> ## Documentation Index
> Fetch the complete documentation index at: https://docs.truemath.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# Statistics

> Aggregation, dispersion, and combinatorics functions: sum, average, min/max, median, quartiles, standard deviation, variance, factorial, permutations, and combinations.

Statistics functions operate over a [table](/math/types/tables).

## Reading the table

Most take a single-column table, or a multi-column table plus a `column` index to operate on. The first column is index `1`, and when `column` is omitted it defaults to `1`; a `column` out of range returns an `Invalid array dimensions` error.

## Aggregation

*These read a table column — see [Reading the table](#reading-the-table).*

### Count \[count]

```
count(table)
```

The number of rows in `table`.

```
count([2; 3; 4])   => 3
```

### Sum \[sum]

```
sum(table)
sum(table; column)
```

The sum of the first column of `table`, or of the given `column`.

```
sum([3; 4; 5])                     => 12
sum([[3;10]; [4;20]; [5;30]]; 2)   => 60
```

### Average \[avg, mean, average]

```
avg(table)
avg(table; column)
```

The average of the values. `mean` and `average` are aliases.

```
avg([3; 4; 5])                     => 4
avg([[3;10]; [4;20]; [5;30]]; 2)   => 20
```

### Product \[product]

```
product(list)
```

The product of the values in a one-dimensional `list`.

## Range and extremes

*These read a table column — see [Reading the table](#reading-the-table).*

### Minimum \[min, minimum]

```
min(table)
min(table; column)
min(valueA; valueB; ...)
```

The minimum value. Given several equal-length single-column tables, returns the row-wise minimum as a table.

```
min([3; 4; 5; 9])                       => 3
min(3; 4; 5; 9)                         => 3
min([3;50]; [4;25]; [5;40]; [9;10])     => [3; 4; 5; 9]
```

### Maximum \[max, maximum]

```
max(table)
max(table; column)
max(valueA; valueB; ...)
```

The maximum value, with the same forms as `min`.

```
max([3; 4; 5; 9])                          => 9
max([[3;50]; [4;25]; [5;40]; [9;10]]; 2)   => 50
```

### Range \[range]

```
range(table)
range(table; column)
```

The difference between the maximum and minimum.

```
range([3; 4; 5; 9])   => 6
```

## Dispersion

*These read a table column — see [Reading the table](#reading-the-table).*

### Standard deviation \[stddev, stddevp]

```
stddev(table)
stddevp(table)
stddev(table; column)
stddevp(table; column)
```

`stddev` is the **sample** standard deviation; `stddevp` is the **population** standard deviation.

```
stddev([3; 4; 5; 6; 7; 8])    => 1.8708
stddevp([3; 4; 5; 6; 7; 8])   => 1.7078
```

### Variance \[variance, variancep]

```
variance(table)
variancep(table)
variance(table; column)
variancep(table; column)
```

`variance` is the **sample** variance; `variancep` is the **population** variance.

```
variance([3; 4; 5; 6; 7; 8])    => 3.5
variancep([3; 4; 5; 6; 7; 8])   => 2.9167
```

### Median and quartiles \[median, quartile1, quartile3]

```
median(table)
median(table; column)
quartile1(table)
quartile1(table; column)
quartile3(table)
quartile3(table; column)
```

The median, first quartile, and third quartile.

```
median([3; 4; 5; 5; 6])         => 5
quartile1([3; 4; 5; 6; 7; 8])   => 4
quartile3([3; 4; 5; 6; 7; 8])   => 7
```

## Sums of products

*These read a table column — see [Reading the table](#reading-the-table).*

### Sum of products \[sumofproducts]

```
sumofproducts(table)
sumofproducts(table; resultsAsTable)
```

Multiplies the items within each row, then sums those products. This is the operation for a weighted sum or dot product — a table of `(weight; value)` rows totals to `sumofproducts(table)`, where [`sum`](#sum-sum) reads only the first column. With `resultsAsTable` set to `true`, returns the per-row products as a table instead.

```
sumofproducts([[3;4]; [5;6]; [7;8]])         => 98
sumofproducts([[3;4]; [5;6]; [7;8]]; true)   => [12; 30; 56]
```

### Sum of squares \[sum2]

```
sum2(table)
sum2(table; column)
```

The sum of the squares of the values.

```
sum2([3; 4; 5; 6; 7; 8])   => 199
```

### Sum XY \[sumxy]

```
sumxy(table; x_column; y_column)
```

The sum of each row's `x_column` value multiplied by its `y_column` value.

```
sumxy([[3;10]; [4;20]; [5;20]; [6;30]; [7;40]; [8;50]]; 1; 2)   => 1070
```

## Probability

### Factorial \[fact]

```
fact(value)
```

The factorial of `value`, a whole number from 1 to 170. Values outside that range return a `Parameter out of range` error.

```
fact(5)          => 120
fact([2; 3; 4])  => [2; 6; 24]
```

### Permutations \[perm]

```
perm(n; r)
```

The number of permutations of `n` taken `r` at a time (order matters). `n` and `r` must be whole numbers with `n` ≥ `r` > 0.

```
perm(8; 3)            => 336
perm([8;7;6]; 3)      => [336; 210; 120]
```

### Combinations \[comb]

```
comb(n; r)
```

The number of combinations of `n` taken `r` at a time (order does not matter). `n` and `r` must be whole numbers with `n` ≥ `r` > 0.

```
comb(8; 3)            => 56
comb([8;7;6]; 3)      => [56; 35; 20]
```

### Random \[rand]

```
rand()
rand(length)
rand(lower; upper)
rand(lower; upper; length)
```

A random real number between 0 and 1 inclusive, or a random whole number between `lower` and `upper` inclusive. With `length`, returns a table of that many random values.