Skewness and excess kurtosis

Skewness measures asymmetry and excess kurtosis measures tail heaviness relative to the normal distribution (normal = 0 for both).

g_1 = \frac{m_3}{m_2^{3/2}}, \qquad G_1 = g_1\,\frac{\sqrt{n(n-1)}}{n-2}

Skewness: G1 is the engine's default (Excel SKEW).

$m_k$: $k$ -th central moment $\frac{1}{n}\sum (x_i - \xbar)^k$ .
$g_1$: Moment (unadjusted) skewness.
$G_1$: Adjusted sample skewness (needs $n \ge 3$ ).

g_2 = \frac{m_4}{m_2^{2}} - 3, \qquad G_2 = \frac{n-1}{(n-2)(n-3)}\bigl[(n+1)\,g_2 + 6\bigr]

Excess kurtosis: G2 is the engine's default (Excel KURT).

$g_2$: Moment excess kurtosis (normal = 0).
$G_2$: Adjusted sample excess kurtosis (needs $n \ge 4$ ).

When to use

Describing shape numerically next to a histogram: incomes and claim sizes are right-skewed; asset returns often have heavy tails.
Checking whether normal-based methods or the 68–95 rule are reasonable.

How to interpret

Skewness > 0: long right tail (mean usually above median); < 0: long left tail; about 0: roughly symmetric.
Excess kurtosis > 0: heavier tails / more extreme values than a normal; < 0: lighter tails (a uniform distribution has −1.2). Kurtosis is about tails, not "peakedness".
Both are driven by a few extreme values and are unstable in small samples.

Common mistakes

Comparing an "excess" kurtosis with 3 (the non-excess convention). Statistics Quest always reports excess kurtosis, normal = 0.
Mixing $g_1$ and $G_1$ from different software.
Relying on the numbers without looking at a plot.

Example

Data 1, 2, 2, 3, 3, 3, 4, 10: mean 3.5 > median 3. $G_1 = 2.24$ (moment $g_1 = 1.80$ ) — strong right skew from the single 10. $G_2 = 5.67$ ( $g_2 = 2.03$ ): heavy tail.