Measures of Central Tendency and Dispresion

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Measures of Central Tendency and Dispresion
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Content Analysis- Challenges

• Lose some nuance when coding
• How to select material from universe of possible
  material?
• Is material accurate?
• Unintentional problems
• Purposeful distortion
• Ultimately a question of validity
• Are coders accurate?
• Can establish reliability
• Harder to establish validity

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Statistics

• Provides description of a sample or population
• Simplification
• Univariate- Only interested in one attribute at a
  time
• Bivariate- consider relationships between 2
  attributes
• Multivariate- the sky is the limit

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Percentages

• Useful for comparing groups with unequal numbers

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Percentages


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Percentages

• To Compute
• (with trait of interest/total ) X 100
• Example 1- Sample of 4 cats, one is black
• (¼)X100- 25
• Example 2-Sample of 750, 612 approve of the
  president
• (612/750)X100 81.6

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What Constitutes the Denominator?

• Percentage of Total
• Percentage of Valid Cases
• Excludes missing cases
• Typically more appropriate
• Cumulative Percent-what percentage so far have
  reached this level

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An Example
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Measures of Central Tendency

• Mode
• Mean (Average)
• Median

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Computing the Mean

• Requires At least ordinal data
• (Y1 Y2 Y3. Yi)/I
• Example have people with incomes of 10,000,
  15,000, 25,000, 55,000, 32,000, 29,500
• Mean(10,00015,00025,000, 55,000
  32,00029,500)/6 27,750

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Mode

• Most common with nominal data
• Count frequencies, find most common
• Ask 30 1st graders favorite color
• 7 blue
• 3 chartreuse
• 4 purple
• 2 yellow
• 10 red
• 3 green
• 1 Black
• Mode- Red

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Frequencies
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Computing the Median

• Requires at least Ordinal Data
• Put values in order
• If odd number, value half are above, half below
• If even number- Average of two middle cases
• Income Example
• 10,000, 15,000, 25,000, 55,000, 32,000, 29,500
• 10,000, 15,000, 25,000, 29,500, 32,000, 55,000
• Median25,250

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When To Use Which?

• Mode- nominal data
• Better to actually give totals for all if few
  choices, e.g. 33 red, 10 green
• Mean- when appropriate data
• Median- with ordinal data, in cases where there
  are a few values that might cause a skew
• Outlier- Data point with extreme value

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Median vs. Mean

• Created a fake town with 100 residents
• Incomes 19,00-138,000
• Mean57600, Median49,500
• Suppose one person with 30,000 moves away,
  replaced by Millionaire
• Mean67,300, Median55,000
• Replaced by 50,000,000
• Mean557,300 Median 55,000
• Replaced by Bill Gates (50 Billion)
• Mean500Million, Median 55,000

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Measures of Dispersion

• Measure of Central Tendency loses something
• Income example?
• Dispersion
• Measure of how much divergence there is from the
  mean
• Histogram
• Horizontal Axis breaks variable down into ranges
• Vertical Axis-count within each range

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Quantifying Dispersion- Standard Deviation

• Find difference from mean for each observation
• Add them up
• Divide by the number of cases minus1

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Standard Deviation from Previous cases

• Mean 50,024, S.D992.5
• Min46,834, Max52,935

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• Mean50,255 S.D.4792
• Min35,671 Max65,095

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• Mean50,311 S.D.10,124
• Min22,522 Max78,642

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• Mean50,982 S.D.18,898
• Min1591 Max105,957

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Gore Thermometer

• Mean57.4, S.D.25.7
• 04.6, 100 5.6

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George W Bush Thermometer

• Mean56.1 S.D.24.9
• 0 4.4 1004.7

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Clinton Thermometer

• Mean55.2 S.D.29.7
• 09.5 1007.1

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For Next Time

• The Normal Distribution
• Bivariate Relationships
• Get stats assignments