KURTOSIS

KURTOSIS (LET)

Let Subcommand

Compute the kurtosis statistic of a variable.

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.

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).

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).

LET A1 = KURTOSIS Y1
LET A1 = KURTOSIS Y1 SUBSET Y1 > -2
LET A1 = EXCESS KURTOSIS Y1

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

STANDARDIZED FOURTH CENTRAL MOMENT
STANDARDIZED 4TH CENTRAL MOMENT

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.

Distributional Analysis

Pre-1987
2014/12: Added EXCESS KURTOSIS statistic
2014/12: Changed from N-1 to N in the formula

 
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