Measures of Central Tendency and Dispersion

These univariate statistics help describe the distribution of a variable.  They each tell you something different about what the distribution looks like.
Measures of Central Tendency
Tell you the value that the "typical" case in the distribution has for your variable.

Measures of Dispersion
Tell you how much variation you have across cases in the distibution of your variable.

Central Tendency

  • Nominal measure
  • Most frequently occuring value in the distribution
  • NOT the number of times that value occurs
  • Ordinal measure
  • The "middle" of the distribution when all cases are arranged in rank order
  • The value of the case that lies at the point at which 50% have a lower value and 50% have a higher value.
  • Interval/Ratio
  • The arithmetic mean, or what is typically referred to as the average
  • Σ x / n
  • Sum (Σ) of all values in the distribution (x) divided by the number of cases (n)
  • Denoted as an x with a bar over it (for sample data) or μ (for population data)

Measures of Dispersion

Variation Ratio
  • Nominal
  • Simply the proportion of cases that are not in the modal category
  • 1-p[Mode]
  • 1 minus the proportion of all cases in the modal category
  • Limited utility
  • Ordinal
  • Difference between the two extremes of the distribution
  • Highest value found in the distribution minus the lowest value found in the distribution
  • Limited utility
Standard Deviation
  • Interval/Ratio
  • Roughly the "average" amount of deviation in a distribution
  • Actually the square root of the average sum of squares.
  • denoted as s (for sample data) or σ (for population data)
    • Take the difference (deviation) between the value of each case from the mean and square it
    • Sum all the squared deviations.  This result is called the variation, or sum of squares.
    • Divide by the number of cases, n.  (Use n − 1 for sample data.)  This result is called the variance.
    • Take the square root.