# Central Moment

## Introduction

The central moment is a statistical measure that describes the shape of a probability distribution. It measures the deviation of values in a dataset from the mean, and is often used in fields such as physics, engineering, and finance.

## Implementation

```
(* Calculate Central Moment *)
let central_moment n x =
let m = float_of_int n in
let u = mean x in
let x = Array.map (fun x -> (x -. u) ** m) x in
let a = Array.fold_left ( +. ) 0. x in
a /. float_of_int (Array.length x)
```

The OCaml code implements an algorithm to calculate the nth central moment of a dataset given as an array.

## Step-by-step Explanation

- The function
`central_moment`

takes two parameters - ‘n’ and an array ‘x’. - ‘n’ is the order of the moment to be calculated.
- The mean of the array ‘x’ is calculated using the built-in ‘mean’ function and assigned to ‘u’.
- The values in the array ‘x’ are then modified to represent their deviation from the mean: each value is subtracted by ‘u’ and then raised to the ‘n’th power.
- The resulting array is assigned to ‘x’.
- The sum of all values in ‘x’ is calculated using the built-in ‘fold_left’ function, and assigned to ‘a’.
- The value of ‘a’ is divided by the length of ‘x’ to determine the average of the modified values, which represents the nth central moment.

## Complexity Analysis

The code has a time complexity of O(n), where ‘n’ is the length of the input array ‘x’. This is due to the need to iterate over the entire array twice - once to calculate the mean, and again to modify the deviation of each value from the mean. The built-in ‘fold_left’ function is also O(n) in complexity. The space complexity of the algorithm is proportional to the size of the input array ‘x’, and is therefore O(n) as well.