SAS Elementary Statistics Procedures 
Simple Statistics 
The base SAS procedures use a standardized set of keywords to refer to statistics. You specify these keywords in SAS statements to request the statistics to be displayed or stored in an output data set.
In the following notation, summation is over observations that contain nonmissing values of the analyzed variable and, except where shown, over nonmissing weights and frequencies of one or more:
is the nonmissing value of the analyzed variable for observation i.
is the frequency that is associated with if you use a FREQ statement. If you omit the FREQ statement, then for all i.
is the weight that is associated with if you use a WEIGHT statement. The base procedures automatically exclude the values of with missing weights from the analysis.
By default, the base procedures treat a negative weight as if it is equal to zero. However, if you use the EXCLNPWGT option in the PROC statement, then the procedure also excludes those values of with nonpositive weights. Note that most SAS/STAT procedures, such as PROC TTEST and PROC GLM, exclude values with nonpositive weights by default.
If you omit the WEIGHT statement, then for all i.
is the number of nonmissing values of , . If you use the EXCLNPWGT option and the WEIGHT statement, then is the number of nonmissing values with positive weights.
is the mean
is the variance
where is the variance divisor (the VARDEF= option) that you specify in the PROC statement. Valid values are as follows:When VARDEF=  equals . . . 
N 

DF 

WEIGHT 

WDF 

The default is DF.
is the standardized variable
The standard keywords and formulas for each statistic follow. Some formulas use keywords to designate the corresponding statistic.
Statistic  PROC MEANS and SUMMARY  PROC UNIVARIATE  PROC TABULATE  PROC REPORT  PROC CORR  PROC SQL  

Number of missing values  X  X  X  X 

X  
Number of nonmissing values  X  X  X  X  X  X  
Number of observations  X  X 



X  
Sum of weights  X  X  X  X  X  X  
Mean  X  X  X  X  X  X  
Sum  X  X  X  X  X  X  
Extreme values  X  X 





Minimum  X  X  X  X  X  X  
Maximum  X  X  X  X  X  X  
Range  X  X  X  X 

X  
Uncorrected sum of squares  X  X  X  X  X  X  
Corrected sum of squares  X  X  X  X  X  X  
Variance  X  X  X  X  X  X  
Covariance 




X 


Standard deviation  X  X  X  X  X  X  
Standard error of the mean  X  X  X  X 

X  
Coefficient of variation  X  X  X  X 

X  
Skewness  X  X  X 




Kurtosis  X  X  X 




Confidence Limits 







of the mean  X  X  X 




of the variance 

X 





of quantiles 

X 





Median  X  X  X  X  X 


Mode  X  X  X  X 



Percentiles/Deciles/Quartiles  X  X  X  X 



t test 








for mean=0  X  X  X  X 

X 

for mean= 

X 




Nonparametric tests for location 

X 





Tests for normality 

X 





Correlation coefficients 




X 


Cronbach's alpha 




X 

Descriptive Statistics 
The keywords for descriptive statistics are
is the sum of squares corrected for the mean, computed as
is the percent coefficient of variation, computed as
is the kurtosis, which measures heaviness of tails. When VARDEF=DF, the kurtosis is computed as
where is . The weighted kurtosis is computed asWhen VARDEF=N, the kurtosis is computed as
and the weighted kurtosis is computed as where is . The formula is invariant under the transformation . When you use VARDEF=WDF or VARDEF=WEIGHT, the kurtosis is set to missing.Note: PROC MEANS and PROC TABULATE do not compute weighted kurtosis.
is the maximum value of .
is the arithmetic mean .
is the minimum value of .
is the most frequent value of .
Note: When QMETHOD=P2, PROC REPORT, PROC MEANS, and PROC TABULATE do not compute MODE.
is the number of values that are not missing. Observations with less than one and equal to missing or (when you use the EXCLNPWGT option) are excluded from the analysis and are not included in the calculation of N.
is the number of values that are missing. Observations with less than one and equal to missing or (when you use the EXCLNPWGT option) are excluded from the analysis and are not included in the calculation of NMISS.
is the total number of observations and is calculated as the sum of N and NMISS. However, if you use the WEIGHT statement, then NOBS is calculated as the sum of N, NMISS, and the number of observations excluded because of missing or nonpositive weights.
is the range and is calculated as the difference between maximum value and minimum value.
is skewness, which measures the tendency of the deviations to be larger in one direction than in the other. When VARDEF=DF, the skewness is computed as
where is . The weighted skewness is computed asWhen VARDEF=N, the skewness is computed as
and the weighted skewness is computed as The formula is invariant under the transformation . When you use VARDEF=WDF or VARDEF=WEIGHT, the skewness is set to missing.Note: PROC MEANS and PROC TABULATE do not compute weighted skewness.
is the standard deviation s and is computed as the square root of the variance, .
is the standard error of the mean, computed as
when VARDEF=DF, which is the default. Otherwise, STDERR is set to missing.is the sum, computed as
is the sum of the weights, , computed as
is the uncorrected sum of squares, computed as
is the variance .
Quantile and Related Statistics 
The keywords for quantiles and related statistics are
is the middle value.
is the 1^{st} percentile.
is the 5^{th} percentile.
is the 10^{th} percentile.
is the 90^{th} percentile.
is the 95^{th} percentile.
is the 99^{th} percentile.
is the lower quartile (25^{th} percentile).
is the upper quartile (75^{th} percentile).
is interquartile range and is calculated as
When you use the WEIGHT statement, the tth percentile is computed as
where is the weight associated with and is the sum of the weights. When the observations have identical weights, the weighted percentiles are the same as the unweighted percentiles with QNTLDEF=5.QNTLDEF=  Description  Formula  

1  weighted average at 


where is taken to be  
2  observation numbered closest to 

if 

if and is even  

if and is odd  
where i is the integer part of  
3  empirical distribution function 

if 

if  
4  weighted average aimed at 


where is taken to be  
5  empirical distribution function with averaging 

if 

if 
Hypothesis Testing Statistics 
The keywords for hypothesis testing statistics are
is the Student's t statistic to test the null hypothesis that the population mean is equal to and is calculated as
By default, is equal to zero. You can use the MU0= option in the PROC UNIVARIATE statement to specify . You must use VARDEF=DF, which is the default variance divisor, otherwise T is set to missing.By default, when you use a WEIGHT statement, the procedure counts the values with nonpositive weights in the degrees of freedom. Use the EXCLNPWGT option in the PROC statement to exclude values with nonpositive weights. Most SAS/STAT procedures, such as PROC TTEST and PROC GLM automatically exclude values with nonpositive weights.
is the twotailed pvalue for Student's t statistic, T, with degrees of freedom. This value is the probability under the null hypothesis of obtaining a more extreme value of T than is observed in this sample.
Confidence Limits for the Mean 
The keywords for confidence limits are
is the twosided confidence limit for the mean. A twosided percent confidence interval for the mean has upper and lower limits
where is , is the ( ) critical value of the Student's t statistics with degrees of freedom, and is the value of the ALPHA= option which by default is 0.05. Unless you use VARDEF=DF, which is the default variance divisor, CLM is set to missing.is the onesided confidence limit below the mean. The onesided percent confidence interval for the mean has the lower limit
Unless you use VARDEF=DF, which is the default variance divisor, LCLM is set to missing.is the onesided confidence limit above the mean. The onesided percent confidence interval for the mean has the upper limit
Unless you use VARDEF=DF, which is the default variance divisor, UCLM is set to missing.Using Weights 
For more information on using weights and an example, see WEIGHT.
Data Requirements for Summarization Procedures 
The following are the minimal data requirements to compute unweighted statistics and do not describe recommended sample sizes. Statistics are reported as missing if VARDEF=DF (the default) and the following requirements are not met:
N and NMISS are computed regardless of the number of missing or nonmissing observations.
SUM, MEAN, MAX, MIN, RANGE, USS, and CSS require at least one nonmissing observation.
VAR, STD, STDERR, CV, T, PRT, and PROBT require at least two nonmissing observations.
SKEWNESS requires at least three nonmissing observations.
KURTOSIS requires at least four nonmissing observations.
SKEWNESS, KURTOSIS, T, PROBT, and PRT require that STD is greater than zero.
CV requires that MEAN is not equal to zero.
CLM, LCLM, UCLM, STDERR, T, PRT, and PROBT require that VARDEF=DF.
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