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