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skewness (skewed distribution)

Lack of symmetry in a distribution; with a long right tail (typical for revenue) the mean is above the median.

Skewness describes how lopsided a distribution is. In a symmetric distribution the two halves mirror each other and the mean sits right next to the median. In a skewed one, most values bunch up on one side and the rest trail off in a long tail on the other. The direction is named after the tail, not after the hump.

Right (positive) skew means a long tail towards large values. It is the default shape of money and time metrics: 90-day spend, order value, session length, cost per acquisition. They can't go below zero but have no ceiling, so a few big values drag the mean up while the median stays with the typical customer: mean > median. Left (negative) skew is the mirror image, a tail towards small values and mean < median. You see it in app-store ratings or course completion, where most people sit near the maximum.

A quick way to put a number on it is Pearson's median skewness:

Sk=3(xˉ−median)s\text{Sk} = \frac{3(\bar{x} - \text{median})}{s}

It is positive for right skew, negative for left skew and close to zero for symmetric data. Spreadsheets (Excel's SKEW) use a different, moment-based formula, roughly the average of cubed z-scores. Both give the same sign but on different scales, so only compare numbers from the same formula.

Why a PM should care: skew decides which summary you report (median for "a typical customer", mean for totals and forecasts), whether the 68–95–99.7 rule and the "3 SD" outlier rule make sense, and whether a t-test on a small sample is safe or a rank-based test such as Mann–Whitney is the better choice. Look at a histogram or box plot before you pick a number.

Example

Right skew: 90-day spend. CatChow's sample of 24 new customers (from the outliers lesson) has a mean of $50, a median of $35 and an SD of $47.66.

Sk=3(50−35)47.66=4547.66≈0.94\text{Sk} = \frac{3(50 - 35)}{47.66} = \frac{45}{47.66} \approx 0.94

Positive, so the tail is on the right. The quick ratio agrees: mean / median =50/35≈1.43= 50 / 35 \approx 1.43. The tail is real money: the top 2 customers spent $350 of the $1,200 total, or 29%. Excel's SKEW on the same 24 values returns about 3.0: a different scale, the same verdict.

Left skew: app ratings. Ten ratings for the CatChow app: 5, 5, 5, 5, 5, 5, 4, 4, 3, 1.

  • Mean =(6×5+4+4+3+1)/10=42/10=4.2= (6 \times 5 + 4 + 4 + 3 + 1) / 10 = 42 / 10 = 4.2
  • Median: sorted, the 5th and 6th values are both 5, so the median is 5
  • SD ≈1.32\approx 1.32 (sum of squared deviations 15.6, divided by 9, square root)

Sk=3(4.2−5)1.32≈−1.82\text{Sk} = \frac{3(4.2 - 5)}{1.32} \approx -1.82

Negative: one angry 1-star review drags the mean below the median. Saying "our typical rating is 4.2" understates how most people rate the app; the median of 5 is closer to the truth, and the 1-star tail is worth reading on its own.

Common mistakes

  • Calling the mean "the typical customer" for right-skewed revenue. With a long tail most customers spend less than the mean; report the median alongside it.
  • Naming the skew after the hump. "Skewed right" means the long tail points right, even though the pile of values sits on the left.
  • Deleting the tail to make the data look normal. The biggest customers are often a large share of revenue; segment them or use robust summaries instead of removing them.
  • Using normal-curve rules on skewed data. The 68–95–99.7 rule and "3 SD from the mean" misjudge what is unusual when one tail is long.
  • Comparing skewness numbers from different tools or tiny samples. Pearson's coefficient and Excel's SKEW use different scales, and with 10–20 values any skewness estimate is noisy.

Learn it in the course