8.14. Aggregate Functions
Aggregate functions operate on a set of values to compute a single result.
Except for count()
, count_if()
, max_by()
, min_by()
and
approx_distinct()
, all of these aggregate functions ignore null values
and return null for no input rows or when all values are null. For example,
sum()
returns null rather than zero and avg()
does not include null
values in the count. The coalesce
function can be used to convert null into
zero.
Some aggregate functions such as array_agg()
produce different results
depending on the order of input values. This ordering can be specified by writing
an ORDER BY Clause within the aggregate function:
array_agg(x ORDER BY y DESC)
array_agg(x ORDER BY x, y, z)
General Aggregate Functions

arbitrary
(x) → [same as input] Returns an arbitrary nonnull value of
x
, if one exists.

array_agg
(x) → array<[same as input]> Returns an array created from the input
x
elements.

avg
(x) → double Returns the average (arithmetic mean) of all input values.

avg
(time interval type) → time interval type Returns the average interval length of all input values.

bool_and
(boolean) → boolean Returns
TRUE
if every input value isTRUE
, otherwiseFALSE
.

bool_or
(boolean) → boolean Returns
TRUE
if any input value isTRUE
, otherwiseFALSE
.

checksum
(x) → varbinary Returns an orderinsensitive checksum of the given values.

count
(*) → bigint Returns the number of input rows.

count
(x) → bigint Returns the number of nonnull input values.

count_if
(x) → bigint Returns the number of
TRUE
input values. This function is equivalent tocount(CASE WHEN x THEN 1 END)
.

every
(boolean) → boolean This is an alias for
bool_and()
.

geometric_mean
(x) → double Returns the geometric mean of all input values.

max_by
(x, y) → [same as x] Returns the value of
x
associated with the maximum value ofy
over all input values.

max_by
(x, y, n) → array<[same as x]> Returns
n
values ofx
associated with then
largest of all input values ofy
in descending order ofy
.

min_by
(x, y) → [same as x] Returns the value of
x
associated with the minimum value ofy
over all input values.

min_by
(x, y, n) → array<[same as x]> Returns
n
values ofx
associated with then
smallest of all input values ofy
in ascending order ofy
.

max
(x) → [same as input] Returns the maximum value of all input values.

max
(x, n) → array<[same as x]> Returns
n
largest values of all input values ofx
.

min
(x) → [same as input] Returns the minimum value of all input values.

min
(x, n) → array<[same as x]> Returns
n
smallest values of all input values ofx
.

reduce_agg
(inputValue T, initialState S, inputFunction(S, T, S), combineFunction(S, S, S)) → S Reduces all input values into a single value.
`inputFunction
will be invoked for each input value. In addition to taking the input value,inputFunction
takes the current state, initiallyinitialState
, and returns the new state.combineFunction
will be invoked to combine two states into a new state. The final state is returned:SELECT id, reduce_agg(value, 0, (a, b) > a + b, (a, b) > a + b) FROM ( VALUES (1, 2), (1, 3), (1, 4), (2, 20), (2, 30), (2, 40) ) AS t(id, value) GROUP BY id;  (1, 9)  (2, 90) SELECT id, reduce_agg(value, 1, (a, b) > a * b, (a, b) > a * b) FROM ( VALUES (1, 2), (1, 3), (1, 4), (2, 20), (2, 30), (2, 40) ) AS t(id, value) GROUP BY id;  (1, 24)  (2, 24000)
The state type must be a boolean, integer, floatingpoint, or date/time/interval.

set_agg
(x) → array<[same as input]> Returns an array created from the distinct input
x
elements.

set_union
(array(T)) > array(T) Returns an array of all the distinct values contained in each array of the input
Example:
SELECT set_union(elements) FROM ( VALUES ARRAY[1, 2, 3], ARRAY[2, 3, 4] ) AS t(elements);
Returns ARRAY[1, 2, 3, 4]

sum
(x) → [same as input] Returns the sum of all input values.
Bitwise Aggregate Functions

bitwise_and_agg
(x) → bigint Returns the bitwise AND of all input values in 2’s complement representation.

bitwise_or_agg
(x) → bigint Returns the bitwise OR of all input values in 2’s complement representation.
Map Aggregate Functions

histogram
(x) > map(K, bigint) Returns a map containing the count of the number of times each input value occurs.

map_agg
(key, value) > map(K, V) Returns a map created from the input
key
/value
pairs.

map_union
(x(K, V)) > map(K, V) Returns the union of all the input maps. If a key is found in multiple input maps, that key’s value in the resulting map comes from an arbitrary input map.

multimap_agg
(key, value) > map(K, array(V)) Returns a multimap created from the input
key
/value
pairs. Each key can be associated with multiple values.
Approximate Aggregate Functions

approx_distinct
(x) → bigint Returns the approximate number of distinct input values. This function provides an approximation of
count(DISTINCT x)
. Zero is returned if all input values are null.This function should produce a standard error of 2.3%, which is the standard deviation of the (approximately normal) error distribution over all possible sets. It does not guarantee an upper bound on the error for any specific input set.

approx_distinct
(x, e) → bigint Returns the approximate number of distinct input values. This function provides an approximation of
count(DISTINCT x)
. Zero is returned if all input values are null.This function should produce a standard error of no more than
e
, which is the standard deviation of the (approximately normal) error distribution over all possible sets. It does not guarantee an upper bound on the error for any specific input set. The current implementation of this function requires thate
be in the range of[0.0040625, 0.26000]
.

approx_percentile
(x, percentage) → [same as x] Returns the approximate percentile for all input values of
x
at the givenpercentage
. The value ofpercentage
must be between zero and one and must be constant for all input rows.

approx_percentile
(x, percentage, accuracy) → [same as x] As
approx_percentile(x, percentage)
, but with a maximum rank error ofaccuracy
. The value ofaccuracy
must be between zero and one (exclusive) and must be constant for all input rows. Note that a lower “accuracy” is really a lower error threshold, and thus more accurate. The default accuracy is0.01
.

approx_percentile
(x, percentages) → array<[same as x]> Returns the approximate percentile for all input values of
x
at each of the specified percentages. Each element of thepercentages
array must be between zero and one, and the array must be constant for all input rows.

approx_percentile
(x, percentages, accuracy) → array<[same as x]> As
approx_percentile(x, percentages)
, but with a maximum rank error ofaccuracy
.

approx_percentile
(x, w, percentage) → [same as x] Returns the approximate weighed percentile for all input values of
x
using the peritem weightw
at the percentagep
. The weight must be an integer value of at least one. It is effectively a replication count for the valuex
in the percentile set. The value ofp
must be between zero and one and must be constant for all input rows.

approx_percentile
(x, w, percentage, accuracy) → [same as x] As
approx_percentile(x, w, percentage)
, but with a maximum rank error ofaccuracy
.

approx_percentile
(x, w, percentages) → array<[same as x]> Returns the approximate weighed percentile for all input values of
x
using the peritem weightw
at each of the given percentages specified in the array. The weight must be an integer value of at least one. It is effectively a replication count for the valuex
in the percentile set. Each element of the array must be between zero and one, and the array must be constant for all input rows.

approx_percentile
(x, w, percentages, accuracy) → array<[same as x]> As
approx_percentile(x, w, percentages)
, but with a maximum rank error ofaccuracy
.

approx_set
(x) → HyperLogLog

merge
(x) → HyperLogLog

khyperloglog_agg
(x) → KHyperLogLog

merge
(qdigest(T)) > qdigest(T)

qdigest_agg
(x) → qdigest<[same as x]>

qdigest_agg
(x, w) → qdigest<[same as x]>

qdigest_agg
(x, w, accuracy) → qdigest<[same as x]>

numeric_histogram
(buckets, value, weight) → map<double, double> Computes an approximate histogram with up to
buckets
number of buckets for allvalue
s with a peritem weight ofweight
. The keys of the returned map are roughly the center of the bin, and the entry is the total weight of the bin. The algorithm is based loosely on [BenHaimTomTov2010].buckets
must be abigint
.value
andweight
must be numeric.

numeric_histogram
(buckets, value) → map<double, double> Computes an approximate histogram with up to
buckets
number of buckets for allvalue
s. This function is equivalent to the variant ofnumeric_histogram()
that takes aweight
, with a peritem weight of1
. In this case, the total weight in the returned map is the count of items in the bin.
Statistical Aggregate Functions

corr
(y, x) → double Returns correlation coefficient of input values.

covar_pop
(y, x) → double Returns the population covariance of input values.

covar_samp
(y, x) → double Returns the sample covariance of input values.

entropy
(c) → double Returns the log2 entropy of count inputvalues.
\[\mathrm{entropy}(c) = \sum_i \left[ {c_i \over \sum_j [c_j]} \log_2\left({\sum_j [c_j] \over c_i}\right) \right].\]c
must be abigint
column of nonnegative values.The function ignores any
NULL
count. If the sum of nonNULL
counts is 0, it returns 0.

kurtosis
(x) → double Returns the excess kurtosis of all input values. Unbiased estimate using the following expression:
\[\mathrm{kurtosis}(x) = {n(n+1) \over (n1)(n2)(n3)} { \sum[(x_i\mu)^4] \over \sigma^4} 3{ (n1)^2 \over (n2)(n3) },\]where \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

regr_intercept
(y, x) → double Returns linear regression intercept of input values.
y
is the dependent value.x
is the independent value.

regr_slope
(y, x) → double Returns linear regression slope of input values.
y
is the dependent value.x
is the independent value.

skewness
(x) → double Returns the skewness of all input values.

stddev
(x) → double This is an alias for
stddev_samp()
.

stddev_pop
(x) → double Returns the population standard deviation of all input values.

stddev_samp
(x) → double Returns the sample standard deviation of all input values.

variance
(x) → double This is an alias for
var_samp()
.

var_pop
(x) → double Returns the population variance of all input values.

var_samp
(x) → double Returns the sample variance of all input values.
Classification Metrics Aggregate Functions
The following functions each measure how some metric of a binary confusion matrix changes as a function of classification thresholds. They are meant to be used in conjunction.
For example, to find the precisionrecall curve, use
WITH recall_precision AS ( SELECT CLASSIFICATION_RECALL(10000, correct, pred) AS recalls, CLASSIFICATION_PRECISION(10000, correct, pred) AS precisions FROM classification_dataset ) SELECT recall, precision FROM recall_precision CROSS JOIN UNNEST(recalls, precisions) AS t(recall, precision)
To get the corresponding thresholds for these values, use
WITH recall_precision AS ( SELECT CLASSIFICATION_THRESHOLDS(10000, correct, pred) AS thresholds, CLASSIFICATION_RECALL(10000, correct, pred) AS recalls, CLASSIFICATION_PRECISION(10000, correct, pred) AS precisions FROM classification_dataset ) SELECT threshold, recall, precision FROM recall_precision CROSS JOIN UNNEST(thresholds, recalls, precisions) AS t(threshold, recall, precision)
To find the ROC curve, use
WITH fallout_recall AS ( SELECT CLASSIFICATION_FALLOUT(10000, correct, pred) AS fallouts, CLASSIFICATION_RECALL(10000, correct, pred) AS recalls FROM classification_dataset ) SELECT fallout recall, FROM recall_fallout CROSS JOIN UNNEST(fallouts, recalls) AS t(fallout, recall)

classification_miss_rate
(buckets, y, x, weight) → array<double> Computes the missrate with up to
buckets
number of buckets. Returns an array of missrate values.y
should be a boolean outcome value;x
should be predictions, each between 0 and 1;weight
should be nonnegative values, indicating the weight of the instance.The missrate is defined as a sequence whose \(j\)th entry is
\[{ \sum_{i \;\; x_i \leq t_j \bigwedge y_i = 1} \left[ w_i \right] \over \sum_{i \;\; x_i \leq t_j \bigwedge y_i = 1} \left[ w_i \right] + \sum_{i \;\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right] },\]where \(t_j\) is the \(j\)th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)th entries of
y
,x
, andweight
, respectively.

classification_miss_rate
(buckets, y, x) → array<double> This function is equivalent to the variant of
classification_miss_rate()
that takes aweight
, with a peritem weight of1
.

classification_fall_out
(buckets, y, x, weight) → array<double> Computes the fallout with up to
buckets
number of buckets. Returns an array of fallout values.y
should be a boolean outcome value;x
should be predictions, each between 0 and 1;weight
should be nonnegative values, indicating the weight of the instance.The fallout is defined as a sequence whose \(j\)th entry is
\[{ \sum_{i \;\; x_i > t_j \bigwedge y_i = 0} \left[ w_i \right] \over \sum_{i \;\; y_i = 0} \left[ w_i \right] },\]where \(t_j\) is the \(j\)th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)th entries of
y
,x
, andweight
, respectively.

classification_fall_out
(buckets, y, x) → array<double> This function is equivalent to the variant of
classification_fall_out()
that takes aweight
, with a peritem weight of1
.

classification_precision
(buckets, y, x, weight) → array<double> Computes the precision with up to
buckets
number of buckets. Returns an array of precision values.y
should be a boolean outcome value;x
should be predictions, each between 0 and 1;weight
should be nonnegative values, indicating the weight of the instance.The precision is defined as a sequence whose \(j\)th entry is
\[{ \sum_{i \;\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right] \over \sum_{i \;\; x_i > t_j} \left[ w_i \right] },\]where \(t_j\) is the \(j\)th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)th entries of
y
,x
, andweight
, respectively.

classification_precision
(buckets, y, x) → array<double> This function is equivalent to the variant of
classification_precision()
that takes aweight
, with a peritem weight of1
.

classification_recall
(buckets, y, x, weight) → array<double> Computes the recall with up to
buckets
number of buckets. Returns an array of recall values.y
should be a boolean outcome value;x
should be predictions, each between 0 and 1;weight
should be nonnegative values, indicating the weight of the instance.The recall is defined as a sequence whose \(j\)th entry is
\[{ \sum_{i \;\; x_i > t_j \bigwedge y_i = 1} \left[ w_i \right] \over \sum_{i \;\; y_i = 1} \left[ w_i \right] },\]where \(t_j\) is the \(j\)th smallest threshold, and \(y_i\), \(x_i\), and \(w_i\) are the \(i\)th entries of
y
,x
, andweight
, respectively.

classification_recall
(buckets, y, x) → array<double> This function is equivalent to the variant of
classification_recall()
that takes aweight
, with a peritem weight of1
.

classification_thresholds
(buckets, y, x) → array<double> Computes the thresholds with up to
buckets
number of buckets. Returns an array of threshold values.y
should be a boolean outcome value;x
should be predictions, each between 0 and 1.The thresholds are defined as a sequence whose \(j\)th entry is the \(j\)th smallest threshold.
Differential Entropy Functions
The following functions approximate the binary differential entropy. That is, for a random variable \(x\), they approximate
where \(f(x)\) is the partial density function of \(x\).

differential_entropy
(sample_size, x) Returns the approximate log2 differential entropy from a random variable’s sample outcomes. The function internally creates a reservoir (see [Black2015]), then calculates the entropy from the sample results by approximating the derivative of the cumulative distribution (see [Alizadeh2010]).
sample_size
(long
) is the maximal number of reservoir samples.x
(double
) is the samples.For example, to find the differential entropy of
x
ofdata
using 1000000 reservoir samples, useSELECT differential_entropy(1000000, x) FROM data
Note
If \(x\) has a known lower and upper bound, prefer the versions taking
(bucket_count, x, 1.0, "fixed_histogram_mle", min, max)
, or(bucket_count, x, 1.0, "fixed_histogram_jacknife", min, max)
, as they have better convergence.

differential_entropy
(sample_size, x, weight) Returns the approximate log2 differential entropy from a random variable’s sample outcomes. The function internally creates a weighted reservoir (see [Efraimidis2006]), then calculates the entropy from the sample results by approximating the derivative of the cumulative distribution (see [Alizadeh2010]).
sample_size
is the maximal number of reservoir samples.x
(double
) is the samples.weight
(double
) is a nonnegative double value indicating the weight of the sample.For example, to find the differential entropy of
x
with weightsweight
ofdata
using 1000000 reservoir samples, useSELECT differential_entropy(1000000, x, weight) FROM data
Note
If \(x\) has a known lower and upper bound, prefer the versions taking
(bucket_count, x, weight, "fixed_histogram_mle", min, max)
, or(bucket_count, x, weight, "fixed_histogram_jacknife", min, max)
, as they have better convergence.

differential_entropy
(bucket_count, x, weight, method, min, max) → double Returns the approximate log2 differential entropy from a random variable’s sample outcomes. The function internally creates a conceptual histogram of the sample values, calculates the counts, and then approximates the entropy using maximum likelihood with or without Jacknife correction, based on the
method
parameter. If Jacknife correction (see [Beirlant2001]) is used, the estimate is\[n H(x)  (n  1) \sum_{i = 1}^n H\left(x_{(i)}\right)\]where \(n\) is the length of the sequence, and \(x_{(i)}\) is the sequence with the \(i\)th element removed.
bucket_count
(long
) determines the number of histogram buckets.x
(double
) is the samples.method
(varchar
) is either'fixed_histogram_mle'
(for the maximum likelihood estimate) or'fixed_histogram_jacknife'
(for the jacknifecorrected maximum likelihood estimate).min
andmax
(bothdouble
) are the minimal and maximal values, respectively; the function will throw if there is an input outside this range.weight
(double
) is the weight of the sample, and must be nonnegative.For example, to find the differential entropy of
x
, each between0.0
and1.0
, with weights 1.0 ofdata
using 1000000 bins and jacknife estimates, useSELECT differential_entropy(1000000, x, 1.0, 'fixed_histogram_jacknife', 0.0, 1.0) FROM data
To find the differential entropy of
x
, each between2.0
and2.0
, with weightsweight
ofdata
using 1000000 buckets and maximumlikelihood estimates, useSELECT differential_entropy(1000000, x, weight, 'fixed_histogram_mle', 2.0, 2.0) FROM data
Note
If \(x\) doesn’t have known lower and upper bounds, prefer the versions taking
(sample_size, x)
(unweighted case) or(sample_size, x, weight)
(weighted case), as they use reservoir sampling which doesn’t require a known range for samples.Otherwise, if the number of distinct weights is low, especially if the number of samples is low, consider using the version taking
(bucket_count, x, weight, "fixed_histogram_jacknife", min, max)
, as jacknife bias correction, is better than maximum likelihood estimation. However, if the number of distinct weights is high, consider using the version taking(bucket_count, x, weight, "fixed_histogram_mle", min, max)
, as this will reduce memory and running time.
[Alizadeh2010]  (1, 2) Alizadeh Noughabi, Hadi & Arghami, N. (2010). “A New Estimator of Entropy”. 
[Beirlant2001]  Beirlant, Dudewicz, Gyorfi, and van der Meulen, “Nonparametric entropy estimation: an overview”, (2001) 
[BenHaimTomTov2010]  Yael BenHaim and Elad TomTov, “A streaming parallel decision tree algorithm”, J. Machine Learning Research 11 (2010), pp. 849–872. 
[Black2015]  Black, Paul E. (26 January 2015). “Reservoir sampling”. Dictionary of Algorithms and Data Structures. 
[Efraimidis2006]  Efraimidis, Pavlos S.; Spirakis, Paul G. (20060316). “Weighted random sampling with a reservoir”. Information Processing Letters. 97 (5): 181–185. 