Kurtosis
SAS
g1g2.sas;
• data EDA; infile
'C:\Users\Vati\Documents\StatData\EDA.d
at'; input Y;
• proc means mean skewness kurtosis N;
var Y; run;
Analysis Variable : Y
Mean
Skewness
72.5104167 0.5255689
Kurtosis
0.0323668
N
96
Standardize the Scores
• PROC STANDARD data=eda mean=0
std=1 out=z_scores; run;
• proc means mean skewness kurtosis N;
var Y; run;
Analysis Variable : Y
Mean
Skewness
-4.09395E-16 0.5255689
Kurtosis
0.0323668
N
96
Note: No change in values of Skewness and
Kurtosis.
Compute z3 & z4
• data z34; set z_scores;
• Z3=Y**3; Z4=Y**4;
• proc means data=z34 noprint; var Z3 Z4;
output out=sumZ34 N=N N4 sum=sumZ3
sumZ4; run;
Compute g1 & g2
• data skew; set sumz34; G1=N/(n-1)/(n2)*sumZ3;
• G2=N*(n+1)/(n-1)/(n-2)/(n-3)*sumZ4 3*(n-1)*(n-1)/(n-2)/(n-3); proc print; run;
N
96
N4
96
sumZ3
48.8889
sumZ4
279.103
G1
G2
0.52557 0.032367
Same values as computed with Proc Means.
Create Large Uniform Sample
• DATA uniform; DROP N; DO N=1 TO
500000; X=UNIFORM(0); OUTPUT; END;
• PROC MEANS mean std skewness
kurtosis; VAR X; run;
Analysis Variable : X
Mean
Std Dev
0.5002771
0.2888593
Skewness
0.0012401
Kurtosis
-1.2009090
Skewness and Kurtosis have their expected values, 0 and -1.2.
*Kurtosis-T.sas;
***************************************************************
options formdlim='-' pageno=min nodate;
TITLE 'T ON 9 DF, T COMPUTED ON EACH OF
500,000 SAMPLES';
TITLE2 'EACH WITH 10 SCORES FROM A
STANDARD NORMAL POPULATION'; run;
DATA T9; DROP N; DO SAMPLE=1 TO 500000;
DO N=1 TO 10; X=NORMAL(0);
OUTPUT; END; END;
PROC MEANS NOPRINT; OUTPUT OUT=TS T=T; VAR
X; BY SAMPLE;
PROC MEANS MEAN STD N KURTOSIS; VAR T; run;
Student’s t on 9 df
Analysis Variable : T
Mean
Std Dev N
Kurtosis
-0.0007 1.13545 500000 1.23472
DATA T16; DROP N; DO SAMPLE=1 TO 500000; DO
N=1 TO 17; X=NORMAL(0);
OUTPUT; END; END;
PROC MEANS NOPRINT; OUTPUT OUT=TS T=T; VAR
X; BY SAMPLE;
TITLE 'T ON 16 DF, SAMPLING DISTRIBUTION OF
500,000 TS'; run;
PROC MEANS MEAN STD N KURTOSIS; VAR T; run;
Analysis Variable : T
Mean
Std Dev N
0.000424 1.06708 500000
Kurtosis
0.47135
DATA T28; DROP N; DO SAMPLE=1 TO 500000; DO
N=1 TO 29; X=NORMAL(0);
OUTPUT; END; END;
PROC MEANS NOPRINT; OUTPUT OUT=TS T=T; VAR
X; BY SAMPLE;
TITLE 'T ON 28 DF, SAMPLING DISTRIBUTION OF
500,000 TS'; run;
PROC MEANS MEAN STD N KURTOSIS; VAR T; run;
Analysis Variable : T
Mean
Std Dev N
0.000321 1.03727 500000
Kurtosis
0.24623
*Kurtosis_Beta2.sas;
*Illustrates the computation of population kurtosis;
*Using data from the handout Skewness, Kurtosis, and
the Normal Curve;
data A; do s=1 to 20; X=5; output; X=15; output; end;
*Creates distribution with 20 scores of 5 and 20 of
15;
data ZA; set A; Z=(X-10)/5; Z4A=Z**4;
*Changes score to z scores and then raises them to
the 4th power;
proc means mean; var Z4A; run;
*Finds the mean (aka ‘expected value’) of z**4;
Analysis Variable: Z4A
2
2
Mean
1.0000000
 = 1,  =
2 -3 = -2
• The rest of the program uses the same
definition of 2 to verify the values of 2
reported in the handout.
*Kurtosis-Normal.sas;
**************************************************************************
options formdlim='-' pageno=min nodate;
TITLE 'Skewness and Kurtosis for 100,000 Samples of 1000
Scores Drawn From a Normal (0,1) Distribution'; run;
DATA normal; DROP N; DO SAMPLE=1 TO 100000; DO N=1 TO
1000; X=NORMAL(0);
OUTPUT; END; END;
PROC MEANS NOPRINT; OUTPUT OUT=SK_KUR
SKEWNESS=SKEWNESS KURTOSIS=KURTOSIS; VAR X; BY
SAMPLE;
PROC MEANS MEAN STD N; VAR skewness kurtosis; run;