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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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