SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
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
        

Privacy Policy/Security Notice
Disclaimer | FOIA

NIST is an agency of the U.S. Commerce Department.

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

Please email comments on this WWW page to alan.heckert.gov.