample: true },
      { icon: '👩', sample: false },
      { icon: '👧', sample: false },
      { icon: '👦', sample: false }
    ];

    const grid = document.getElementById('unit-grid');
    population.forEach((p, idx) => {
      const el = document.createElement('div');
      el.className = 'unit selected';
      el.id = `unit-${idx}`;
      el.textContent = p.icon;
      grid.appendChild(el);
    });

    let currentMode = 'census';

    function setMode(mode) {
      currentMode = mode;
      
      const isCensus = mode === 'census';
      document.getElementById('btn-census').classList.toggle('active', isCensus);
      document.getElementById('btn-sample').classList.toggle('active', !isCensus);
      
      document.getElementById('pop-title').textContent = isCensus ? 'Population (100%)' : 'Representative Subset';
      document.getElementById('cost-val').textContent = isCensus ? 'High' : 'Low';
      document.getElementById('coverage-val').textContent = isCensus ? 'All (24)' : 'Subset (6)';
      
      document.getElementById('cost-bar').style.width = isCensus ? '100%' : '25%';
      document.getElementById('coverage-bar').style.width = isCensus ? '100%' : '25%';

      population.forEach((p, idx) => {
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          el.classList.add('selected');
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        }
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    // Stop autoplay on manual interaction
    ['btn-census', 'btn-sample'].forEach(id => {
      document.getElementById(id).addEventListener('click', () => {
        clearInterval(timer);
      });
    });
  </script>
</body>
</html>

If a sample only includes a fraction of the population, how do you make sure its results reflect the whole group accurately?

## Sampling Strategies and Representativeness

**Sampling bias** occurs when some members of a population are systematically more likely to be selected than others, distorting the investigation's results.

In **Simple Random Sampling** (SRS), every individual has an equal chance of selection, such as drawing names randomly from a complete roster.

In **Stratified Sampling**, you divide the population into subgroups (strata) and sample from each so stratum proportions match the population: $\frac{n_i}{N_i} = \frac{n}{N}$.

In **Systematic Sampling**, you choose individuals at fixed regular intervals from an ordered list, such as selecting every $k^{\text{th}}$ person who enters a school gate.

<!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>Stratified Sampling</title>
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      width: 100%;
      max-width: 480px;
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      gap: 12px;
      flex-grow: 1;
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    .section {
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    .header-row {
      display: flex;
      justify-content: space-between;
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      margin-bottom: 8px;
      font-size: 13px;
      font-weight: 700;
      color: #374151;
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    .strata-container {
      display: grid;
      grid-template-columns: 5fr 3fr 2fr;
      gap: 8px;
    }

    .stratum {
      border-radius: 8px;
      padding: 6px;
      display: flex;
      flex-direction: column;
      align-items: center;
      transition: opacity 0.3s ease;
    }

    .stratum-a { background: #eff6ff; border: 1px solid #bfdbfe; }
    .stratum-b { background: #ecfdf5; border: 1px solid #a7f3d0; }
    .stratum-c { background: #fffbeb; border: 1px solid #fde68a; }

    .pct-tag {
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      font-weight: 700;
      margin-bottom: 4px;
    }
    .stratum-a .pct-tag { color: #1d4ed8; }
    .stratum-b .pct-tag { color: #047857; }
    .stratum-c .pct-tag { color: #b45309; }

    .dot-grid {
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      flex-wrap: wrap;
      gap: 4px;
      justify-content: center;
      align-content: center;
      min-height: 42px;
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    .dot {
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      border-radius: 50%;
      transition: transform 0.3s ease, opacity 0.3s ease, box-shadow 0.3s ease;
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    .connector {
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      justify-content: center;
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      color: #9ca3af;
      font-size: 16px;
      height: 14px;
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    .sample-area {
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      transform: translateY(-4px);
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</head>
<body>

  <div class="toolbar">
    <button class="btn active" id="btnStep1" onclick="setStep(1)">1. Population</button>
    <button class="btn" id="btnStep2" onclick="setStep(2)">2. Strata</button>
    <button class="btn" id="btnStep3" onclick="setStep(3)">3. Sample</button>
  </div>

  <div class="main-stage">
    
    <div class="section">
      <div class="header-row">
        <span>Population</span>
        <span style="font-weight: 500; color: #6b7280;">N = 20</span>
      </div>
      <div class="strata-container">
        <div class="stratum stratum-a" id="popA">
          <span class="pct-tag">50% (10)</span>
          <div class="dot-grid" id="gridPopA"></div>
        </div>
        <div class="stratum stratum-b" id="popB">
          <span class="pct-tag">30% (6)</span>
          <div class="dot-grid" id="gridPopB"></div>
        </div>
        <div class="stratum stratum-c" id="popC">
          <span class="pct-tag">20% (4)</span>
          <div class="dot-grid" id="gridPopC"></div>
        </div>
      </div>
    </div>

    <div class="connector">↓</div>

    
    <div class="section sample-area" id="sampleBox">
      <div class="header-row">
        <span>Stratified Sample</span>
        <span style="font-weight: 500; color: #6b7280;" id="sampleCountLabel">n = 10</span>
      </div>
      <div class="strata-container">
        <div class="stratum stratum-a">
          <span class="pct-tag">50% (5)</span>
          <div class="dot-grid" id="gridSampleA"></div>
        </div>
        <div class="stratum stratum-b">
          <span class="pct-tag">30% (3)</span>
          <div class="dot-grid" id="gridSampleB"></div>
        </div>
        <div class="stratum stratum-c">
          <span class="pct-tag">20% (2)</span>
          <div class="dot-grid" id="gridSampleC"></div>
        </div>
      </div>
    </div>
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A common mistake is assuming that a huge sample automatically fixes bad collection methods.

Surveying only readily accessible people is **Convenience Sampling**—a non-probability method where even a huge sample size cannot overcome selection bias.

Maintaining high **Data Quality** requires actively mitigating sampling bias (undercoverage and overcoverage), **non-response bias**, and **measurement bias** caused by leading survey wording.

A robust plan also enforces ethical research protocols, including **informed consent**, participant **anonymity**, and strict data privacy protections.

Once you have selected your question, your variables, and your sampling method, how do you organize everything before starting?

Maintaining high **Data Quality** requires actively mitigating sampling bias (undercoverage and overcoverage), **non-response bias**, and **measurement bias** caused by leading or ambiguous questions.

Ethical investigation plans must also protect participants by establishing protocols for **informed consent**, respondent **anonymity**, and secure **data privacy**.

A **data collection table** acts as the blueprint for your investigation, standardizing how each piece of data is gathered before collection begins.

Each row represents a single **observational unit** (one participant or trial), and each column records one specific variable along with its exact unit of measurement.

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      <span>↔</span> Row (Unit)
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      <span>↕</span> Column (Variable)
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  <div class="table-container mode-row" id="matrix">
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          <th style="width: 22%;">Unit<span class="unit">Participant</span></th>
          <th class="col-1" onclick="setCol(1)">Height<span class="unit">(cm)</span></th>
          <th class="col-2 target-col" onclick="setCol(2)">Reaction<span class="unit">(s)</span></th>
          <th class="col-3" onclick="setCol(3)">Sleep<span class="unit">(h)</span></th>
        </tr>
      </thead>
      <tbody>
        <tr class="row-1" onclick="setRow(1)">
          <td class="unit-label">#1</td>
          <td class="col-1">162</td>
          <td class="col-2 target-col">0.28</td>
          <td class="col-3">7.5</td>
        </tr>
        <tr class="row-2 target-row" onclick="setRow(2)">
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          <td class="col-1">175</td>
          <td class="col-2 target-col target-cell">0.31</td>
          <td class="col-3">8.0</td>
        </tr>
        <tr class="row-3" onclick="setRow(3)">
          <td class="unit-label">#3</td>
          <td class="col-1">158</td>
          <td class="col-2 target-col">0.25</td>
          <td class="col-3">6.5</td>
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Always run a **pilot test** by collecting data from 2 or 3 individuals first to spot confusing survey questions or broken measurement tools before launching your full study.

math

Planning a Statistical Investigation. Design a statistical investigation by formulating a question, identifying variables and planning data collection.

Statistical Investigations: Introduction to the PPDAC Cycle and Posing Investigative Questions

Study Design: Statistical Questions, Types of Variables, and Sampling Methods

Conduct Statistical Investigations: Year 9 Curriculum Planning Tool (AC9M9ST05)

To develop deep competency in statistical investigation design, anchor lessons around real-world, student-centered inquiries using authentic secondary data (e.g., ABS CensusAtSchool) or primary school-based data. Explicitly teach the distinction between overarching investigative questions and survey items, using collaborative peer-audits where student pairs stress-test each other's sampling plans for potential biases, ethical considerations, and clear operational definitions before any data collection begins.