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Mathematics · Ch 7 — Statistics

Summary

Summary

  • Measures of dispersion quantify the spread of data: range, mean deviation, variance, and standard deviation.
  • Range = maximum value −- minimum value; simplest but least reliable.
  • Mean deviation about a central value AA (mean or median) for ungrouped data:

MD(A)=1n∑i=1n∣xi−A∣\text{MD}(A) = \frac{1}{n} \sum_{i=1}^{n} |x_i - A|

For grouped data, use class marks xix_i and frequencies fif_i:

MD(A)=1N∑i=1nfi∣xi−A∣,N=∑fi\text{MD}(A) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - A|, \quad N = \sum f_i

  • Variance (σ2\sigma^2) and standard deviation (σ\sigma) are the most important measures. For ungrouped data:

σ2=1n∑i=1n(xi−xˉ)2,σ=σ2\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2, \quad \sigma = \sqrt{\sigma^2}

  • For grouped data (discrete or continuous):

σ2=1N∑i=1nfi(xi−xˉ)2\sigma^2 = \frac{1}{N} \sum_{i=1}^{n} f_i (x_i - \bar{x})^2

  • Shortcut formula for variance (avoids calculating deviations from mean):

σ2=1N∑fixi2−(1N∑fixi)2\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - \left( \frac{1}{N} \sum f_i x_i \right)^2

  • Coefficient of variation (CV) compares relative variability across datasets:

CV=σxˉ×100%\text{CV} = \frac{\sigma}{\bar{x}} \times 100\%

Lower CV implies more consistency (less relative spread).

  • For two or more groups, the combined mean and combined variance can be computed using:

xˉc=n1xˉ1+n2xˉ2n1+n2\bar{x}_c = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

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