Variability

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Variability

• Taking into account the spread of a distribution
  of scores

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Concepts

• Variability
• See Fig 4.11
• Range
• Range URL X(max) LRL X(min)
• Consider 4, 8, 7, 9, 6, 11
• Range 11.5 3.5 8
• How much information does the range provide about
  the variability of scores?
• Consider 4, 5, 5, 5, 5, 11
• How is the range different for the above?
• Interquartile range

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Interquartile Range (IQR)

• Range for the middle 50 of the observations (Q2)
• Chop off top 25 of observations (Q3)
• Chop off bottom 25 of observations(Q1)
• See Fig 4.3

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Procedure for determining IQR

• 1. Order observations from least to most
• 2. Find the positions in the distribution that
  divide observations into quarters (number of
  observations 1)/4
• 3. Now we can remove the top and bottom quarters.
• 4. Count from the bottom up for first quartile
  (Q1)
• 5. Count from the top down for third quartile
  (Q3)
• 6. IQR Q3 Q1

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Advantage of IRQ over range

• One more thing statisticians more often use
  semi-interquartile range (SIQR IRQ/2)
• Conveys the same information the variability of
  the scores in the middle of the distribution.
• Not sensitive to extreme values

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Variance

• If we want to describe how observations deviate
  from the mean, why not find the average of the
  deviations?
• Deviation X (X-bar)

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note n should be N in the formula below
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note n should be N in the formula below and s
should be small sigma (population parameter)
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note n should be N in the formula below and s
should be small sigma (population parameter)
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note s should be small sigma (population
parameter)
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note n should be N in the formula below and s
should be small sigma (population parameter)
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Standard Deviation
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note n should be N in the formula below and s
should be small sigma (population parameter)
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note n should be N in the formula below and s
should be small sigma (population parameter)
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Standard Deviation and Variance for Samples

• A sample statistic is said to be biased if, on
  the average, it consistently overestimates or
  underestimates the corresponding population
  parameter
• Generally, the sample statistic underestimates
  the variability of the population parameter
• See Fig 4.6

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Computing unbiased statistics

• Instead of n, use degrees of freedom (df) or
  (n-1) to calculate the sample variance and sample
  statistic
• What would that look like?
• The notation for sample variance is s, not sigma.

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Factors that affect variability

• Extreme scores
• The range is most affected by extreme scores
• The SIQR is least affected by this
• Sample size
• The range is directly related to sample size
• Stability under sampling
• The range will change unpredictably
• Open-ended distributions
• Cannot compute range or standard deviation
• Only available measure of variability is SIQR.

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The role of variability in inferential statistics

• The question is whether the sample data reflects
  patterns that exist in the population, or are the
  sample data simply showing random fluctuations
  that occur by chance.
• The average of the statistic across samples
  should be the same as the statistic of the
  population.
• See Table 4.1