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## 5.1 Reductions

The following operations take arrays as input and produce single output values. The boundary values for size 0 arrays are the unit with respect to the combination operation (min, max, sum, or product).

### 5.1.1 Minimum and maximum

real min(array[] real x)
The minimum value in x, or $$+\infty$$ if x is size 0.
Available since 2.0

int min(array[] int x)
The minimum value in x, or error if x is size 0.
Available since 2.0

real max(array[] real x)
The maximum value in x, or $$-\infty$$ if x is size 0.
Available since 2.0

int max(array[] int x)
The maximum value in x, or error if x is size 0.
Available since 2.0

### 5.1.2 Sum, product, and log sum of exp

int sum(array[] int x)
The sum of the elements in x, defined for $$x$$ of size $$N$$ by $\text{sum}(x) = \begin{cases} \sum_{n=1}^N x_n & \text{if} N > 0 \\[4pt] 0 & \text{if} N = 0 \end{cases}$
Available since 2.1

real sum(array[] real x)
The sum of the elements in x; see definition above.
Available since 2.0

real prod(array[] real x)
The product of the elements in x, or 1 if x is size 0.
Available since 2.0

real prod(array[] int x)
The product of the elements in x, $\text{product}(x) = \begin{cases} \prod_{n=1}^N x_n & \text{if} N > 0 \\[4pt] 1 & \text{if} N = 0 \end{cases}$
Available since 2.0

real log_sum_exp(array[] real x)
The natural logarithm of the sum of the exponentials of the elements in x, or $$-\infty$$ if the array is empty.
Available since 2.0

### 5.1.3 Sample mean, variance, and standard deviation

The sample mean, variance, and standard deviation are calculated in the usual way. For i.i.d. draws from a distribution of finite mean, the sample mean is an unbiased estimate of the mean of the distribution. Similarly, for i.i.d. draws from a distribution of finite variance, the sample variance is an unbiased estimate of the variance.2 The sample deviation is defined as the square root of the sample deviation, but is not unbiased.

real mean(array[] real x)
The sample mean of the elements in x. For an array $$x$$ of size $$N > 0$$, $\text{mean}(x) \ = \ \bar{x} \ = \ \frac{1}{N} \sum_{n=1}^N x_n.$ It is an error to the call the mean function with an array of size $$0$$.
Available since 2.0

real variance(array[] real x)
The sample variance of the elements in x. For $$N > 0$$, $\text{variance}(x) \ = \ \begin{cases} \frac{1}{N-1} \sum_{n=1}^N (x_n - \bar{x})^2 & \text{if } N > 1 \\[4pt] 0 & \text{if } N = 1 \end{cases}$ It is an error to call the variance function with an array of size 0.
Available since 2.0

real sd(array[] real x)
The sample standard deviation of elements in x. $\text{sd}(x) = \begin{cases} \sqrt{\, \text{variance}(x)} & \text{if } N > 1 \\[4pt] 0 & \text{if } N = 0 \end{cases}$ It is an error to call the sd function with an array of size 0.
Available since 2.0

### 5.1.4 Euclidean distance and squared distance

real distance(vector x, vector y)
The Euclidean distance between x and y, defined by $\text{distance}(x,y) \ = \ \sqrt{\textstyle \sum_{n=1}^N (x_n - y_n)^2}$ where N is the size of x and y. It is an error to call distance with arguments of unequal size.
Available since 2.2

real distance(vector x, row_vector y)
The Euclidean distance between x and y
Available since 2.2

real distance(row_vector x, vector y)
The Euclidean distance between x and y
Available since 2.2

real distance(row_vector x, row_vector y)
The Euclidean distance between x and y
Available since 2.2

real squared_distance(vector x, vector y)
The squared Euclidean distance between x and y, defined by $\mathrm{squared\_distance}(x,y) \ = \ \text{distance}(x,y)^2 \ = \ \textstyle \sum_{n=1}^N (x_n - y_n)^2,$ where N is the size of x and y. It is an error to call squared_distance with arguments of unequal size.
Available since 2.7

real squared_distance(vector x, row_vector y)
The squared Euclidean distance between x and y
Available since 2.26

real squared_distance(row_vector x, vector y)
The squared Euclidean distance between x and y
Available since 2.26

real squared_distance(row_vector x, row_vector y)
The Euclidean distance between x and y
Available since 2.26

### 5.1.5 Quantile

Produces sample quantiles corresponding to the given probabilities. The smallest observation corresponds to a probability of 0 and the largest to a probability of 1.

Implements algorithm 7 from Hyndman, R. J. and Fan, Y., Sample quantiles in Statistical Packages (R’s default quantile function).

real quantile(data array[] real x, data real p)
The p-th quantile of x
Available since 2.27

array[] real quantile(data array[] real x, data array[] real p)
An array containing the quantiles of x given by the array of probabilities p
Available since 2.27

1. Dividing by $$N$$ rather than $$(N-1)$$ produces a maximum likelihood estimate of variance, which is biased to underestimate variance.↩︎