← Glossaryskewness (skewed distribution)
Lack of symmetry in a distribution; with a long right tail (typical for revenue) the mean is above the median.
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.
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=s3(xˉ−median)
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.
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=47.663(50−35)=47.6645≈0.94
Positive, so the tail is on the right. The quick ratio agrees: mean / median =50/35≈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
- Median: sorted, the 5th and 6th values are both 5, so the median is 5
- SD ≈1.32 (sum of squared deviations 15.6, divided by 9, square root)
Sk=1.323(4.2−5)≈−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.
