Describing Variability

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Chapter 5

• Describing Variability

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We have the mean Shape so
Question The distributions of scores are normal
in both Psy101 and Organic Chemistry. The mean of
Psy101 and Organic Chemistry is 60 (mPsy101
60 mOrgChem). So, is Psy101 the same
difficulty as Organic Chemistry?
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Variability

• So we have the mean, do all values equal the
  mean?
• Nope, we have variability!
• What is variability?
• How much the scores vary around the mean.
• How spread out the scores are.
• Variability is a measure of spread!
• High lots of spread
• Low packed together

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Who Varies?

• Where does variability come from?
• Other factors that influence an individuals
  score!
• Example Class Scores vary!
• IQ
• Time spent studying
• Previous experience
• Taken class before
• Test taking skills
• Etc.

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Who Varies?

• Where does variability come from?
• Example Drunkeness Ratings!
• Amount Drunk
• Proof of Liquor
• Weight
• Tolerance
• Amount ate that day
• Etc.
• Since different individuals have differences in
  other factors, their scores will be different
  (vary)

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Explaining Variability

• Goal of science is to explain Variability/Variance
  .
• Local Goal Determine if a factor/IV changes the
  scores (DV).
• Allows us to explain why two scores are different
  (Explain the variability)
• Global Goal
• Determine every factor/IV that affects the score.
• If know every factor, can predict exactly what
  score an individual will have (Explain all of the
  variability)

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Explaining Variability

• Random Variability
• Random variability is not truly random.
• Systematic changes caused by other factors that
  were uncontrolled.
• Randomness is relative.
• Variability not explained by the IV
• Factors uncontrolled in experiment
• Allowed to be random amongst the individuals
  being measured.

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Measures of Variability

• Range
• Overall Spread
• Interquartile Range (IQR)
• Middle section of the range
• Variance
• Mean of the squared deviations
• Standard Deviation
• Mean of the deviations

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Measures of Variability

• Measures of variability cannot be negative.
• Lowest amount of spread possible is no spread

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Range and the IQR

• Distance between high and low scores.

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Home, home on the.

• Range
• Distance between highest and lowest score.
• Formula
• Range Highest Score - Lowest Score
• Construction
• 1) Order scores from lowest to highest.
• 2) Subtract Lowest from Highest

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The Range

• Problems with the range.
• As sample size increases, range of sample tends
  to increase.
• As draw more observations, increase the chance of
  getting more extreme observations.
• Easily influenced by outliers!

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The Range and Outliers
Drunkenness Scores After 5 Drinks
Range Highest - Lowest
72
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The IQR

• Interquartile Range (IQR)
• Goal Modify range to reduce the influence of
  outliers.
• How Take the range of the middle 50 of the
  scores
• Take range of middle quarters
• Ignore Outer Quarters
• 75th Percentile - 25th Percentile

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The IQR

• Calculating
• 1) Order data from lowest to highest
• 2) Determine positions you need to count in order
  to find beginning and end of middle 50
• (n 1)/4
• Round to nearest
• 3) Beginning at highest observations, count down
  the number of positions from step 2
• 4) Mark that number as the 75th percentile

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The IQR

• Calculating
• 5) Beginning at lowest observations, count up the
  number of positions from step 2
• 6) Mark that number as the 25th percentile
• 7) IQR from Step 4 - from Step 6

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The IQR
Drunkenness Ratings after 3 shots of Bacardi 151
15, 55, 60, 60, 63, 63, 63, 63, 65, 65, 70
Range 55
Steps 56
Steps 34

• Step 2
• (n 1)/4
• (11 1)/4
• 12/4
• 3

Step 7
5
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The Range vs IQR
15, 55, 60, 60, 63, 63, 63, 63, 65, 65, 70
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Range IQR

• Problems w/ Range IQR
• Only considers 2 values
• Blind to variability within the range

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Variance Standard Deviation

• Average of the Deviations

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Why Mean of Deviations

• Logic of Measure
• Variability is how scores vary around the mean.
• Measure how much each score differs from the mean
• Known as a Deviation (x - ?)
• Measure variability as the average difference
  from the mean.
• Add up deviations and divide by total
• S(x - ?)/n

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Why Mean of Deviations

• Problem w/ Simple Mean of Deviations
• S(x - ?)/n always equals 0
• Mean is balance point
• Point where positive deviations and negative
  deviations balance each other out
• S(x - ?) 0

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How the Mean Works
A Deviation (x - ?)
What does it balance?
S(x - ?) 0
Balances the deviations
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How the Mean Works
? 10
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Why Mean of Deviations

• Problem w/ Simple Mean of Deviations
• Resolution to the problem
• Find a way to keep value of deviation, but remove
  sign
• Square them

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The Variance

• Variance
• S(x - ?)2/n
• Mean of the squared deviations
• Problem Not very useful for describing data
• Example Previous data set
• 2, 4, 6, 8, 10, 12, 14, 16, 18
• The variance is 26.67
• Variance is in Squared Units
• Solution Take the square root of the variance

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The Standard Deviation

• Standard Deviation
• Square root of the variance
• ?(S(x - ?)2/n)
• If the variance is the mean of the squared
  deviations, then the standard deviation is the
  mean of the deviations
• Typical deviation gt Typical difference from the
  mean
• In real units

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Formulas Calculations

• The fun stuff..

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Measures of Variability
Important Symbols and Terminology
Population
Sample
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Definition Formulas for Variance
Population Variance
Sample Variance
Population Standard Deviation
Sample Standard Deviation
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Definition Formulas for Variance
Why is it called definition formula? Because it
defines what it is measuring.
Mean of x-scores
Mean of squared deviations
Mean
m S(x) N
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Calculating
Calculate the Variance and Standard Deviation for
the following sample of Drunkenness Scores.
2, 4, 6, 8, 10, 12, 14, 16, 18
n 9
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S2
S (x - ?)2
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S2
S (x - ?)2
(x - ?)
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S2
S (x - ?)2
(x - ?)
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S2
S (x - ?)2
(x - ?)
10
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S2
S (x - ?)2
(x - ?)
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S 5.16
S2
26.67
S (x - ?)2
(x - ?)
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Computational Formulas for Variance Standard
Deviation
Population Variance
Sample Variance
Population Standard Deviation
Sample Standard Deviation
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Sum of Squares
S(x - ?)2

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Calculating
Calculate the Variance and Standard Deviation for
the following sample of Drunkenness Scores.
2, 4, 6, 8, 10, 12, 14, 16, 18
n 9
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S2
S (x2)
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S2
S (x2)
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S2
S (x2)
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S2
S (x2)
26.67
S 5.16
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Standard Deviation

• Final Notes

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Notes on Standard Deviation

• Rough mean of deviations, not true mean of
  deviations.
• The squaring of the deviations inflates the final
  results

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S 5.16
S2
26.67
S (x - ?)2
(x - ?)
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Notes on Standard Deviation

• The majority of the distribution lies between one
  standard deviation above the mean and one
  standard deviation below the mean.
• 68.25
• Minority lies outside 1 standard deviation of the
  mean

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Notes on Standard Deviation

• Measure of distance
• The mean is a measure of position
• Position of the center
• The standard deviation is a measure of distance
• Distance from the mean
• Standard deviation cannot be negative.
• No such thing as a negative distance
• Sum of Squares can not be negative
• All squares are positive