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Understanding central tendency properties in statistics
 

Understanding central tendency properties in statistics

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    Understanding central tendency properties in statistics Understanding central tendency properties in statistics Presentation Transcript

    • Understanding Central Tendency Properties in Statistics
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    • Numerical Data
      Properties
      RelativeStanding
      Central Tendency
      Variation
      Mean
      Range
      Percentiles
      Interquartile Range
      Median
      Z–scores
      Variance
      Mode
      Standard Deviation
      Numerical DataProperties & Measures
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    • Measure of central tendency
      Most common measure
      Acts as ‘balance point’
      Affected by extreme values (‘outliers’)
      Formula (sample mean)
      Mean
      n

      X
      i
      X
      X
      X





      n
      1
      2
      1
      i
      X


      n
      n
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    • Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7

      X
      X
      X
      X
      X
      X
      X





      1
      2
      3
      4
      5
      6

      1
      X


      n
      6





      10
      8
      9
      11
      6
      3
      .
      .
      .
      .

      6

      30
      .
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    • Numerical DataProperties & Measures
      Numerical Data
      Properties
      RelativeStanding
      Central
      Variation
      Tendency
      Percentiles
      Mean
      Range
      Median
      Interquartile Range
      Z–scores
      Mode
      Variance
      Standard Deviation
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    • Measure of central tendency
      Middle value in ordered sequence
      If n is odd, middle value of sequence
      If n is even, average of 2 middle values
      Position of median in sequence
      Not affected by extreme values
      Median

      n
      1
      Positioning
      Point

      2
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    • Raw Data: 24.1 22.6 21.5 23.7 22.6
      Ordered: 21.5 22.6 22.6 23.7 24.1
      Position: 1 2 3 4 5
      Median Example Odd-Sized Sample


      n
      1
      5
      1
      Positioning
      Point



      3
      0
      .
      2
      2
      Median

      22
      6
      .
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    • Median Example Even-Sized Sample
      Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
      Ordered: 4.9 6.3 7.78.9 10.3 11.7
      Position: 1 2 34 5 6


      n
      1
      6
      1
      Positioning
      Point



      3
      5
      .
      2
      2

      7
      7
      8
      9
      .
      .
      Median


      8
      30
      .
      2
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    • Numerical DataProperties & Measures
      Numerical Data
      Properties
      RelativeStanding
      Central
      Variation
      Tendency
      Range
      Mean
      Percentiles
      Interquartile Range
      Median
      Z–scores
      Mode
      Variance
      Standard Deviation
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    • Measure of central tendency
      Value that occurs most often
      Not affected by extreme values
      May be no mode or several modes
      May be used for quantitative or qualitative data
      Mode
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    • No ModeRaw Data: 10.3 4.9 8.9 11.7 6.3 7.7
      One ModeRaw Data: 6.3 4.9 8.9 6.3 4.9 4.9
      More Than 1 ModeRaw Data: 21 28 28 41 4343
      Mode Example
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    • Thinking Challenge
      You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11.
      Describe the stock pricesin terms of central tendency.
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    • Central Tendency Solution
      Mean
      n

      X
      i
      X
      X
      X




      1
      2
      8
      i

      1
      X


      n
      8







      17
      16
      21
      18
      13
      16
      12
      11

      8

      15
      5
      .
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    • Central Tendency Solution
      Median
      • Raw Data: 17 16 21 18 13 16 12 11
      • Ordered: 11 12 13 16 16 17 18 21
      • Position: 1 2 3 4 5 6 7 8


      n
      1
      8
      1



      4
      5
      .
      Positioning Point
      2
      2

      16
      16
      Median


      16
      2
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    • Mode
      Raw Data: 17 16 21 18 13 16 12 11
      Mode = 16
      Central Tendency Solution
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    • Summary of Central Tendency Measures
      Measure
      Formula
      Description
      Mean
      Balance Point
      
      X
      /
      n
      i
      Median
      (
      n
      +1)
      Middle Value
      Position
      2
      When Ordered
      Mode
      none
      Most Frequent
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