Measures of Dispersion Overview

Measures of Dispersion

  • Measures of dispersion indicate the variability in a sample.

  • Range (R): Difference between the largest and smallest values in the sample.

    • Formula: X = XL - XS

Mean and Deviations

  • When the center is the arithmetic mean, deviations from it can be summarized.

  • The sum of deviations from the mean is always zero

Mean Deviation, Variance, and Standard Deviation

  • Mean Deviation: Can summarize the spread but is less common.

  • Sample Variance (s2s^2): Average of the squared deviations from the mean.

  • Sample Standard Deviation (SD or ss): Measures the spread around the mean.

    • A larger ss signifies more variability.

    • ss can equal 0 if there is no spread.

    • Units of ss are the same as the data units.

Degrees of Freedom

  • Variance is divided by n−1n - 1 (degrees of freedom) rather than nn.

  • Explanation:

    • The sum of deviations from the mean is always zero.

    • If n−1n-1 data points are known, the last one is determined automatically.

Empirical Rule

  • Applicable to unimodal, moderately skewed distributions.

  • States:

    • Approximately 68% of data lies between extMean−sext{Mean} - s and extMean+sext{Mean} + s.

    • Approximately 95% lies between extMean−2sext{Mean} - 2s and extMean+2sext{Mean} + 2s.

    • Over 99% lies between extMean−3sext{Mean} - 3s and extMean+3sext{Mean} + 3s.

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  • Standard deviation is preferred as it shares units with the data.

Coefficient of Variation (CV)

  • The CV compares relative variation between datasets with different units.

  • CV is defined as: s/x̅ * 100%

  • Remains consistent regardless of unit changes; both mean and standard deviation change similarly, keeping the ratio the same.

  • Example: Comparing cholesterol (mg/100 ml) with body weight (lbs) using CV.

Displaying Data

  • Numeric and graphic methods help summarize and present data effectively:

    • Numeric Summary includes measures of location (mean, median) and measures of spread (standard deviation, range).

    • Graphic Methods include bar graphs, stem-and-leaf plots, and box plots.