Statistical Methods

• Branch of Applied Mathematics
• Descriptive Statistics – summary statistics
• Quantitative – measures along a continuum
• Qualitative – categories
• Correlation Statistics – relationships
• Inferential Statistics – based on theories of probability

Frequency Distributions

• Group or Set of all Scores on a Variable
• Described by Location on a Scale of Measurement
• Measures of Central Tendency
• Described by Dispersion of Measurements on the Scale
• Measures of Variability
• Described by Shape of the Dispersion of Measurements
• Normal Distribution – symmetric bell-shaped curve (typically assumed – Y Frequency, X Value of Obs.)
• Bimodal Distribution – have two distinct points around which scores cluster
• Skewed Distribution – asymmetric
• Positively Skewed – scores bunch up at low end and tail off at high end
• Negatively Skewed – scores bunch up at high end

• Mode – score that occurs most frequently
• Median – the 50^th percentile of a distribution (i.e., half of the observations fall below; half fall above)
• Odd Number of Scores – median is the middle score when scores arranged in rank order
• Even Number of Scores – median is the point halfway between the two central values when scores arranged in rank order
• Mean (average) – sum of the observations divided by the number of observations
  
• Unimodal, Symmetric Distributions (e.g., normal distribution) – mean, median, and mode are all the same
• Symmetric Distributions (even bimodal) – mean and median are the same
• Skewed Distributions – mean is pulled toward the tail; median falls between the mean and the mode

• Deviation Scores – reflect variation in a set of scores
  
• Sum of Squares
• Sum of Deviation Scores is Zero – useless as a measure of variability
• Absolute Value – (i.e., positive value) can be used to determine mean deviations; rarely used as results in mathematical problems with inferential statistic
• Sum of Squares – square deviation scores and sum
  

• Variance of a Population (lowercase Greek sigma squared)
• Mean Square – mean of the squared deviation scores
  

• Variance Estimate from a Sample – s²
• Sum of Squares of Sample is less than Sum of Squares of Population (i.e., fewer observations are in sample)
• Degrees of Freedom = n-1
  

• Standard Deviation – s
• Positive Square Root of the Variance
  

• Normal Distribution
• 68.3% of observations will fall within 1 standard deviation of the mean
• 95.4% of observations will fall within 2 standard deviations of the mean
• 99.7% of observations will fall within 3 standard deviations of the mean

• Transformation of Original Scores
• Allows Comparing Scores from Different Scales
• Unit of Measurement is the Standard Deviation
• Score Expressed in Standard Deviation Units from the Mean
• Distribution of Standard Scores has the Same Shape as the Original Distribution
• Mean of Zero
• Standard Deviation of 1
  

• Correlation Coefficient – a measurement of the relationship between two variables
• Values from –1.00 to +1.00
• Zero Indicates no Relationship
• Rule of Thumb – anything less than ± 0.70 implies weak or no relationship
• Prediction – estimation of one variable from knowledge of another
• Accuracy increases as correlation increases
• Pearson Product-Moment – both variables on interval scales (means of distributions must be computed)
• Interval Scale – distance between any two scores is of a known magnitude (e.g., temperature)
• Coefficient of Contingency – both variables on nominal scales
• Nominal Scale – classifies elements into two or more categories, but not according to order or magnitude (e.g., gender)