Distinction Between Absolute and Relative Measures of Dispersion

Absolute Measure of Dispersion: An absolute measure expresses the dispersion in the same units as the original data. Examples include range, variance, standard deviation, and mean deviation. These measures are useful when comparing datasets with similar means and units.

Relative Measure of Dispersion: A relative measure is a dimensionless quantity expressed as a ratio or percentage of a central tendency measure (usually the mean). Examples include coefficient of variation, coefficient of range, and coefficient of quartile deviation. These measures are useful for comparing the consistency of datasets with different means or different units.

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Computer Consistency Analysis

For Company A:

Mean: $\bar{x}_A = \frac{1 \times 5 + 3 \times 16 + 5 \times 13 + 7 \times 7 + 9 \times 5 + 11 \times 4}{50} = \frac{5 + 48 + 65 + 49 + 45 + 44}{50} = \frac{256}{50} = 5.12$ seconds

Variance: $\sigma_A^2 = \frac{\sum f(x - \bar{x})^2}{n} = \frac{5(1-5.12)^2 + 16(3-5.12)^2 + 13(5-5.12)^2 + 7(7-5.12)^2 + 5(9-5.12)^2 + 4(11-5.12)^2}{50} = 6.8896$

Standard Deviation: $\sigma_A = \sqrt{6.8896} = 2.624$ seconds

Coefficient of Variation: $CV_A = \frac{\sigma_A}{\bar{x}_A} \times 100 = \frac{2.624}{5.12} \times 100 = 51.25%$

For Company B:

Mean: $\bar{x}_B = \frac{1 \times 2 + 3 \times 7 + 5 \times 12 + 7 \times 19 + 9 \times 9 + 11 \times 1}{50} = \frac{2 + 21 + 60 + 133 + 81 + 11}{50} = \frac{308}{50} = 6.16$ seconds

Variance: $\sigma_B^2 = \frac{2(1-6.16)^2 + 7(3-6.16)^2 + 12(5-6.16)^2 + 19(7-6.16)^2 + 9(9-6.16)^2 + 1(11-6.16)^2}{50} = 5.8704$

Standard Deviation: $\sigma_B = \sqrt{5.8704} = 2.423$ seconds

Coefficient of Variation: $CV_B = \frac{\sigma_B}{\bar{x}_B} \times 100 = \frac{2.423}{6.16} \times 100 = 39.37%$

Conclusion: Since $CV_B (39.37%) < CV_A (51.25%)$, Company B's computers are more consistent as they have a lower coefficient of variation, indicating less relative variability in execution time.