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Dyalog APL, (削除) 24 (削除ここまで) (削除) 23 (削除ここまで) (削除) 21 (削除ここまで) (削除) 20 (削除ここまで) 19 bytes

×ばつ(M←+/÷≢)

This defines an unnamed, monadic function train, which is equivalent to the following function.

{.5*⍨M(×ばつ⍨⍵)×ばつ(M←{(+/⍵)÷≢⍵})⍵}

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How it works

The code consists of several trains.

M←+/÷≢

This defines a monadic 3-train (fork) M that executes +/ (sum of all elements) and ≢ (length) for the right argument, then applies ÷ (division) to the results, returning the arithmetic mean of the input.

×ばつM

This is another fork that applies M to the right argument, repeats this a second time, and applies ×ばつ (product) to the results, returning μ².

×ばつ⍨-(×ばつM)

This is yet another fork that calculates the square of the arithmetic mean as explained before, applies ×ばつ⍨ (product with itself) to the right argument, and finally applies - (difference) to the results.

For input (x₁, ..., x_N), this function returns (x₁ - μ², ..., x_N - μ²).

*∘.5∘M

This composed function is applies M to its right argument, then *∘.5. The latter uses right argument currying to apply map input a to a*0.5 (square root of a).

(*∘.5∘M)(×ばつ⍨-(×ばつM))

Finally, we have this monadic 2-train (atop), which applies the right function first, then the left to its result, calculating the standard deviation as follows.

formula