Measures of Dispersion and Location

• Dispersion


  The difference between the actual value and the average value
• Measures of Dispersion


  • Range
  • Average Deviation
  • Variance
  • Standard Deviation
• Measures of Location


  • Quartiles
  • Deciles
  • Percentiles
  • Midhinge
  • Interquartile Range
  • Quartile Deviation
• Range


  The difference of the highest value and the lowest value in the data set
• Average Deviation


  The absolute difference between the element and a given point
• Calculating Average Deviation for Ungrouped Data


  1. AD = Σ|X - X̄| / N

  2. AD = Σ|X - X̄| / n
• Calculating Average Deviation for Grouped Data


  1. AD = Σ|X - X̄|f / N

  2. AD = Σ|X - X̄|f / n
• Standard Deviation


  A statistical term that provides a good indication of volatility, it measures how widely values are dispersed from the average, and is calculated as the square root of variance
• Range Rule of Thumb


  Used to approximate or give a rough estimate of the standard deviation, s = range/4
• Variance


  The mathematical expectation of the average squared deviations from the mean
• Calculating Variance for Ungrouped Data


  1. s^2 = Σ(X - X̄)^2 / (n-1)

  2. s^2 = Σ(X - X̄)^2 / n
• Calculating Variance for Grouped Data


  1. s^2 = Σ(X - X̄)^2f / (n-1)

  2. s^2 = Σ(X - X̄)^2f / n
• Population Variance


  σ^2 = Σ(X - μ)^2 / N
• Population Standard Deviation


  σ = √(Σ(X - μ)^2 / N)
• Variance


  A measure of the spread or dispersion of a set of data values from the mean
• Standard Deviation


  A measure of the amount of variation or dispersion of a set of data values
• Solution 2


  1. Class Limits

  2. f

  3. fX

  4. fX^2
• Class Limits


  • 18 – 26
  • 27 – 35
  • 36 – 44
  • 45 – 53
  • 54 – 62
  • 63 – 71
  • 72 – 80
• f


  • 3
  • 5
  • 9
  • 14
  • 11
  • 6
  • 2
• fX


  • 66
  • 155
  • 360
  • 686
  • 638
  • 402
  • 152
• fX^2


  • 129,761
• Population Variance


  ∑(X - X̄)^2 / N
• Population Standard Deviation


  √(∑(X - X̄)^2 / N)
• Population Mean


  ∑X / N
• Population


  The entire group being studied
• Individual Value


  A single data point within the population
• Solution for Variance & SD

  1. Compute for the mean

  2. Compute for (X - X̄)^2

  3. Compute for Variance

  4. Compute for Standard Deviation
• Deciles


  Dividing a dataset into ten equal parts
• Percentiles


  Dividing a dataset into one hundred equal parts
• Midhinge


  The mean of the first (Q1) and third (Q3) quartiles in the data set
• Interquartile Range (IQR)


  A measure of statistical dispersion, being equal to the difference between the third and first quartiles
• Quartile Deviation (QD)


  A measure of absolute dispersion that ignores the observations on the tails
• Coefficient of Variation (CV)

  A measure of the relative dispersion of a dataset, calculated as the ratio of the standard deviation to the mean
• V


  Coefficient of Variation
• Coefficient of variations can be applied when one is interested to compare standard deviations of two different units
• Since the coefficient of variation is larger for age, the ages are more variable than the salary
• Since the coefficient of variation is larger for sales, the sales are more variable than the commissions
• Chebychev's Theorem


  For any set of observations, the proportion of the values that lie within k standard deviations of the mean is at least 1 - 1/k^2, where k is any constant greater than 1
• Chebychev's theorem states that 88.89% of the data values will fall within 3 standards of the mean
• Empirical Rule


  68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations