Lies, Damn Lies, and Statistics

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Lies, Damn Lies, and Statistics

• An introduction to the mathematical concepts and
  operations used in testing and assessment

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Fundamentals

• Data collected from all members of a population
  are designated as population statistics. Such
  statistics are distinguished by the use of
  lower-case letters.
• Statistics taken from a sample of members within
  a population are designated as sample statistics.
  Such statistics are distinguished by the use of
  upper-case letters.
• All statistics are based on two concepts
• Central Tendency
• Variance

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Central Tendency

• There are three primary measures of central
  tendency mean, median and mode.
• The mean, or arithmetic average, is the most used
  statistic and will be covered in the next slide.
• The median is the middle number when all values
  are arranged in order, i.e. if we had 11 data
  points and arranged them from largest to
  smallest, the median would be the 6th number in
  the ranked order.
• The mode is the number that appears the most
  frequently.

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The mean (AKA arithmetic average)

• Population (µ), sample

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We Need More Information!

• The mean, though a popular and very useful piece
  of information, is rarely sufficient by itself to
  describe the data.
• In order to put the mean into perspective, we
  need to know what the variance, or spread, in
  scores is. In order to do this the Standard
  Deviation was invented.

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Describing Score Spread

• Compute the mean of the individual scores

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Describing Score Spread

• Subtract the mean from each individual score to
  get the deviation score.

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Describing Score Spread

• Subtract the mean from each individual score to
  get the deviation score.
• If the deviation scores sum to zero, you did it
  right. If they dont, check your math.

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Describing Score Spread

• Square each deviation score to get the squared
  deviation

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Describing Score Spread

• Square each deviation score to get the squared
  deviation
• Add up the squared deviations to get the sum of
  squares.

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Describing Score Spread

• Compute the average of the squared deviations by
  dividing the sum of squares by n. This number is
  known as the variance (s2) .

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Describing Score Spread

• Since we had to square the deviations to avoid a
  zero average, we must now undo the squaring
  process by taking the square root of the
  variance. We have now successfully computed the
  standard deviation.

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Standard Deviation

• Population (s)

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Standard Deviation

• Sample (S)

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Review of SD computation

• Find the mean of the scores
• Subtract the mean from each score (deviation
  score)
• Square the deviation scores.
• Sum the squared deviation scores and divide by n
  (pop) or n-1 (sample) to get the variance.
• Take the square root of the variance.