Applied StatisticsEvidence that can survive scrutiny

Module 01 Two independent groups

A mean difference is simple. Defending it isn’t.

See what pooled, Welch, and HC2 inference change—and what none of them can establish about causality.

Open the live lab Inspect the assumptions

The same estimand, written two ways

ŷ = α̂ + β̂G β̂ = ȳB − ȳA
exactly

The binary-regressor coefficient is the raw difference in group means. The model changes the language—not the quantity.

Learn itFrom estimand to estimatorIntuition, algebra, and the exact regression equivalence. Run itStress the assumptionsChange sample size, spread, and design; watch the intervals respond. Defend itChoose and reportSeparate mathematical assumptions from identification claims. Teach itRebuild the argumentAn original exercise, checks, and neutral facilitation notes.

Learn it One contrast, three uncertainty models

Start with the estimand. Then argue about the standard error.

All three methods estimate the sample mean of Group B minus the sample mean of Group A. They differ in how they model uncertainty and calibrate the reference distribution.

P

Pooled t

Combines the sample variances under a common population-variance restriction. Its standard error can be smaller or larger than Welch’s, depending on the sample sizes and variances.

SE = sp√(1/nA + 1/nB)

W

Welch t

Keeps each group’s contribution separate and adjusts the degrees of freedom. Equal variances are not required.

SE = √(sA²/nA + sB²/nB)

H2

Binary OLS + HC2

Expresses the same comparison as a regression. With an intercept and one binary regressor, HC2 reproduces Welch’s standard error exactly.

VarHC2(β̂) = sA²/nA + sB²/nB

Run it Synthetic workflow-time case

Move the evidence. Watch the argument change.

The observations are generated locally from the moments you set. The case is synthetic by design: no personal, client, or institutional data.

Group A
Group B
Sample size
Mean
Standard deviation
A mean38.00n = 20 · s = 5.00
B − A+5.00sample mean contrast
B mean43.00n = 16 · s = 9.00
Generated observationsDots are deterministic for a given setting.
Generated observations by group A dot plot comparing the generated Group A and Group B values.
95% confidence intervalsAll rows target B − A; the vertical line marks zero.
Confidence intervals for all three methods Horizontal confidence intervals compare pooled, Welch, and binary OLS with HC2 inference.

Welch t

+5.00

95% CI [−0.21, 10.21]

Standard error
2.512
Degrees of freedom
22.3
p-value
0.059

What this method asks you to defend

Welch t

Assumption
Computation
Calibration
Use

Real public-data case World Development Indicators · 2024

Life expectancy across two current World Bank income classifications

This unweighted descriptive comparison includes every economy classified by the World Bank API as low income or upper-middle income at retrieval with a reported 2024 value.

Low income64.41years · n = 25
Upper-middle income74.01years · n = 59
UMC − LIC+9.60years · Welch SE = 1.05
Welch 95% CI[7.48, 11.72]0 missing of 84 economies

What the estimate describes. In this snapshot, the mean 2024 life expectancy among the 59 upper-middle-income economies is 9.60 years higher than among the 25 low-income economies.

What it does not identify. The grouping is observational and current classifications were applied to 2024 outcomes. The contrast is not a causal effect of income classification and is not population weighted.

Source: World Bank, World Development Indicators, retrieved 2026-08-18, CC BY 4.0. Audit the derived CSV and snapshot record with direct API URLs and hashes.

Defend it Estimation, assumptions, identification

A smaller p-value does not repair a weak design.

Inference quantifies sampling uncertainty under a model. Identification explains why the contrast answers a causal question. One cannot substitute for the other.

Current design reading · descriptive

Estimation is not identification

Exact algebraic check

Regression is a re-expression, not a different contrast.

Mean difference
5.000000
Binary coefficient
5.000000
Welch SE
2.512469
HC2 SE
2.512469

Equal within machine precision

Method comparison for the current synthetic sample
MethodEstimateSEdf95% CIp

A defensible sentence

01

Name the estimand

State the direction, units, population, and time frame. “Difference” is incomplete without them.

02

Justify the variance model

Explain why pooling is warranted—or retain separate variance contributions with Welch or HC2.

03

Report the convention

Name the standard-error correction and reference distribution. Software defaults are not a methods section.

04

Bound the interpretation

Connect causal language to assignment, sampling, interference, attrition, and measurement—not statistical significance.

Teach it A reusable original exercise

Make the method choice before revealing the answer.

This prompt is original and instructor-neutral. It can be used in a workshop, research meeting, or self-study session without carrying a named course or learner.

Decision prompt · 15 minutes

Two workflows, unequal variation

A team records completion times under two historically adopted workflows. Group A has n = 20, mean = 38, s = 5; Group B has n = 16, mean = 43, s = 9.

  1. Write the estimand in words and symbols.
  2. Choose pooled, Welch, or binary OLS + HC2. State what your choice assumes.
  3. Explain why the observed contrast does or does not identify a causal effect.
  4. Translate the result into one sentence that a decision-maker can audit.
Open deterministic answer checks

Continue without duplication

Neighboring experiences

Pictures of InferenceVisual intuition across statistics, calculus, and econometrics → American Policy AtlasA public-data observatory for applied policy questions →