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Dataplot Vol 2 Vol 1

KURTOSIS

Name:
    KURTOSIS (LET)
Type:
    Let Subcommand
Purpose:
    Compute the kurtosis statistic of a variable.
Description:
    The kurtosis is the standardized fourth central moment. The formula is:

      \[ \mbox{kurtosis} = \frac{\sum_{i=1}^{N}(y_{i} - \bar{y})^{4}/n} {s^{4}} \]
    where \(\bar{y}\) is the mean, s is the standard deviation, and n is the number of data points. Note that in computing the kurtosis, the standard deviation is computed using n in the denominator rather than n - 1.

    Kurtosis is a measure of how heavy tailed a distribution is. A normal distribution has a kurtosis of 3. A kurtosis greater than 3 indicates that the data is heavy tailed (i.e., more spread out) relative to a normal distribution while a kurtosis value less than 3 indicates that the data is light tailed (i.e., more compressed) relative to a normal distribution.

    Some sources subtract 3 from the kurtosis value in order to make the kurtosis 0 for a normal distribution. We refer to this as the "excess kurtosis" statistic.

Syntax 1:
    LET <par> = KURTOSIS <y>       <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <par> is a parameter where the calculated kurtosis is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax returns the conventional kurtosis statistic (i.e., 3 is not subtracted).

Syntax 2:
    LET <par> = EXCESS KURTOSIS <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <par> is a parameter where the calculated kurtosis is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax returns the excess kurtosis statistic (i.e., 3 is subtracted).

Examples:
    LET A1 = KURTOSIS Y1
    LET A1 = KURTOSIS Y1 SUBSET Y1 > -2
    LET A1 = EXCESS KURTOSIS Y1
Note:
    Dataplot statistics can be used in a number of commands. For details, enter

Default:
    None
Synonyms:
    STANDARDIZED FOURTH CENTRAL MOMENT
    STANDARDIZED 4TH CENTRAL MOMENT
Related Commands:
    MEAN = Compute the mean of a variable.
    STANDARD DEVIATION = Compute the standard deviation of a variable.
    SKEWNESS = Compute the skewness of a variable.
    MEDIAN = Compute the median of a variable.
    RANGE = Compute the range of a variable.
Applications:
    Distributional Analysis
Implementation Date:
    Pre-1987
    2014/12: Added EXCESS KURTOSIS statistic
    2014/12: Changed from N-1 to N in the formula
Program:
     
    LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 100
    LET Y2 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 100
    LET Y3 = SLASH RANDOM NUMBERS FOR I = 1 1 100
    LET Y4 = CAUCHY RANDOM NUMBERS FOR I = 1 1 100
    LET A1 = KURTOSIS Y1
    LET A2 = KURTOSIS Y2
    LET A3 = KURTOSIS Y3
    LET A4 = KURTOSIS Y4
    SET WRITE DECIMALS 3
    PRINT A1 A2 A3 A4
        
    The following output is generated
     
     PARAMETERS AND CONSTANTS--
    
        A1      --          4.217
        A2      --          3.948
        A3      --         29.870
        A4      --         52.326
        

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Date created: 12/14/2014
Last updated: 12/14/2014

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