standard-deviation

name	standard-deviation
description	Compute and interpret standard deviation (and related spread measures — variance, IQR, MAD, CV) for numeric columns in a dataset. Handles sample vs. population formulas, grouped/stratified computation, and flags columns where SD is misleading (heavy skew, outliers, near-constant values).

Standard Deviation

Compute standard deviation for one or more numeric columns, plus the context needed to actually use the number: sample vs. population formula, comparison to related spread measures, and warnings when SD is the wrong summary.

Inputs

• Path to a dataset (CSV / Parquet / Excel / DuckDB table).
• Optional: specific columns. Default: all numeric columns.
• Optional: grouping column — compute SD within each group.
• Optional: formula — sample (n-1 denominator, default) or population (n denominator). Default is sample, because almost all real data is a sample of something.

Recommended CLI tooling

• duckdb — built-in stddev_samp(), stddev_pop(), variance(), quantile_cont().
• uv run --with pandas --with scipy python -c '...' — MAD, trimmed SD, bootstrap CI for SD.

Procedure

Step 1 — Pick the right columns

For each candidate numeric column:

• Skip if it's an ID, code, or row index (monotonic increasing, all-unique integer). SD is meaningless.
• Skip if it's a boolean-coded 0/1 column unless the user asks (SD = sqrt(p(1-p)), rarely the useful summary).
• Include continuous measurements, counts, ratios, currency, scores.

Report any column you skipped and why.

Step 2 — Compute the core statistics

For each column (and each group, if grouping):

Statistic	What it tells you
n (non-null count)	Sample size the SD is based on.
mean, median	Centre. If they differ substantially, distribution is skewed.
stddev_samp	Standard deviation, n-1 denominator. Default report value.
variance_samp	Square of SD. Report if user explicitly wants it.
min, max	Range. Flag if max is >10× the 99th percentile — outlier pulling SD up.
q25, q75, IQR	Robust spread — compare to SD.
mad (median absolute deviation)	Robust SD analogue. MAD × 1.4826 ≈ SD if data is normal.
cv (coefficient of variation) = SD / mean	Dimensionless spread. Only meaningful when mean > 0 and the column has a natural zero (not temperatures-in-C).

SQL sketch in DuckDB:

SELECT
  count(col)                                    AS n,
  avg(col)                                      AS mean,
  median(col)                                   AS median,
  stddev_samp(col)                              AS sd,
  quantile_cont(col, 0.25)                      AS q25,
  quantile_cont(col, 0.75)                      AS q75,
  quantile_cont(col, 0.75) - quantile_cont(col, 0.25) AS iqr,
  min(col)                                      AS min,
  max(col)                                      AS max
FROM '<file>';

Step 3 — Diagnose whether SD is trustworthy

For each column, check:

• n < 30: SD estimate is unstable. Report with a warning and a bootstrap 95% CI if the user wants precision.
• Skew > 1 or < -1: the distribution is asymmetric; SD will overstate typical deviation. Prefer IQR or MAD.
• Kurtosis > 5: heavy tails; a few outliers dominate SD. Report trimmed SD (drop top/bottom 2.5%) alongside the raw SD.
• SD ≈ 0 (relative to mean): column is near-constant. Check for an imputation spike or a pre-aggregated series.
• |MAD × 1.4826 − SD| / SD > 0.3: robust and classical spread disagree by >30%, meaning outliers are inflating SD. Flag.
• mean ≈ 0 on CV request: CV is undefined/unstable; don't report CV.

Step 4 — Grouped / stratified (if grouping column provided)

Compute Step 2 statistics per group. Additionally surface:

• Groups with substantially larger SD than others (e.g. > 2× the median group SD). These may have different variance regimes or hidden subgroups.
• Groups with n < 30 — SD not reliable.
• The ratio of between-group variance to within-group variance (a one-way ANOVA F would formalise this; for a quick read, compare stddev(group_means) to mean(group_sds)).

Step 5 — Report

Write outputs/standard-deviation/report.md:

• Summary table of all reported columns with n, mean, median, SD, IQR, MAD, CV, and a trust flag (ok / skewed / heavy-tailed / small-n / near-constant).
• Per-column notes for anything flagged in Step 3.
• Grouped table (if applicable).
• Interpretation hint for each "ok" column: "≈68% of values fall within ±1 SD of the mean (assuming roughly normal)" — include this only for columns where the normality assumption is tenable.

Keep it concise. SD is a primitive; this skill should feed other skills, not produce a wall of prose.

Notes

• Default to sample SD (n-1). Only use population SD when the user explicitly has the whole population (e.g. every country in a fixed list, not a sample).
• If the user asks for "standard deviation" of a categorical or string column, push back — they may want something else (frequency variance, entropy).
• Pairs with multivariate-analysis (which needs standardised inputs) and forensic-sweep (which uses SD patterns to detect tampering).