What does the sign of sample skewness tell you about the shape of a distribution?

## The definition Skewness is the **third standardised moment** — a single dimensionless number describing how asymmetric a distribution is around its own centre. For a sample `x1 ... xn` with mean `xbar` and standard deviation `s`, the simplest (population-style, or *biased*) sample skewness is `g1 = (1/n) * sum( ((xi - xbar) / s)^3 )` Read that inside out. `(xi - xbar)` is a deviation from the centre. Dividing by `s` turns it into a z-score, so the quantity is expressed in standard deviations rather than in dollars or seconds. Cubing does two things at once: it **preserves the sign** (a negative deviation cubed stays negative) and it **amplifies distance** (a point 3 SD out contributes 27 units; a point 1 SD out contributes 1). Averaging those cubed z-scores gives one number whose sign reports which side of the centre carries the far-out points. Most reporting conventions use a small-sample-corrected version, the adjusted Fisher-Pearson coefficient `G1 = sqrt(n(n-1)) / (n-2) * g1` which inflates `g1` slightly on small samples. The two agree closely once `n` is large, and both carry the same sign, so the interpretation below is identical. ## Reading the sign - **Positive skewness (right-skewed, positively skewed):** the tail on the high side is long and thin. Session durations, house prices, income, and file sizes all behave this way — a floor at zero, a dense cluster of ordinary values, and a handful of very large ones that have nowhere to hide. - **Negative skewness (left-skewed):** the tail on the low side is long and thin. Think of a test where nearly everyone scores between 85 and 100 and a few score 30. - **Near zero:** consistent with symmetry. A normal distribution has population skewness exactly 0, and so does any symmetric distribution whose third moment exists. A useful mnemonic: the sign names the direction the *tail* points, not where the bulk of the data sits. Right-skewed data has *most* of its observations on the left, packed together — the name comes from the sparse tail, not the dense body. ## Zero skewness does not prove symmetry This is the classic follow-up. Symmetry implies zero skewness, but the converse fails: you can build asymmetric distributions whose positive and negative cubed deviations happen to cancel exactly. Zero is *consistent with* symmetry, not proof of it. Treat skewness as one summary of shape, alongside a look at the actual distribution, rather than a verdict. ## Units, scaling, and reflection Because every deviation is divided by `s`, skewness is **unit-free**. Multiply every value by a positive constant, or add a constant to every value, and skewness does not move: converting session durations from seconds to minutes, or from a timestamp offset to elapsed time, leaves the number identical. This is exactly why it is standardised — it makes shape comparable across variables measured on completely different scales. Multiplying by a *negative* constant reflects the distribution: the magnitude stays the same and the sign flips. A right-skewed variable becomes left-skewed when you negate it. ## Sensitivity The cube gives distant points enormous leverage. A single observation 6 SD from the mean contributes 216 units to the sum while a typical point contributes about 1. So sample skewness can swing substantially when one extreme record enters or leaves the data, and it is a noisy statistic on small samples. On tens of thousands of rows the sign is usually stable and worth reporting; on twenty rows, treat both the size and the sign with suspicion. ## Rough magnitude guidance There is no universal threshold, and interviewers do not expect one, but practitioners commonly treat |skewness| below about 0.5 as near-symmetric, 0.5 to 1 as moderately skewed, and above 1 as strongly skewed. Some reference values help calibrate: an exponential distribution has population skewness exactly 2; any symmetric distribution has 0. Quote these as orientation, not as a decision rule. ## What to do with the answer Knowing the sign and rough size tells you that a symmetric-shape assumption is questionable and that a plain mean-plus-SD summary is hiding structure. It is also the trigger for considering a transform on strictly positive data — taking logs pulls in a long right tail and can bring skewness close to zero, which is why heavily right-skewed measurements are so often summarised or modelled on a log scale.