Measures of Dispersion

Lesson Overview

• Focus on measures of dispersion

• Goals for the lesson:

  • Compute range, variance, and standard deviation

  • Understand standard deviation and variance

  • Calculate coefficient of variation and compare variations

  • Use the empirical rule and Chebyshev's theorem to describe data

  • Calculate variance and standard deviation of grouped data

Importance of Measures of Dispersion

• Measures of center determine where data is centered on a number line.

• Variability provides insight into the shape of the distribution.

  • Example: Average wait time at a doctor’s office is 20 minutes.

  • Key question: Is the wait time consistent for all patients?

Range

• Definition: Difference between the largest and smallest values in a dataset.

• Formula:

  • Range = Maximum Data Value - Minimum Data Value.

• Limitation: Not as descriptive as variance and standard deviation; does not reflect how data is spread around the mean.

Variance

• Definition: Measure of how far data values are spread from the mean; a squared measurement.

• Population Variance Formula:

  • ( \sigma^2 = \frac{\Sigma (X_i - \mu)^2}{N} )

    • Where:

      • (X_i) = ith value in the population

      • (\mu) = population mean

      • N = number of values in the population

• Sample Variance Formula:

  • ( s^2 = \frac{\Sigma (X_i - \bar{x})^2}{n - 1} )

    • Where:

      • (X_i) = ith value in the sample

      • (\bar{x}) = sample mean

      • n = number of data values in the sample

• Rounding Rule: Round to 1 more decimal place than the largest number of decimal places in the data.

• Variance units are squared, which can make interpretation less straightforward.

Standard Deviation

• Definition: Measure of expected deviation from the mean; provides scale for variation.

• Population Standard Deviation Formula:

  • ( \sigma = \sqrt{\frac{\Sigma (X_i - \mu)^2}{N}} )

• Sample Standard Deviation Formula:

  • ( s = \sqrt{\frac{\Sigma (X_i - \bar{x})^2}{n - 1}} )

• Rounding: Same rule as variance.

• Difference in formulas arises due to the need for correction in sample standard deviation (biased estimator).

Coefficient of Variation (CV)

• Definition: Ratio of standard deviation to mean expressed as a percentage.

• Population Formula:

  • ( CV = \frac{\sigma}{\mu} \times 100% )

• Sample Formula:

  • ( CV = \frac{s}{\bar{x}} \times 100% )

• Purpose: Allows comparison of spreads between different datasets.

Empirical Rule

• Applicable to bell-shaped distributions:

  • Approximately 68% of data within 1 standard deviation from the mean.

  • Approximately 95% of data within 2 standard deviations from the mean.

  • Approximately 99.7% of data within 3 standard deviations from the mean.

Chebyshev’s Theorem

• Useful for all distributions, not just bell-shaped.

• Provides a minimum estimate of data within k standard deviations of the mean:

  • Formula: Proportion = 1 - (1/k^2) for k > 1.

  • For k = 2: At least 75% of data within 2 standard deviations.

  • For k = 3: At least 88.9% of data within 3 standard deviations.

Variance and Standard Deviation of Grouped Data

• When given a frequency distribution without original data:

  • Use class midpoints as representative values.

• Formula for Standard Deviation of Grouped Data:

  • ( s = \sqrt{\frac{n \sum f_i x_i^2 - (\sum f_i x_i)^2}{n(n - 1)}} )

    • Where:

      • n = sample size

      • f_i = frequency of class i

      • x_i = midpoint of class i.

• Estimate variance using relationship between variance and standard deviation.

Conclusion

• Understanding measures of dispersion helps in accurately describing data distributions.