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Updated: Oct 3 2020

Statistical Distribution

  • Descriptive Statistics
    • Mean
      • average of all observation
      • mean = (sum of all observations)/(sample size)
    • Median
      • the middle value of all observations
      • if sample size is odd
        • median = ((n+1)/2)th largest value
      • if the sample size is even
        • median = the average of the (n/2)th and ((n/2)+1)th largest value
    • Mode
      • the most commonly occurring value
      • if there is more than 1 most commonly occurring value, there are as many modes as most commonly occurring values
      • in decreasing order of resistance to outliers, mode > median > mean
  • Types of Distributions
    • Normal
      • aka Gaussian, bell-shaped
      • for continuous variables
      • mean = median = mode
    • Bi-modal
      • distribution has 2 humps (each being a relative mode)
      • if symmetrical, mean = median
    • Skewed
      • positive skew
        • asymmetrical with tail trailing off to right
        • mean > median > mode
      • negative skew
        • asymmetrical with tail trailing off to left
        • mean < median < mode
      • mean very sensitive to skew
      • median somewhat resistant to skew
      • mode very resistant to skew
    • Other
      • non-continuous variable types have their own distributions
        • e.g., binary, categorical, ordinal, binomial, and count variables
  • Characteristics of the Normal Distribution
    • For continuous variables
    • Defined entirely by 2 parameters
    • Mean (µ)
      • standard deviation (σ)
    • A certain percentage of all observations will always fall within +/- certain standard deviations of the mean
      • +/- 1 standard deviation = 68%
      • +/- 2 standard deviations = 95%
      • +/- 3 standard deviations = 99.7%
  • Regression to the Mean
    • Phenomenon in which sample points which were initially extreme often become closer to the mean in future measurements
    • Most points will fall near on the average; therefore, extreme points are often a result of "luck" (e.g., a student performs particularly poor on an exam but normally performs at the average level)
    • Has significance for study design
      • e.g., patients with high blood pressure may improve after taking an experimental anti-hypertensive, but that improvement on the next measurement may be due to regression to the mean rather than the treatmentt
        • the solution is to compare a control and experimental group.
  • Measures of Variability
    • Standard deviation
      • a statistical measure that demonstrates how close together or spread apart the data is
        • if data is closer together, the standard deviation will be smaller (and vice versa)
      • often designated by σ
      • equation
        • square root[(sum of the differences between each data point and the mean squared)/n]
    • Standard error
      • a statistical measure that demonstrates how far the sample mean is from the true population mean
        • helps determine confidence intervals
      • equation
        • standard deviation/square root of n
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