Standard Deviation Day 5

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Standard Deviation Day 5
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Standard Deviation

• Another common measure of spread is the Standard
  Deviation a measure of the average deviation
  of all observations from the mean.
• To calculate Standard Deviation
• Calculate the mean.
• Determine each observations deviation (x -
  xbar).
• Average the squared-deviations by dividing the
  total squared deviation by (n-1).
• This quantity is the Variance.
• Square root the result to determine the Standard
  Deviation.

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Standard Deviation

• Variance
• Standard Deviation
• Example 1.16 (p.85) Metabolic Rates

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Standard Deviation
Metabolic Rates mean1600
What does this value, s, mean?
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Standard Deviation

• Deviations show how spread out from the mean the
  data is.
• Variance is large if the observation are largely
  spread out from the mean. Small if the
  observations are close to the mean.

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Standard Deviation

• Because all the values of deviations will sum to
  zero, if we know n-1 values we can always find
  the next value. For that reason we divide the
  variation by n-1.
• N-1 is called degrees of freedom.

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Standard Deviation

• Properties of degrees of freedom
• S measures spread from the mean so it should only
  be used when mean is the measure of center.
• S0 when there is no spread.
• S is not resistant, strong skewness or outliers
  can make s very large.

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Standard Deviation

• How do we decide between mean, s, and five number
  summary?
• If the data is largely skewed use five number
  summary because no one number will describe the
  data.
• Use x and s for reasonably symmetric data.
• ALWAYS, ALWAYS, ALWAYS graph your data!

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Linear Transformations

• Variables can be measured in different units
  (feet vs meters, pounds vs kilograms, etc)
• When converting units, the measures of center and
  spread will change.
• Linear Transformations (xnewabx) do not change
  the shape of a distribution.
• Multiplying each observation by b multiplies both
  the measure of center and spread by b.
• Adding a to each observation adds a to the
  measure of center, but does not affect spread.

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Linear Transformations

• Example 1.15 and handout.