## Sunday, March 23, 2014

### Statistics

Some more basic functions, this time focusing on sequences of numbers. First sum.
let sum t =
(*http://en.wikipedia.org/wiki/Sum

Summation is the operation of adding a sequence of numbers

*)
List.fold_left (fun acc x -> acc +. x) 0.0 t

Now, arithmetic_mean follows trivially.
let arithmetic_mean t =
(*http://en.wikipedia.org/wiki/Arithmetic_mean

Simply the mean or average when the context is clear, is the sum
of a collection of numbers divided by the number of numbers in the
collection.

*)
(sum t)/.(float_of_int (List.length t))

This formulation shows clearly the relationship between standard deviation and arithmetic mean.
let standard_deviation t =
(*http://en.wikipedia.org/wiki/Standard_deviation

For a finite set of numbers, the standard deviation is found by
taking the square root of the average of the squared differences
of the values from their average value.

*)
let av = arithmetic_mean t in
let squared_diffs = List.fold_left
(fun acc xi -> ((xi -. av) *. (xi -. av)) :: acc) [] t
in sqrt (arithmetic_mean squared_diffs)

The harmonic_mean function is, I find, a more interesting function.
let harmonic_mean t =
(*http://en.wikipedia.org/wiki/Harmonic_mean

The harmonic mean is the preferable method for averaging
multiples, such as the price/earning ratio, in which price is in
the numerator. If these ratios are averaged using an arithmetic
mean (a common error), high data points are given greater weights
than low data points. The harmonic mean, on the other hand, gives
equal weight to each data point.

*)
let n = float_of_int (List.length t) in
let denom=List.fold_left
(fun acc xi -> acc +. 1.0 /. xi) 0.0 t
in n /. denom