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The Percentile Extension
Table Of Contents

# 1. Overview

The percentile extension provides four aggregate functions that compute a percentile score and/or the median value for a distribution. The percentile extension is not in the SQLite core. It is not part of the amalgamation. The percentile extension is a loadable extension that must be compiled and linked separately from the core SQLite.

# 2. Obtaining and Compiling The Percentile Extension

The code for the percentile extension is found in the ext/misc/percentile.c file of the main SQLite source tree. It may be compiled into an SQLite loadable extension using a command like:

    gcc -g -fPIC -shared percentile.c -o percentile.so

Alternatively, the percentile.c file may be compiled into the application. In that case, the following function should be invoked to register the extension with each new database connection:

    int sqlite3_percentile_init(sqlite3 *db, void*, void*);

The first argument passed should be the database handle to register the extension with. The second and third arguments should both be passed 0.

Percentile is included in most builds of the command-line shell.

# 3. Aggregate Functions Implemented By Percentile

The percentile extension implements the aggregate SQL functions described below. The algorithms used by all of these functions use O(N) space and O(NlogN) time, where N is the number of non-NULL inputs.

## 3.1. The median(Y) aggregate function

The median(Y) function is an aggregate that computes the median value of all non-NULL inputs Y. If any Y input to median() is not NULL and is not a numeric value, an error is raised. If there are no non-NULL, numeric inputs, then the result of median() is NULL.

The median is the value of the center element when all the inputs are sorted and the number of inputs is odd. If there are an even number of inputs is even, then the median is the average of the two center inputs.

The median(Y) function is equivalent to percentile(Y,50).

## 3.2. The percentile(Y,P) aggregate function

The percentile(Y,P) aggregate function computes an answer X which is a value that is greater than or equal to P percent of the non-NULL inputs and which is less than or equal to 100-P percent of the inputs. The parameter P must be a number between 0.0 and 100.0. The value of P must be the same for all terms of the aggregate and may not be NULL. Y inputs must be either NULL or numeric. NULL values for Y are ignored. Any non-NULL Y input that is not numeric causes an error to be raised.

The percentile() function works by sorting the non-NULL inputs and then computing the input or inputs that are closest to P percent from the first to the last. The return value is the weighted average of the two closest inputs.

## 3.3. The percentile_cont(Y,P) aggregate function

The percentile_cont(Y,P) function works like percentile(Y,P) except that the P value spans the range of 0.0 to 1.0 instead of 0.0 to 100.0. Thus the result of percentile_cont(Y,P) is the same as percentile(Y,P*100).

The percentile_cont() function is defined by SQL standards. However, instead of being a simple function call "percentile_cont(Y,P)", the SQL-standard syntax goes like this:

    SELECT percentile_cont(P) WITHIN GROUP (ORDER BY Y) FROM tab;

That is a lot of syntax to mean exactly the same thing as:

    SELECT percentile_cont(Y,P) FROM tab;

SQLite will support the SQL-standard syntax, but only if it is compiled (from canonical sources, not from the amalgamation) using the -DSQLITE_ENABLE_ORDERED_SET_AGGREGATES=1 compile-time option. Without that compile-time option, only the simpler "percentile_cont(Y,P)" form is supported. Since there are no advantages to the prolix SQL-standard format, and dramatic readability disadvantages, and because the SQLITE_ENABLE_ORDERED_SET_AGGREGATES compile-time option causes the SQLite library to be larger, that option is omitted from most builds.

This author believes that the "_cont" suffix on this function name is an abbreviation for "continuous" and reflects the fact that the return value is a weighted average of the two closest input value to the actual percentile rank. The name is an SQL standard, not something choosen by the SQLite developers.

## 3.4. The percentile_disc(Y,P) aggregate function

The percentile_disc(Y,P) function works like percentile_cont(Y,P) except that instead of doing a weighted average of the closest available inputs, it always returns a value that is one of the input values - the smaller of the two possible choices. The percentile_disc(Y,P) function is defined by SQL standards. As with percentile_cont(), the prolix ordered-set aggregate syntax is required, but that syntax is only supported by SQLite when SQLite is compiled using the SQLITE_ENABLE_ORDERED_SET_AGGREGATES compile-time option.

This author believes that the "_disc" suffix on this function name is an abbreviation for "discrete". The name is an SQL standard, not something choosen by the SQLite developers.

# 4. Design Requirements

The following requirements define the percentile extension.

1. The percentile(Y,P) function is an aggregate function taking exactly two arguments.

2. If the P argument to percentile(Y,P) is not the same for every row in the aggregate then an error is thrown. The word "same" in the previous sentence means that the value differ by less than 0.001.

3. If the P argument to percentile(Y,P) evaluates to anything other than a number in the range of 0.0 to 100.0 inclusive then an error is thrown.

4. If any Y argument to percentile(Y,P) evaluates to a value that is not NULL and is not numeric then an error is thrown.

5. If any Y argument to percentile(Y,P) evaluates to plus or minus infinity then an error is thrown. (SQLite always interprets NaN values as NULL.)

6. Both Y and P in percentile(Y,P) can be arbitrary expressions, including CASE WHEN expressions.

7. The percentile(Y,P) aggregate is able to handle inputs of at least one million (1,000,000) rows.

8. If there are no non-NULL values for Y, then percentile(Y,P) returns NULL.

9. If there is exactly one non-NULL value for Y, the percentile(Y,P) returns the one Y value.

10. If there N non-NULL values of Y where N is two or more and the Y values are ordered from least to greatest and a graph is drawn from 0 to N-1 such that the height of the graph at J is the J-th Y value and such that straight lines are drawn between adjacent Y values, then the percentile(Y,P) function returns the height of the graph at P*(N-1)/100.

11. The percentile(Y,P) function always returns either a floating point number or NULL.

12. The percentile(Y,P) is implemented as a single C99 source-code file that compiles into a shared-library or DLL that can be loaded into SQLite using the sqlite3_load_extension() interface.

13. A separate median(Y) function is the equivalent percentile(Y,50).

14. A separate percentile_cont(Y,P) function is equivalent to percentile(Y,P/100.0). In other words, the fraction value in the second argument is in the range of 0 to 1 instead of 0 to 100.

15. A separate percentile_disc(Y,P) function is like percentile_cont(Y,P) except that instead of returning the weighted average of the nearest two input values, it returns the next lower value. So the percentile_disc(Y,P) will always return a value that was one of the inputs.

16. All of median(), percentile(Y,P), percentile_cont(Y,P) and percentile_disc(Y,P) can be used as window functions.