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Deviation from Mean (Average)  
  • Two steps: (1) calculate arithmetic mean, (2)  subtract average from actual and divide by total measurements
  • Excel::Statistical
    STDEV
    STDEVA
    STDEVP
    STDEVPA

Factor Analysis  
  • Suppose a psychologist proposes a theory that there are two kinds of intelligence, "verbal intelligence" and "mathematical intelligence". Note that these are inherently unobservable. Evidence for the theory is sought in the examination scores, from each of 10 different academic fields, of 1000 students. If each student is chosen randomly from a large population, then each student's 10 scores are random variables.
  • The psychologist's theory may say that, for each of the 10 academic fields, the score averaged over the group of all students who share some common pair of values for verbal and mathematical "intelligences" is some constant times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.e., it is a linear combination of those two "factors". The numbers, for this particular subject, by which the two kinds of intelligence are multiplied to obtain the expected score, are posited by the theory to be the same for all intelligence level pairs, and are called "factor loadings" for this subject.
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Frequency Distribution (Histogram)  
  • Assign useful division of data (i.e. students who scored 90-95% could be a division, and 96-100% might be another, as well as below 60%
    a third division, etc).
  • Record the frequency of each division within the data (i.e. 4 students in the 90-95% division, 1 student in the 96-100% division, 2 students in the below 60% division, etc).
  • Take frequencies and determine their overall occurrences (i.e. 0.12 of students were in the 90-95% division, etc).
  • Display results in a chart or a graph
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Mean (1): Arithmetic Mean
Easily distorted by outliers
More a measure of "central tendency"
 
  • This calculation is totally linear: add the totals of each measurement and divide by the number of measurements
  • So,
    (3+7+1+24+11+32)/(6)=13
  • While the mean is often used to report central tendency, it may not be appropriate for describing skewed distributions, because it is easily misinterpreted.
  • The arithmetic mean is greatly influenced by outliers.
  • These distortions can occur when the mean is different from the median.
  • When this happens the median may be a better description of central tendency.
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or

Mean (2): Geometric Mean  
  • Geometric mean finds the "N'th" root of the product of all the measurements.
  • So, if we have 8 measurements, we multiply the 8 values together and find the "8th" root of that value, which we call the Geometric Mean.
  • Shortcomings: The geometric mean only applies to positive numbers, in order to avoid taking the root of a negative product, which would result in imaginary numbers.
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