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Math · 19 · Problem-Solving and Data Analysis

Descriptive Statistics II

Notes

Key points

  • Range = max − min: one outlier changes it a lot.
  • Standard deviation = typical distance from the mean. Compare it; never compute it.
  • Values bunched near the mean → small SD; spread toward the ends → large SD.
  • Box plot: min, Q1, median, Q3, max. Box = middle 50%.
  • Histogram / dot plot: count to the middle position to find the median.
  • Add kk to all values → SD, range unchanged. Multiply by kk → both ×∣k∣|k|.
  • Comparing distributions: compare a centre (mean/median) and a spread (SD/range/IQR) separately.

Formulas

Rangemax−min\text{max}-\text{min}
Interquartile rangeIQR=Q3−Q1\text{IQR}=Q_3-Q_1Spread of the middle 50%
Shift and scalex→x+k: σ samex→kx: σ→∣k∣ σx\to x+k:\ \sigma\ \text{same}\qquad x\to kx:\ \sigma\to|k|\,\sigma

Common traps

  • Reading a longer box or whisker as more data points — it's more spread.
  • Thinking a higher mean means a larger SD. Centre and spread are separate.
  • Removing an outlier and assuming the median changes as much as the mean.

SAT tip

SD questions never need the formula: the set whose values cluster tighter around its mean has the smaller standard deviation.

Test yourself

10 official SAT questions on Descriptive Statistics II, easiest first.

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