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KURTOSIS_POP and KURTOSIS_SAMP Analytic Functions

This article gives an overview of the KURTOSIS_POP and KURTOSIS_SAMP analytic functions. If you are new to analytic functions you should probably read this introduction to analytic functions first.

Related articles.

Basics

You can read a basic introduction to kurtosis here.

The KURTOSIS_POP and KURTOSIS_SAMP aggregate and analytic functions were added in Oracle 21c describe the "tailedness" or shape of a probability distribution. In both cases they return a numeric value with the following meaning.

The KURTOSIS_POP and KURTOSIS_SAMP functions will not return the same results, but both should be representative of the distribution of data. The larger the data set, the more similar their results will be.

Setup

We need some data with various distribution patterns to measure using the KURTOSIS_POP and KURTOSIS_SAMP aggregate functions. The following table contains columns that represent skewed data and a normal distribution.

-- drop table t1 purge;

create table t1 (
  id      number generated always as identity,
  skew    number,
  normal  number
);


insert into t1 (skew, normal)
select case
         when level > 9800 then 1
         else dbms_random.value(400, 500)
       end,
       dbms_random.normal
from   dual
connect by level <= 10000;
commit;

We will use the following table to demonstrate the use of the KURTOSIS_POP and KURTOSIS_SAMP analytic functions. The sample size is too small, but the EMP table will be familiar to people practicing analytic function using the articles on this site.

create table emp (
  empno    number(4) constraint pk_emp primary key,
  ename    varchar2(10),
  job      varchar2(9),
  mgr      number(4),
  hiredate date,
  sal      number(7,2),
  comm     number(7,2),
  deptno   number(2) );

insert into emp values (7369,'SMITH','CLERK',7902,to_date('17-12-1980','dd-mm-yyyy'),800,NULL,20);
insert into emp values (7499,'ALLEN','SALESMAN',7698,to_date('20-2-1981','dd-mm-yyyy'),1600,300,30);
insert into emp values (7521,'WARD','SALESMAN',7698,to_date('22-2-1981','dd-mm-yyyy'),1250,500,30);
insert into emp values (7566,'JONES','MANAGER',7839,to_date('2-4-1981','dd-mm-yyyy'),2975,NULL,20);
insert into emp values (7654,'MARTIN','SALESMAN',7698,to_date('28-9-1981','dd-mm-yyyy'),1250,1400,30);
insert into emp values (7698,'BLAKE','MANAGER',7839,to_date('1-5-1981','dd-mm-yyyy'),2850,NULL,30);
insert into emp values (7782,'CLARK','MANAGER',7839,to_date('9-6-1981','dd-mm-yyyy'),2450,NULL,10);
insert into emp values (7788,'SCOTT','ANALYST',7566,to_date('13-JUL-87','dd-mm-rr')-85,3000,NULL,20);
insert into emp values (7839,'KING','PRESIDENT',NULL,to_date('17-11-1981','dd-mm-yyyy'),5000,NULL,10);
insert into emp values (7844,'TURNER','SALESMAN',7698,to_date('8-9-1981','dd-mm-yyyy'),1500,0,30);
insert into emp values (7876,'ADAMS','CLERK',7788,to_date('13-JUL-87', 'dd-mm-rr')-51,1100,NULL,20);
insert into emp values (7900,'JAMES','CLERK',7698,to_date('3-12-1981','dd-mm-yyyy'),950,NULL,30);
insert into emp values (7902,'FORD','ANALYST',7566,to_date('3-12-1981','dd-mm-yyyy'),3000,NULL,20);
insert into emp values (7934,'MILLER','CLERK',7782,to_date('23-1-1982','dd-mm-yyyy'),1300,NULL,10);
commit;

KURTOSIS_POP and KURTOSIS_SAMP as Aggregate Functions

The following query uses the KURTOSIS_POP aggregate function to display the "tailedness" of the data in the SKEW and NORMAL columns.

select kurtosis_pop(skew) as skew,
       kurtosis_pop(normal) as normal
from   t1;

      SKEW     NORMAL
---------- ----------
30.0819301 .0453792598

SQL>

As expected the SKEW column returns a non-zero value, and the NORMAL column returns a near-zero value

The KURTOSIS_POP function uses a sample size of 100% of the rows, which can represent an overhead for large data sets. In contrast the KURTOSIS_SAMP function uses a smaller sample size, making it more efficient for large data sets, whilst still returning representative results.

select kurtosis_samp(skew) as skew,
       kurtosis_samp(normal) as normal
from   t1;

      SKEW     NORMAL
---------- ----------
30.0975767 .0460021977

SQL>

Using DISTINCT or UNIQUE keywords mean only unique values in the expression are used for the calculation. The ALL keyword is that same as the default action.

select kurtosis_samp(distinct skew) as skew_distinct,
       kurtosis_samp(unique skew) as skew_unique,
       kurtosis_samp(all skew) as skew_all,
       kurtosis_samp(skew) as skew
from   t1;

SKEW_DISTINCT SKEW_UNIQUE   SKEW_ALL       SKEW
------------- ----------- ---------- ----------
   4.44000461  4.44000461 30.0975767 30.0975767

SQL>

SQL>

The DISTINCT, UNIQUE and ALL keywords are also available for the analytic functions.

KURTOSIS_POP Analytic Function

Using an empty OVER clause turns the KURTOSIS_POP function into an analytic function. The lack of a partitioning clause means the whole result set is treated as a single partition. The following query uses the KURTOSIS_POP analytic function to display the "tailedness" of the data in the SAL column, as well as all the original data.

select empno,
       ename,
       deptno,
       sal,
       round(kurtosis_pop(sal) over (),2) as sal_kurtosis 
from   emp;

     EMPNO ENAME          DEPTNO        SAL SAL_KURTOSIS
---------- ---------- ---------- ---------- ------------
      7369 SMITH              20        800          .49
      7499 ALLEN              30       1600          .49
      7521 WARD               30       1250          .49
      7566 JONES              20       2975          .49
      7654 MARTIN             30       1250          .49
      7698 BLAKE              30       2850          .49
      7782 CLARK              10       2450          .49
      7788 SCOTT              20       3000          .49
      7839 KING               10       5000          .49
      7844 TURNER             30       1500          .49
      7876 ADAMS              20       1100          .49
      7900 JAMES              30        950          .49
      7902 FORD               20       3000          .49
      7934 MILLER             10       1300          .49

SQL>

Adding the partitioning clause allows us to display the salary kurtosis per department, along with the employee data for each department.

select empno,
       ename,
       deptno,
       sal,
       round(kurtosis_pop(sal) over (partition by deptno),2) as sal_kurtosis_by_dept
from   emp;

     EMPNO ENAME          DEPTNO        SAL SAL_KURTOSIS_BY_DEPT
---------- ---------- ---------- ---------- --------------------
      7782 CLARK              10       2450                    0
      7839 KING               10       5000                    0
      7934 MILLER             10       1300                    0
      7566 JONES              20       2975                -1.77
      7902 FORD               20       3000                -1.77
      7876 ADAMS              20       1100                -1.77
      7369 SMITH              20        800                -1.77
      7788 SCOTT              20       3000                -1.77
      7521 WARD               30       1250                  .46
      7844 TURNER             30       1500                  .46
      7499 ALLEN              30       1600                  .46
      7900 JAMES              30        950                  .46
      7698 BLAKE              30       2850                  .46
      7654 MARTIN             30       1250                  .46

SQL>

KURTOSIS_SAMP Analytic Function

Using an empty OVER clause turns the KURTOSIS_SAMP function into an analytic function. The lack of a partitioning clause means the whole result set is treated as a single partition. The following query uses the KURTOSIS_SAMP analytic function to display the "tailedness" of the data in the SAL column, as well as all the original data.

select empno,
       ename,
       deptno,
       sal,
       round(kurtosis_samp(sal) over (),2) as sal_kurtosis 
from   emp;

     EMPNO ENAME          DEPTNO        SAL SAL_KURTOSIS
---------- ---------- ---------- ---------- ------------
      7369 SMITH              20        800         1.32
      7499 ALLEN              30       1600         1.32
      7521 WARD               30       1250         1.32
      7566 JONES              20       2975         1.32
      7654 MARTIN             30       1250         1.32
      7698 BLAKE              30       2850         1.32
      7782 CLARK              10       2450         1.32
      7788 SCOTT              20       3000         1.32
      7839 KING               10       5000         1.32
      7844 TURNER             30       1500         1.32
      7876 ADAMS              20       1100         1.32
      7900 JAMES              30        950         1.32
      7902 FORD               20       3000         1.32
      7934 MILLER             10       1300         1.32

SQL>

Adding the partitioning clause allows us to display the salary kurtosis per department, along with the employee data for each department.

select empno,
       ename,
       deptno,
       sal,
       round(kurtosis_samp(sal) over (partition by deptno),2) as sal_kurtosis_by_dept
from   emp;

     EMPNO ENAME          DEPTNO        SAL SAL_KURTOSIS_BY_DEPT
---------- ---------- ---------- ---------- --------------------
      7782 CLARK              10       2450                    0
      7839 KING               10       5000                    0
      7934 MILLER             10       1300                    0
      7566 JONES              20       2975                 -3.1
      7902 FORD               20       3000                 -3.1
      7876 ADAMS              20       1100                 -3.1
      7369 SMITH              20        800                 -3.1
      7788 SCOTT              20       3000                 -3.1
      7521 WARD               30       1250                 3.84
      7844 TURNER             30       1500                 3.84
      7499 ALLEN              30       1600                 3.84
      7900 JAMES              30        950                 3.84
      7698 BLAKE              30       2850                 3.84
      7654 MARTIN             30       1250                 3.84

SQL>

Quick Links

The "*" indicates the function supports the full analytic syntax, including the windowing clause.

AVG * BIT_AND_AGG * BIT_OR_AGG * BIT_XOR_AGG * CHECKSUM *
CLUSTER_DETAILS CLUSTER_DISTANCE CLUSTER_ID CLUSTER_SET CORR *
COUNT * COVAR_POP * COVAR_SAMP * CUME_DIST DENSE_RANK
FEATURE_DETAILS FEATURE_ID FEATURE_SET FEATURE_VALUE FIRST
FIRST_VALUE * KURTOSIS_POP * KURTOSIS_SAMP * LAG LAST
LAST_VALUE * LEAD LISTAGG MATCH_RECOGNIZE MAX *
MEDIAN MIN * NTH_VALUE * NTILE PERCENT_RANK
PERCENTILE_CONT PERCENTILE_DISC PREDICTION PREDICTION_COST PREDICTION
PREDICTION_COST PREDICTION_DETAILS PREDICTION_PROBABILITY PREDICTION_SET RANK
RATIO_TO_REPORT REGR_ (Linear Regression) Functions * ROW_NUMBER SKEWNESS_POP * SKEWNESS_SAMP *
STDDEV * STDDEV_POP * STDDEV_SAMP * SUM * VAR_POP *
VAR_SAMP * VARIANCE * String Aggregation Top-N Queries  

For more information see:

Hope this helps. Regards Tim...

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