Putting a Number on the Tail

Saying a distribution "has a long right tail" is useful, but two datasets can both lean right by very different amounts. The skewness coefficient turns the lean into a single number. For values $x_1, \dots, x_n$ with mean $\bar{x}$ :

g = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{3/2}}

The numerator cubes each distance from the mean. Cubing keeps the sign, so a value far above the mean pushes $g$ up and a value far below pushes it down, and it lets the farthest values dominate.
The denominator is the population standard deviation $\sigma$ raised to the power $3$ . That makes $g$ unit-free: the same data recorded in seconds instead of minutes gives the same $g$ .

Task: write skewness(values) that returns a tuple (g, tail):

g is the coefficient above, rounded to 4 decimal places.
tail names the direction of the longer tail, decided from the rounded g: "right" if g > 0, "left" if g < 0, and "symmetric" if g is exactly 0.
If every value is identical there is no spread to divide by: return (0.0, "symmetric").

values is a non-empty list of numbers in any order.

Putting a Number on the Tail

Examples

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