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84 lines
3 KiB
Markdown
84 lines
3 KiB
Markdown
+++
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title = "Measures of dispersion"
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description = "Interpret range, interquartile range, variance, and standard deviation for comparing data sets."
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template = "chapter.html"
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weight = 31
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[extra]
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subject = "math"
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chapter_id = "measures-dispersion"
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strand = "Data Handling"
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+++
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<div class="book-prose">
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<p>
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Dispersion describes how spread out data is. Two datasets can have the same average but very
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different variability.
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</p>
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<h2 id="range-iqr" class="book-prose__heading">Range and interquartile range</h2>
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<p class="book-formula">\[
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\text{Range}=\text{max}-\text{min},\qquad
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\text{IQR}=Q_3-Q_1.
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\]</p>
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<p>
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IQR is less affected by outliers than range.
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</p>
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<h2 id="variance-sd" class="book-prose__heading">Variance and standard deviation</h2>
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<p>
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Variance measures average squared distance from mean.
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Standard deviation is square root of variance and has same unit as data.
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</p>
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<p class="book-formula">\[
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\sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}}
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\]</p>
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<h2 id="comparing-data" class="book-prose__heading">Comparing datasets</h2>
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<ul>
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<li>Smaller spread statistics \(\Rightarrow\) more consistent data.</li>
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<li>Use median + IQR for skewed data.</li>
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<li>Use mean + standard deviation for roughly symmetric data.</li>
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</ul>
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<h2 id="boxplot-reading" class="book-prose__heading">Box plot interpretation</h2>
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<p>
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Box shows middle 50% (\(Q_1\) to \(Q_3\)); median line marks center.
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Longer whisker/box indicates larger spread.
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</p>
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<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
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<ul>
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<li>State both center and spread when comparing sets.</li>
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<li>Do not confuse variance with standard deviation.</li>
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<li>Check if grouped-data values are estimates.</li>
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<li>Interpret spread in context (consistency/risk).</li>
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</ul>
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</div>
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<div class="book-prose">
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">Find range of 12, 9, 15, 20, 13.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(20-9=11\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">If \(Q_1=18\), \(Q_3=31\), find IQR.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> 13.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Dataset A and B have same mean. A has SD 2, B has SD 8. Which is more consistent?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> A (smaller SD).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Why is IQR preferred over range when outliers exist?</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> IQR focuses on middle 50% and is less distorted by extreme values.</p>
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</div>
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</details>
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</div>
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