Understanding central tendency properties in statistics

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  • 1. Understanding Central Tendency Properties in Statistics
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  • 2. 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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  • 3. 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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  • 4. 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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  • 5. 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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  • 6. 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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  • 7. 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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  • 8. 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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  • 9. 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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  • 10. 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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  • 11. 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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  • 12. 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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  • 13. 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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  • 14. Central Tendency Solution
    Median
    • Raw Data: 17 16 21 18 13 16 12 11
    • 15. Ordered: 11 12 13 16 16 17 18 21
    • 16. 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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  • 17. Mode
    Raw Data: 17 16 21 18 13 16 12 11
    Mode = 16
    Central Tendency Solution
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  • 18. 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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