The simplest kind of extended statistics tracks functional dependencies, a concept used in definitions of database normal forms. We say that column b is functionally dependent on column a if knowledge of the value of a is sufficient to determine the value of b, that is there are no two rows having the same value of a but different values of b. In a fully normalized database, functional dependencies should exist only on primary keys and superkeys. However, in practice many data sets are not fully normalized for various reasons; intentional denormalization for performance reasons is a common example. Even in a fully normalized database, there may be partial correlation between some columns, which can be expressed as partial functional dependency.

The existence of functional dependencies directly affects the accuracy of estimates in certain queries. If a query contains conditions on both the independent and the dependent column(s), the conditions on the dependent columns do not further reduce the result size; but without knowledge of the functional dependency, the query planner will assume that the conditions are independent, resulting in underestimating the result size.

To inform the planner about functional dependencies, ANALYZE can collect measurements of cross-column dependency. Assessing the degree of dependency between all sets of columns would be prohibitively expensive, so data collection is limited to those groups of columns appearing together in a statistics object defined with the dependencies option. It is advisable to create dependencies statistics only for column groups that are strongly correlated, to avoid unnecessary overhead in both ANALYZE and later query planning.

Here is an example of collecting functional-dependency statistics:

CREATE STATISTICS stts (dependencies) ON city, zip FROM zipcodes;

ANALYZE zipcodes;

SELECT stxname, stxkeys, stxddependencies
  FROM pg_statistic_ext join pg_statistic_ext_data on (oid = stxoid)
  WHERE stxname = 'stts';
 stxname | stxkeys |             stxddependencies             
---------+---------+------------------------------------------
 stts    | 1 5     | {"1 => 5": 1.000000, "5 => 1": 0.423130}
(1 row)

Here it can be seen that column 1 (zip code) fully determines column 5 (city) so the coefficient is 1.0, while city only determines zip code about 42% of the time, meaning that there are many cities (58%) that are represented by more than a single ZIP code.

When computing the selectivity for a query involving functionally dependent columns, the planner adjusts the per-condition selectivity estimates using the dependency coefficients so as not to produce an underestimate.

SELECT * FROM zipcodes WHERE city = 'San Francisco' AND zip = '94105';

SELECT * FROM zipcodes WHERE city = 'San Francisco' AND zip = '90210';

Single-column statistics store the number of distinct values in each column. Estimates of the number of distinct values when combining more than one column (for example, for GROUP BY a, b) are frequently wrong when the planner only has single-column statistical data, causing it to select bad plans.

To improve such estimates, ANALYZE can collect n-distinct statistics for groups of columns. As before, it's impractical to do this for every possible column grouping, so data is collected only for those groups of columns appearing together in a statistics object defined with the ndistinct option. Data will be collected for each possible combination of two or more columns from the set of listed columns.

Continuing the previous example, the n-distinct counts in a table of ZIP codes might look like the following:

CREATE STATISTICS stts2 (ndistinct) ON city, state, zip FROM zipcodes;

ANALYZE zipcodes;

SELECT stxkeys AS k, stxdndistinct AS nd
  FROM pg_statistic_ext join pg_statistic_ext_data on (oid = stxoid)
  WHERE stxname = 'stts2';
-[ RECORD 1 ]--------------------------------------------------------
k  | 1 2 5
nd | {"1, 2": 33178, "1, 5": 33178, "2, 5": 27435, "1, 2, 5": 33178}
(1 row)

This indicates that there are three combinations of columns that have 33178 distinct values: ZIP code and state; ZIP code and city; and ZIP code, city and state (the fact that they are all equal is expected given that ZIP code alone is unique in this table). On the other hand, the combination of city and state has only 27435 distinct values.

It's advisable to create ndistinct statistics objects only on combinations of columns that are actually used for grouping, and for which misestimation of the number of groups is resulting in bad plans. Otherwise, the ANALYZE cycles are just wasted.

Another type of statistics stored for each column are most-common value lists. This allows very accurate estimates for individual columns, but may result in significant misestimates for queries with conditions on multiple columns.

To improve such estimates, ANALYZE can collect MCV lists on combinations of columns. Similarly to functional dependencies and n-distinct coefficients, it's impractical to do this for every possible column grouping. Even more so in this case, as the MCV list (unlike functional dependencies and n-distinct coefficients) does store the common column values. So data is collected only for those groups of columns appearing together in a statistics object defined with the mcv option.

Continuing the previous example, the MCV list for a table of ZIP codes might look like the following (unlike for simpler types of statistics, a function is required for inspection of MCV contents):

CREATE STATISTICS stts3 (mcv) ON city, state FROM zipcodes;

ANALYZE zipcodes;

SELECT m.* FROM pg_statistic_ext join pg_statistic_ext_data on (oid = stxoid),
                pg_mcv_list_items(stxdmcv) m WHERE stxname = 'stts3';

 index |         values         | nulls | frequency | base_frequency 
-------+------------------------+-------+-----------+----------------
     0 | {Washington, DC}       | {f,f} |  0.003467 |        2.7e-05
     1 | {Apo, AE}              | {f,f} |  0.003067 |        1.9e-05
     2 | {Houston, TX}          | {f,f} |  0.002167 |       0.000133
     3 | {El Paso, TX}          | {f,f} |     0.002 |       0.000113
     4 | {New York, NY}         | {f,f} |  0.001967 |       0.000114
     5 | {Atlanta, GA}          | {f,f} |  0.001633 |        3.3e-05
     6 | {Sacramento, CA}       | {f,f} |  0.001433 |        7.8e-05
     7 | {Miami, FL}            | {f,f} |    0.0014 |          6e-05
     8 | {Dallas, TX}           | {f,f} |  0.001367 |        8.8e-05
     9 | {Chicago, IL}          | {f,f} |  0.001333 |        5.1e-05
   ...
(99 rows)

This indicates that the most common combination of city and state is Washington in DC, with actual frequency (in the sample) about 0.35%. The base frequency of the combination (as computed from the simple per-column frequencies) is only 0.0027%, resulting in two orders of magnitude under-estimates.

It's advisable to create MCV statistics objects only on combinations of columns that are actually used in conditions together, and for which misestimation of the number of groups is resulting in bad plans. Otherwise, the ANALYZE and planning cycles are just wasted.