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In database management, an aggregate function or aggregation function is a function where multiple values are processed together to form a single summary statistic.
Common aggregate functions include:
Average (i.e., arithmetic mean)
Count
Maximum
Median
Minimum
Mode
Range
Sum
Others include:
Nanmean (mean ignoring NaN values, also known as "nil" or "null")
Stddev
Formally, an aggregate function takes as input a set, a multiset (bag), or a list from some input domain I and outputs an element of an output domain O. The input and output domains may be the same, such as for SUM, or may be different, such as for COUNT.
Aggregate functions occur commonly in numerous programming languages, in spreadsheets, and in relational algebra.
The listagg function, as defined in the SQL:2016 standard
aggregates data from multiple rows into a single concatenated string.
In the entity relationship diagram, aggregation is represented as seen in Figure 1 with a rectangle around the relationship and its entities to indicate that it is being treated as an aggregate entity.
Decomposable aggregate functions
Aggregate functions present a bottleneck, because they potentially require having all input values at once. In distributed computing, it is desirable to divide such computations into smaller pieces, and distribute the work, usually computing in parallel, via a divide and conquer algorithm.
Some aggregate functions can be computed by computing the aggregate for subsets, and then aggregating these aggregates; examples include COUNT, MAX, MIN, and SUM. In other cases the aggregate can be computed by computing auxiliary numbers for subsets, aggregating these auxiliary numbers, and finally computing the overall number at the end; examples include AVERAGE (tracking sum and count, dividing at the end) and RANGE (tracking max and min, subtracting at the end). In other cases the aggregate cannot be computed without analyzing the entire set at once, though in some cases approximations can be distributed; examples include DISTINCT COUNT (Count-distinct problem), MEDIAN, and MODE.
Such functions are called decomposable aggregation functions or decomposable aggregate functions. The simplest may be referred to as self-decomposable aggregation functions, which are defined as those functions f such that there is a merge operator
⋄
{\displaystyle \diamond }
such that
f
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⊎
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=
f
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⋄
f
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{\displaystyle f(X\uplus Y)=f(X)\diamond f(Y)}
where
⊎
{\displaystyle \uplus }
is the union of multisets (see monoid homomorphism).
For example, SUM:
SUM
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=
x
{\displaystyle \operatorname {SUM} ({x})=x}
, for a singleton;
SUM
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=
SUM
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+
SUM
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{\displaystyle \operatorname {SUM} (X\uplus Y)=\operatorname {SUM} (X)+\operatorname {SUM} (Y)}
, meaning that merge
⋄
{\displaystyle \diamond }
is simply addition.
COUNT:
COUNT
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x
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=
1
{\displaystyle \operatorname {COUNT} ({x})=1}
,
COUNT
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=
COUNT
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+
COUNT
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{\displaystyle \operatorname {COUNT} (X\uplus Y)=\operatorname {COUNT} (X)+\operatorname {COUNT} (Y)}
.
MAX:
MAX
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=
x
{\displaystyle \operatorname {MAX} ({x})=x}
,
MAX
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=
max
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MAX
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,
MAX
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{\displaystyle \operatorname {MAX} (X\uplus Y)=\max {\bigl (}\operatorname {MAX} (X),\operatorname {MAX} (Y){\bigr )}}
.
MIN:
MIN
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=
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{\textstyle \operatorname {MIN} ({x})=x}
,
MIN
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min
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,
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{\displaystyle \operatorname {MIN} (X\uplus Y)=\min {\bigl (}\operatorname {MIN} (X),\operatorname {MIN} (Y){\bigr )}}
.
Note that self-decomposable aggregation functions can be combined (formally, taking the product) by applying them separately, so for instance one can compute both the SUM and COUNT at the same time, by tracking two numbers.
More generally, one can define a decomposable aggregation function f as one that can be expressed as the composition of a final function g and a self-decomposable aggregation function h,
f
=
g
∘
h
,
f
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=
g
(
h
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)
{\displaystyle f=g\circ h,f(X)=g(h(X))}
. For example, AVERAGE=SUM/COUNT and RANGE=MAX−MIN.
In the MapReduce framework, these steps are known as InitialReduce (value on individual record/singleton set), Combine (binary merge on two aggregations), and FinalReduce (final function on auxiliary values), and moving decomposable aggregation before the Shuffle phase is known as an InitialReduce step,
Decomposable aggregation functions are important in online analytical processing (OLAP), as they allow aggregation queries to be computed on the pre-computed results in the OLAP cube, rather than on the base data. For example, it is easy to support COUNT, MAX, MIN, and SUM in OLAP, since these can be computed for each cell of the OLAP cube and then summarized ("rolled up"), but it is difficult to support MEDIAN, as that must be computed for every view separately.
Other decomposable aggregate functions
In order to calculate the average and standard deviation from aggregate data, it is necessary to have available for each group: the total of values (Σxi = SUM(x)), the number of values (N=COUNT(x)) and the total of squares of the values (Σxi2=SUM(x2)) of each groups.
AVG:
AVG
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=
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AVG
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∗
COUNT
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+
AVG
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∗
COUNT
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/
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+
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{\displaystyle \operatorname {AVG} (X\uplus Y)={\bigl (}\operatorname {AVG} (X)*\operatorname {COUNT} (X)+\operatorname {AVG} (Y)*\operatorname {COUNT} (Y){\bigr )}/{\bigl (}\operatorname {COUNT} (X)+\operatorname {COUNT} (Y){\bigr )}}
or
AVG
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=
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SUM
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+
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/
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+
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{\displaystyle \operatorname {AVG} (X\uplus Y)={\bigl (}\operatorname {SUM} (X)+\operatorname {SUM} (Y){\bigr )}/{\bigl (}\operatorname {COUNT} (X)+\operatorname {COUNT} (Y){\bigr )}}
or, only if COUNT(X)=COUNT(Y)
AVG
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⊎
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=
(
AVG
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+
AVG
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)
/
2
{\displaystyle \operatorname {AVG} (X\uplus Y)={\bigl (}\operatorname {AVG} (X)+\operatorname {AVG} (Y){\bigr )}/2}
SUM(x2):
The sum of squares of the values is important in order to calculate the Standard Deviation of groups
SUM
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2
⊎
Y
2
)
=
SUM
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+
SUM
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2
)
{\displaystyle \operatorname {SUM} (X^{2}\uplus Y^{2})=\operatorname {SUM} (X^{2})+\operatorname {SUM} (Y^{2})}
STDDEV:
For a finite population with equal probabilities at all points, we have
STDDEV
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=
s
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=
1
N
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N
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2
=
1
N
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=
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N
x
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−
(
x
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=
SUM
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/
COUNT
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−
AVG
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2
{\displaystyle \operatorname {STDDEV} (X)=s(x)={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}}={\sqrt {{\frac {1}{N}}\left(\sum _{i=1}^{N}x_{i}^{2}\right)-({\overline {x}})^{2}}}={\sqrt {\operatorname {SUM} (x^{2})/\operatorname {COUNT} (x)-\operatorname {AVG} (x)^{2}}}}
This means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value.
STDDEV
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⊎
Y
)
=
SUM
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2
⊎
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2
)
/
COUNT
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⊎
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)
−
AVG
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2
{\displaystyle \operatorname {STDDEV} (X\uplus Y)={\sqrt {\operatorname {SUM} (X^{2}\uplus Y^{2})/\operatorname {COUNT} (X\uplus Y)-\operatorname {AVG} (X\uplus Y)^{2}}}}
STDDEV
(
X
⊎
Y
)
=
(
SUM
(
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2
)
+
SUM
(
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2
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)
/
(
COUNT
(
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)
+
COUNT
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)
−
(
(
SUM
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+
SUM
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)
/
(
COUNT
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+
COUNT
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)
)
2
{\displaystyle \operatorname {STDDEV} (X\uplus Y)={\sqrt {{\bigl (}\operatorname {SUM} (X^{2})+\operatorname {SUM} (Y^{2}){\bigr )}/{\bigl (}\operatorname {COUNT} (X)+\operatorname {COUNT} (Y){\bigr )}-{\bigl (}(\operatorname {SUM} (X)+\operatorname {SUM} (Y))/(\operatorname {COUNT} (X)+\operatorname {COUNT} (Y)){\bigr )}^{2}}}}
See also
Cross-tabulation a.k.a. Contingency table
Data drilling
Data mining
Data processing
Extract, transform, load
Fold (higher-order function)
Group by (SQL), SQL clause
OLAP cube
Online analytical processing
Pivot table
Relational algebra
Utility functions on indivisible goods#Aggregates of utility functions
XML for Analysis
AggregateIQ
MapReduce
References
Literature
Grabisch, Michel; Marichal, Jean-Luc; Mesiar, Radko; Pap, Endre (2009). Aggregation functions. Encyclopedia of Mathematics and its Applications. Vol. 127. Cambridge: Cambridge University Press. ISBN 978-0-521-51926-7. Zbl 1196.00002.
Oracle Aggregate Functions: MAX, MIN, COUNT, SUM, AVG Examples
External links
Aggregate Functions (Transact-SQL)
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AGGREGATE function - Microsoft Support
Returns an aggregate in a list or database. The AGGREGATE function can apply different aggregate functions to a list or database with the option to ignore hidden rows and error values. AGGREGATE (function_num, options, ref1, [ref2], …) AGGREGATE (function_num, options, array, [k]) The AGGREGATE function syntax has the following arguments:
SQL Aggregate functions - GeeksforGeeks
Dec 10, 2024 · SQL Aggregate Functions are used to perform calculations on a set of rows and return a single value. They are often used with the GROUP BY clause in SQL to summarize data for each group. Commonly used aggregate functions include COUNT (), …
How to Use AGGREGATE Function in Excel (13 Examples)
Jun 16, 2024 · In this article, you will learn about the AGGREGATE function in Excel. We will explore its uses, aggregate data, with multiple criteria, combine aggregate with IF function, INDEX function, and aggregate vs subtotal.
Excel AGGREGATE function | Exceljet
The Excel AGGREGATE function returns an aggregate calculation like AVERAGE, COUNT, MAX, etc., optionally ignoring hidden rows and errors. A total of 19 operations are available, specified by function number in the first argument (see table for options).
Aggregate function - Wikipedia
In database management, an aggregate function or aggregation function is a function where multiple values are processed together to form a single summary statistic. (Figure 1) Entity relationship diagram representation of aggregation. Common aggregate functions include: Average (i.e., arithmetic mean) Count; Maximum; Median; Minimum; Mode ...
Aggregate Functions (Transact-SQL) - SQL Server | Microsoft Learn
May 23, 2023 · An aggregate function performs a calculation on a set of values, and returns a single value. Except for COUNT(*), aggregate functions ignore null values. Aggregate functions are often used with the GROUP BY clause of the SELECT statement. All …
SQL Aggregate Functions - W3Schools
SQL Aggregate Functions. An aggregate function is a function that performs a calculation on a set of values, and returns a single value. Aggregate functions are often used with the GROUP BY clause of the SELECT statement.
How to use the Excel AGGREGATE Function (Step-by-Step Guide)
Feb 3, 2025 · In this guide, you’ll learn how to use the AGGREGATE function in Microsoft Excel with real-world examples. We’ll cover how to sum data while ignoring errors, find the maximum value in a filtered list, and count non-empty cells without manual cleanup.
How to Use Conditional AGGREGATE Function in Excel
Jul 28, 2024 · AGGREGATE (14,6, (D5:D16)/ (B5:B16=C18),1) → It gives a value based on the specified function and conditions. 14 → It is the function_num argument. By this number, the LARGE function of Excel is declared. 6 → This refers to the options argument. This number represents the Ignore error values option.
How to use the Excel AGGREGATE function - ExcelFind.com
The Excel AGGREGATE function is a versatile tool for performing a variety of calculations like SUM, AVERAGE, COUNT, MAX, and more, with options to ignore errors, hidden rows, or nested subtotals. It is especially useful in complex datasets where certain elements need to be excluded from calculations.
