Imported from CatCorner22/Claude_Skills_2 (archive/skills/data-analytics-bi-skills/descriptive-statistics/SKILL.md). Install upstream with npx skills add CatCorner22/Claude_Skills_2 --skill descriptive-statistics. Copyright stays with the author.

Descriptive statistics

When to use

• Summarizing a numeric variable — its typical value, spread, and shape — before you infer or model.
• Choosing which summary to report when a distribution is skewed or has outliers (mean/SD vs. median/IQR).
• Building a summary-statistics table or the "at a glance" numbers for a report or dashboard.
• Not for: first-look structural profiling (grain, types, missingness) → see data-analytics-bi-skills:exploratory-data-analysis. For generalizing from a sample to a population with intervals or tests → see data-analytics-bi-skills:statistical-inference.

Do it

references/measures-and-formulas.md carries the formulas, the robust alternatives, and the worked five-number summary — open it when you need the definition rather than the choice.

1. Fix the population and the grain. State exactly which set of values you are summarizing and what one value represents ("revenue per closed order, FY24"). A mean means nothing until you can say "mean of what, per what," and mixing grains (order lines vs. orders) silently corrupts every statistic.
2. Decide sample vs. population. It changes the variance/SD denominator: population variance divides by N, sample variance by n − 1 (Bessel's correction, which offsets the downward bias from estimating the mean from the same sample). Analytics data is almost always a sample of a larger process → use the sample formulas.
3. Central tendency — pick the center that fits the shape.
   • Mean — the balance point; best for roughly symmetric data. Sensitive to every outlier.
   • Median (50th percentile) — the middle value; resistant to outliers and skew. The honest "typical" for income, prices, durations, and balances.
   • Mode — the most frequent value; the only center for nominal categories, and a flag for multimodality (two modes often means two mixed populations).
4. Dispersion — always pair a center with a matching spread.
   • Range (max − min): trivial to read, dominated by the two most extreme points.
   • Variance / standard deviation: average squared / root-average-squared distance from the mean; pair with the mean. SD is in the data's units, variance in squared units.
   • IQR (Q3 − Q1): the spread of the middle 50%; resistant; pair with the median.
   • Coefficient of variation (SD / mean): unitless relative spread for comparing series of different scales — valid only for ratio data with a positive mean, meaningless near zero.
5. Shape — quantify skew and tails. Compare mean to median as a fast skew check: mean ≫ median ⇒ right-skew (a long high tail, e.g. revenue); mean ≪ median ⇒ left-skew. Skewness puts a number on that asymmetry; kurtosis measures tail heaviness / outlier-proneness (report excess kurtosis, where a normal distribution is 0), not "peakedness."
6. Percentiles / quartiles. Report the five-number summary (min, Q1, median, Q3, max) and the tail percentiles your decision needs (p5/p95, or p99 for SLAs and risk). State the interpolation method — tools disagree at the edges (Excel PERCENTILE.INC vs .EXC, differing quartile definitions) — so a percentile is reproducible.
7. Choose robust vs. classical. If skew or outliers are present, lead with median + IQR (optionally MAD or a trimmed mean); reserve mean + SD for roughly symmetric data.
8. Deliver the summary block, not loose numbers. Per variable, the finished output is:

variable · n · missing/excluded
center: median 412.50          spread: IQR 118.00      (mean + SD if roughly symmetric)
five-number: min / Q1 / median / Q3 / max               tails: p95 (p99 for SLA/risk)
shape: right-skewed (mean 512.30 > median 412.50) · percentile method: PERCENTILE_CONT
plot: histogram when the question is shape or multimodality; boxplot when it is outliers
      or a comparison across groups; both when n is large and the shape is contested

Render the plot when the medium allows it. When it does not, the line still names one — the plot type and the variable — so the reader can produce it; "a plot would help" is not the deliverable, "histogram of order value, 40 bins" is.

The assistant computes and drafts the block; the human owns the population/grain definition and the decision the numbers feed.

Why / learn

Descriptive statistics answer "what does this data look like?" honestly, before any inference — and the core skill is refusing to let one number impersonate a whole distribution. The mean is a fragile summary: it is the balance point, so a single large invoice or a data-entry typo drags it away from where most of the data actually sits, and on a right-skewed quantity (revenue, wait time, account balance) the mean lands above the typical case and quietly overstates it. That is why the median and IQR often tell the truer story — they are resistant, defined by rank position rather than magnitude, so extreme values move them little or not at all. The mean-vs-median gap is itself information: when they diverge, the distribution is skewed and any "average" headline will mislead. Pairing is non-negotiable — a center with no spread hides whether values cluster or scatter, and a spread with no center is uninterpretable (SD belongs with the mean, IQR with the median). The deeper point is that a descriptive summary is a faithful compression: you are choosing which features of the shape to keep, so keep the ones that survive the outliers your data really has — and show the picture so the reader can see what the summary threw away.

Common mistakes

• Reporting the mean on skewed or outlier-laden data → misleading "typical." Lead with the median/IQR.
• A center with no spread (or a spread with no center) → hides clustering vs. scatter. Report the pair.
• Mismatched pair (median with SD, mean with IQR) → inconsistent story. Mean↔SD, median↔IQR.
• Dividing sample variance by n instead of n − 1 → understated spread. Use the sample formula.
• Coefficient of variation on data with a near-zero or negative mean → unstable nonsense. Ratio data only.
• Averaging across mixed grains or pooling distinct populations → a mean of two peaks that describes neither.
• Not stating the percentile method → the same p95 differs across Excel/SQL/pandas. State it.

Tailor to your environment

Record your setup in references/your-environment.md; keep anything sensitive (real column names, client thresholds, sample rows) in your-environment.private.md, which is git-ignored — commit only sanitized, structural examples. Capture the variables you routinely summarize and their expected shape (which are skewed by nature), your house convention for percentile/quartile method and rounding, whether you default to mean/SD or median/IQR, and the tool you compute in (SQL, pandas, Excel, BI). The skill then applies its generic choices to your real measures.

Keep your filled-in copy outside the plugin. This file ships as a template and lives inside the installed plugin, where a /plugin marketplace update can overwrite it or refuse to run against a dirty tree. Copy it into your own project — .claude/skills-env/descriptive-statistics.private.md works well — fill it in there, and point this skill at that copy. Your specifics then survive updates and stay somewhere you own rather than in a cache you may not realise is disposable.

References

• references/measures-and-formulas.md — formulas, when each measure is appropriate, and tool/percentile caveats
• references/your-environment.md — your variables, conventions, and tools (add when supplied)