A/B Test Setup

You are an expert in experimentation and A/B testing. Your goal is to help design tests that produce statistically valid, actionable results.

Initial Assessment

Check for product marketing context first: If .claude/product-marketing-context.md exists, read it before asking questions. Use that context and only ask for information not already covered or specific to this task.

Before designing a test, understand:

1. Test Context - What are you trying to improve? What change are you considering?
2. Current State - Baseline conversion rate? Current traffic volume?
3. Constraints - Technical complexity? Timeline? Tools available?

---

Core Principles

1. Start with a Hypothesis

• Not just "let's see what happens"
• Specific prediction of outcome
• Based on reasoning or data

2. Test One Thing

• Single variable per test
• Otherwise you don't know what worked

3. Statistical Rigor

• Pre-determine sample size
• Don't peek and stop early
• Commit to the methodology

4. Measure What Matters

• Primary metric tied to business value
• Secondary metrics for context
• Guardrail metrics to prevent harm

---

Hypothesis Framework

Structure

Because [observation/data],
we believe [change]
will cause [expected outcome]
for [audience].
We'll know this is true when [metrics].

Example

Weak: "Changing the button color might increase clicks."

Strong: "Because users report difficulty finding the CTA (per heatmaps and feedback), we believe making the button larger and using contrasting color will increase CTA clicks by 15%+ for new visitors. We'll measure click-through rate from page view to signup start."

---

Test Types

Type	Description	Traffic Needed
A/B	Two versions, single change	Moderate
A/B/n	Multiple variants	Higher
MVT	Multiple changes in combinations	Very high
Split URL	Different URLs for variants	Moderate

---

Sample Size

Quick Reference

Baseline	10% Lift	20% Lift	50% Lift
1%	150k/variant	39k/variant	6k/variant
3%	47k/variant	12k/variant	2k/variant
5%	27k/variant	7k/variant	1.2k/variant
10%	12k/variant	3k/variant	550/variant

Calculators:

• Evan Miller's
• Optimizely's

For detailed sample size tables and duration calculations: See references/sample-size-guide.md

---

Metrics Selection

Primary Metric

• Single metric that matters most
• Directly tied to hypothesis
• What you'll use to call the test

Secondary Metrics

• Support primary metric interpretation
• Explain why/how the change worked

Guardrail Metrics

• Things that shouldn't get worse
• Stop test if significantly negative

Example: Pricing Page Test

• Primary: Plan selection rate
• Secondary: Time on page, plan distribution
• Guardrail: Support tickets, refund rate

---

Designing Variants

What to Vary

Category	Examples
Headlines/Copy	Message angle, value prop, specificity, tone
Visual Design	Layout, color, images, hierarchy
CTA	Button copy, size, placement, number
Content	Information included, order, amount, social proof

Best Practices

• Single, meaningful change
• Bold enough to make a difference
• True to the hypothesis

---

Traffic Allocation

Approach	Split	When to Use
Standard	50/50	Default for A/B
Conservative	90/10, 80/20	Limit risk of bad variant
Ramping	Start small, increase	Technical risk mitigation

Considerations:

• Consistency: Users see same variant on return
• Balanced exposure across time of day/week

---

Implementation

Client-Side

• JavaScript modifies page after load
• Quick to implement, can cause flicker
• Tools: PostHog, Optimizely, VWO

Server-Side

• Variant determined before render
• No flicker, requires dev work
• Tools: PostHog, LaunchDarkly, Split

---

Running the Test

Pre-Launch Checklist

• Hypothesis documented
• Primary metric defined
• Sample size calculated
• Variants implemented correctly
• Tracking verified
• QA completed on all variants

During the Test

DO:

• Monitor for technical issues
• Check segment quality
• Document external factors

DON'T:

• Peek at results and stop early
• Make changes to variants
• Add traffic from new sources

The Peeking Problem

Looking at results before reaching sample size and stopping early leads to false positives and wrong decisions. Pre-commit to sample size and trust the process.

---

Analyzing Results

Statistical Significance

• 95% confidence = p-value < 0.05
• Means <5% chance result is random
• Not a guarantee—just a threshold

Analysis Checklist

1. Reach sample size? If not, result is preliminary
2. Statistically significant? Check confidence intervals
3. Effect size meaningful? Compare to MDE, project impact
4. Secondary metrics consistent? Support the primary?
5. Guardrail concerns? Anything get worse?
6. Segment differences? Mobile vs. desktop? New vs. returning?

Interpreting Results

Result	Conclusion
Significant winner	Implement variant
Significant loser	Keep control, learn why
No significant difference	Need more traffic or bolder test
Mixed signals	Dig deeper, maybe segment

---

Documentation

Document every test with:

• Hypothesis
• Variants (with screenshots)
• Results (sample, metrics, significance)
• Decision and learnings

For templates: See references/test-templates.md

---

Common Mistakes

Test Design

• Testing too small a change (undetectable)
• Testing too many things (can't isolate)
• No clear hypothesis

Execution

• Stopping early
• Changing things mid-test
• Not checking implementation

Analysis

• Ignoring confidence intervals
• Cherry-picking segments
• Over-interpreting inconclusive results

---

Task-Specific Questions

1. What's your current conversion rate?
2. How much traffic does this page get?
3. What change are you considering and why?
4. What's the smallest improvement worth detecting?
5. What tools do you have for testing?
6. Have you tested this area before?

---

Proactive Triggers

Proactively offer A/B test design when:

1. Conversion rate mentioned — User shares a conversion rate and asks how to improve it; suggest designing a test rather than guessing at solutions.
2. Copy or design decision is unclear — When two variants of a headline, CTA, or layout are being debated, propose testing instead of opinionating.
3. Campaign underperformance — User reports a landing page or email performing below expectations; offer a structured test plan.
4. Pricing page discussion — Any mention of pricing page changes should trigger an offer to design a pricing test with guardrail metrics.
5. Post-launch review — After a feature or campaign goes live, propose follow-up experiments to optimize the result.

---

Output Artifacts

Artifact	Format	Description
Experiment Brief	Markdown doc	Hypothesis, variants, metrics, sample size, duration, owner
Sample Size Calculator Input	Table	Baseline rate, MDE, confidence level, power
Pre-Launch QA Checklist	Checklist	Implementation, tracking, variant rendering verification
Results Analysis Report	Markdown doc	Statistical significance, effect size, segment breakdown, decision
Test Backlog	Prioritized list	Ranked experiments by expected impact and feasibility

---

Communication

All outputs should meet the quality standard: clear hypothesis, pre-registered metrics, and documented decisions. Avoid presenting inconclusive results as wins. Every test should produce a learning, even if the variant loses. Reference marketing-context for product and audience framing before designing experiments.

---

Related Skills

• page-cro — USE when you need ideas for what to test; NOT when you already have a hypothesis and just need test design.
• analytics-tracking — USE to set up measurement infrastructure before running tests; NOT as a substitute for defining primary metrics upfront.
• campaign-analytics — USE after tests conclude to fold results into broader campaign attribution; NOT during the test itself.
• pricing-strategy — USE when test results affect pricing decisions; NOT to replace a controlled test with pure strategic reasoning.
• marketing-context — USE as foundation before any test design to ensure hypotheses align with ICP and positioning; always load first.

---

Referenced Files

The following files are referenced in this skill and included for context.

references/sample-size-guide.md

# Sample Size Guide

Reference for calculating sample sizes and test duration.

## Sample Size Fundamentals

### Required Inputs

1. **Baseline conversion rate**: Your current rate
2. **Minimum detectable effect (MDE)**: Smallest change worth detecting
3. **Statistical significance level**: Usually 95% (α = 0.05)
4. **Statistical power**: Usually 80% (β = 0.20)

### What These Mean

**Baseline conversion rate**: If your page converts at 5%, that's your baseline.

**MDE (Minimum Detectable Effect)**: The smallest improvement you care about detecting. Set this based on:
- Business impact (is a 5% lift meaningful?)
- Implementation cost (worth the effort?)
- Realistic expectations (what have past tests shown?)

**Statistical significance (95%)**: Means there's less than 5% chance the observed difference is due to random chance.

**Statistical power (80%)**: Means if there's a real effect of size MDE, you have 80% chance of detecting it.

---

## Sample Size Quick Reference Tables

### Conversion Rate: 1%

| Lift to Detect | Sample per Variant | Total Sample |
|----------------|-------------------|--------------|
| 5% (1% → 1.05%) | 1,500,000 | 3,000,000 |
| 10% (1% → 1.1%) | 380,000 | 760,000 |
| 20% (1% → 1.2%) | 97,000 | 194,000 |
| 50% (1% → 1.5%) | 16,000 | 32,000 |
| 100% (1% → 2%) | 4,200 | 8,400 |

### Conversion Rate: 3%

| Lift to Detect | Sample per Variant | Total Sample |
|----------------|-------------------|--------------|
| 5% (3% → 3.15%) | 480,000 | 960,000 |
| 10% (3% → 3.3%) | 120,000 | 240,000 |
| 20% (3% → 3.6%) | 31,000 | 62,000 |
| 50% (3% → 4.5%) | 5,200 | 10,400 |
| 100% (3% → 6%) | 1,400 | 2,800 |

### Conversion Rate: 5%

| Lift to Detect | Sample per Variant | Total Sample |
|----------------|-------------------|--------------|
| 5% (5% → 5.25%) | 280,000 | 560,000 |
| 10% (5% → 5.5%) | 72,000 | 144,000 |
| 20% (5% → 6%) | 18,000 | 36,000 |
| 50% (5% → 7.5%) | 3,100 | 6,200 |
| 100% (5% → 10%) | 810 | 1,620 |

### Conversion Rate: 10%

| Lift to Detect | Sample per Variant | Total Sample |
|----------------|-------------------|--------------|
| 5% (10% → 10.5%) | 130,000 | 260,000 |
| 10% (10% → 11%) | 34,000 | 68,000 |
| 20% (10% → 12%) | 8,700 | 17,400 |
| 50% (10% → 15%) | 1,500 | 3,000 |
| 100% (10% → 20%) | 400 | 800 |

### Conversion Rate: 20%

| Lift to Detect | Sample per Variant | Total Sample |
|----------------|-------------------|--------------|
| 5% (20% → 21%) | 60,000 | 120,000 |
| 10% (20% → 22%) | 16,000 | 32,000 |
| 20% (20% → 24%) | 4,000 | 8,000 |
| 50% (20% → 30%) | 700 | 1,400 |
| 100% (20% → 40%) | 200 | 400 |

---

## Duration Calculator

### Formula

Duration (days) = (Sample per variant × Number of variants) / (Daily traffic × % exposed)

### Examples

**Scenario 1: High-traffic page**
- Need: 10,000 per variant (2 variants = 20,000 total)
- Daily traffic: 5,000 visitors
- 100% exposed to test
- Duration: 20,000 / 5,000 = **4 days**

**Scenario 2: Medium-traffic page**
- Need: 30,000 per variant (60,000 total)
- Daily traffic: 2,000 visitors
- 100% exposed
- Duration: 60,000 / 2,000 = **30 days**

**Scenario 3: Low-traffic with partial exposure**
- Need: 15,000 per variant (30,000 total)
- Daily traffic: 500 visitors
- 50% exposed to test
- Effective daily: 250
- Duration: 30,000 / 250 = **120 days** (too long!)

### Minimum Duration Rules

Even with sufficient sample size, run tests for at least:
- **1 full week**: To capture day-of-week variation
- **2 business cycles**: If B2B (weekday vs. weekend patterns)
- **Through paydays**: If e-commerce (beginning/end of month)

### Maximum Duration Guidelines

Avoid running tests longer than 4-8 weeks:
- Novelty effects wear off
- External factors intervene
- Opportunity cost of other tests

---

## Online Calculators

### Recommended Tools

**Evan Miller's Calculator**
https://www.evanmiller.org/ab-testing/sample-size.html
- Simple interface
- Bookmark-worthy

**Optimizely's Calculator**
https://www.optimizely.com/sample-size-calculator/
- Business-friendly language
- Duration estimates

**AB Test Guide Calculator**
https://www.abtestguide.com/calc/
- Includes Bayesian option
- Multiple test types

**VWO Duration Calculator**
https://vwo.com/tools/ab-test-duration-calculator/
- Duration-focused
- Good for planning

---

## Adjusting for Multiple Variants

With more than 2 variants (A/B/n tests), you need more sample:

| Variants | Multiplier |
|----------|------------|
| 2 (A/B) | 1x |
| 3 (A/B/C) | ~1.5x |
| 4 (A/B/C/D) | ~2x |
| 5+ | Consider reducing variants |

**Why?** More comparisons increase chance of false positives. You're comparing:
- A vs B
- A vs C
- B vs C (sometimes)

Apply Bonferroni correction or use tools that handle this automatically.

---

## Common Sample Size Mistakes

### 1. Underpowered tests
**Problem**: Not enough sample to detect realistic effects
**Fix**: Be realistic about MDE, get more traffic, or don't test

### 2. Overpowered tests
**Problem**: Waiting for sample size when you already have significance
**Fix**: This is actually fine—you committed to sample size, honor it

### 3. Wrong baseline rate
**Problem**: Using wrong conversion rate for calculation
**Fix**: Use the specific metric and page, not site-wide averages

### 4. Ignoring segments
**Problem**: Calculating for full traffic, then analyzing segments
**Fix**: If you plan segment analysis, calculate sample for smallest segment

### 5. Testing too many things
**Problem**: Dividing traffic too many ways
**Fix**: Prioritize ruthlessly, run fewer concurrent tests

---

## When Sample Size Requirements Are Too High

Options when you can't get enough traffic:

1. **Increase MDE**: Accept only detecting larger effects (20%+ lift)
2. **Lower confidence**: Use 90% instead of 95% (risky, document it)
3. **Reduce variants**: Test only the most promising variant
4. **Combine traffic**: Test across multiple similar pages
5. **Test upstream**: Test earlier in funnel where traffic is higher
6. **Don't test**: Make decision based on qualitative data instead
7. **Longer test**: Accept longer duration (weeks/months)

---

## Sequential Testing

If you must check results before reaching sample size:

### What is it?
Statistical method that adjusts for multiple looks at data.

### When to use
- High-risk changes
- Need to stop bad variants early
- Time-sensitive decisions

### Tools that support it
- Optimizely (Stats Accelerator)
- VWO (SmartStats)
- PostHog (Bayesian approach)

### Tradeoff
- More flexibility to stop early
- Slightly larger sample size requirement
- More complex analysis

---

## Quick Decision Framework

### Can I run this test?

Daily traffic to page: _____ Baseline conversion rate: _____ MDE I care about: _____

Sample needed per variant: _____ (from tables above) Days to run: Sample / Daily traffic = _____

If days > 60: Consider alternatives If days > 30: Acceptable for high-impact tests If days < 14: Likely feasible If days < 7: Easy to run, consider running longer anyway

references/test-templates.md

# A/B Test Templates Reference

Templates for planning, documenting, and analyzing experiments.

## Test Plan Template

```markdown
# A/B Test: [Name]

## Overview
- **Owner**: [Name]
- **Test ID**: [ID in testing tool]
- **Page/Feature**: [What's being tested]
- **Planned dates**: [Start] - [End]

## Hypothesis

Because [observation/data],
we believe [change]
will cause [expected outcome]
for [audience].
We'll know this is true when [metrics].

## Test Design

| Element | Details |
|---------|---------|
| Test type | A/B / A/B/n / MVT |
| Duration | X weeks |
| Sample size | X per variant |
| Traffic allocation | 50/50 |
| Tool | [Tool name] |
| Implementation | Client-side / Server-side |

## Variants

### Control (A)
[Screenshot]
- Current experience
- [Key details about current state]

### Variant (B)
[Screenshot or mockup]
- [Specific change #1]
- [Specific change #2]
- Rationale: [Why we think this will win]

## Metrics

### Primary
- **Metric**: [metric name]
- **Definition**: [how it's calculated]
- **Current baseline**: [X%]
- **Minimum detectable effect**: [X%]

### Secondary
- [Metric 1]: [what it tells us]
- [Metric 2]: [what it tells us]
- [Metric 3]: [what it tells us]

### Guardrails
- [Metric that shouldn't get worse]
- [Another safety metric]

## Segment Analysis Plan
- Mobile vs. desktop
- New vs. returning visitors
- Traffic source
- [Other relevant segments]

## Success Criteria
- Winner: [Primary metric improves by X% with 95% confidence]
- Loser: [Primary metric decreases significantly]
- Inconclusive: [What we'll do if no significant result]

## Pre-Launch Checklist
- [ ] Hypothesis documented and reviewed
- [ ] Primary metric defined and trackable
- [ ] Sample size calculated
- [ ] Test duration estimated
- [ ] Variants implemented correctly
- [ ] Tracking verified in all variants
- [ ] QA completed on all variants
- [ ] Stakeholders informed
- [ ] Calendar hold for analysis date

---

Results Documentation Template

# A/B Test Results: [Name]

## Summary
| Element | Value |
|---------|-------|
| Test ID | [ID] |
| Dates | [Start] - [End] |
| Duration | X days |
| Result | Winner / Loser / Inconclusive |
| Decision | [What we're doing] |

## Hypothesis (Reminder)
[Copy from test plan]

## Results

### Sample Size
| Variant | Target | Actual | % of target |
|---------|--------|--------|-------------|
| Control | X | Y | Z% |
| Variant | X | Y | Z% |

### Primary Metric: [Metric Name]
| Variant | Value | 95% CI | vs. Control |
|---------|-------|--------|-------------|
| Control | X% | [X%, Y%] | — |
| Variant | X% | [X%, Y%] | +X% |

**Statistical significance**: p = X.XX (95% = sig / not sig)
**Practical significance**: [Is this lift meaningful for the business?]

### Secondary Metrics

| Metric | Control | Variant | Change | Significant? |
|--------|---------|---------|--------|--------------|
| [Metric 1] | X | Y | +Z% | Yes/No |
| [Metric 2] | X | Y | +Z% | Yes/No |

### Guardrail Metrics

| Metric | Control | Variant | Change | Concern? |
|--------|---------|---------|--------|----------|
| [Metric 1] | X | Y | +Z% | Yes/No |

### Segment Analysis

**Mobile vs. Desktop**
| Segment | Control | Variant | Lift |
|---------|---------|---------|------|
| Mobile | X% | Y% | +Z% |
| Desktop | X% | Y% | +Z% |

**New vs. Returning**
| Segment | Control | Variant | Lift |
|---------|---------|---------|------|
| New | X% | Y% | +Z% |
| Returning | X% | Y% | +Z% |

## Interpretation

### What happened?
[Explanation of results in plain language]

### Why do we think this happened?
[Analysis and reasoning]

### Caveats
[Any limitations, external factors, or concerns]

## Decision

**Winner**: [Control / Variant]

**Action**: [Implement variant / Keep control / Re-test]

**Timeline**: [When changes will be implemented]

## Learnings

### What we learned
- [Key insight 1]
- [Key insight 2]

### What to test next
- [Follow-up test idea 1]
- [Follow-up test idea 2]

### Impact
- **Projected lift**: [X% improvement in Y metric]
- **Business impact**: [Revenue, conversions, etc.]

---

Test Repository Entry Template

For tracking all tests in a central location:

| Test ID | Name | Page | Dates | Primary Metric | Result | Lift | Link |
|---------|------|------|-------|----------------|--------|------|------|
| 001 | Hero headline test | Homepage | 1/1-1/15 | CTR | Winner | +12% | [Link] |
| 002 | Pricing table layout | Pricing | 1/10-1/31 | Plan selection | Loser | -5% | [Link] |
| 003 | Signup form fields | Signup | 2/1-2/14 | Completion | Inconclusive | +2% | [Link] |

---

Quick Test Brief Template

For simple tests that don't need full documentation:

## [Test Name]

**What**: [One sentence description]
**Why**: [One sentence hypothesis]
**Metric**: [Primary metric]
**Duration**: [X weeks]
**Result**: [TBD / Winner / Loser / Inconclusive]
**Learnings**: [Key takeaway]

---

Stakeholder Update Template

## A/B Test Update: [Name]

**Status**: Running / Complete
**Days remaining**: X (or complete)
**Current sample**: X% of target

### Preliminary observations
[What we're seeing - without making decisions yet]

### Next steps
[What happens next]

### Timeline
- [Date]: Analysis complete
- [Date]: Decision and recommendation
- [Date]: Implementation (if winner)

---

Experiment Prioritization Scorecard

For deciding which tests to run:

Factor	Weight	Test A	Test B	Test C
Potential impact	30%
Confidence in hypothesis	25%
Ease of implementation	20%
Risk if wrong	15%
Strategic alignment	10%
Total

Scoring: 1-5 (5 = best)

---

Hypothesis Bank Template

For collecting test ideas:

| ID | Page/Area | Observation | Hypothesis | Potential Impact | Status |
|----|-----------|-------------|------------|------------------|--------|
| H1 | Homepage | Low scroll depth | Shorter hero will increase scroll | High | Testing |
| H2 | Pricing | Users compare plans | Comparison table will help | Medium | Backlog |
| H3 | Signup | Drop-off at email | Social login will increase completion | Medium | Backlog |

---

## Skill Companion Files

> Additional files collected from the skill directory layout.

### scripts/sample_size_calculator.py

```python
#!/usr/bin/env python3
"""
sample_size_calculator.py — A/B Test Sample Size Calculator
100% stdlib, no pip installs required.

Usage:
    python3 sample_size_calculator.py                          # demo mode
    python3 sample_size_calculator.py --baseline 0.05 --mde 0.20
    python3 sample_size_calculator.py --baseline 0.05 --mde 0.20 --daily-traffic 500
    python3 sample_size_calculator.py --baseline 0.05 --mde 0.20 --json
"""

import argparse
import json
import math
import sys


# ---------------------------------------------------------------------------
# Z-score approximation (scipy-free, Beasley-Springer-Moro algorithm)
# ---------------------------------------------------------------------------

def _norm_ppf(p: float) -> float:
    """Percent-point function (inverse CDF) of the standard normal.
    Uses rational approximation — accurate to ~1e-9.
    Reference: Abramowitz & Stegun 26.2.17 / Peter Acklam's algorithm.
    """
    if p <= 0 or p >= 1:
        raise ValueError(f"p must be in (0, 1), got {p}")

    # Coefficients for rational approximation
    a = [-3.969683028665376e+01,  2.209460984245205e+02,
         -2.759285104469687e+02,  1.383577518672690e+02,
         -3.066479806614716e+01,  2.506628277459239e+00]
    b = [-5.447609879822406e+01,  1.615858368580409e+02,
         -1.556989798598866e+02,  6.680131188771972e+01,
         -1.328068155288572e+01]
    c = [-7.784894002430293e-03, -3.223964580411365e-01,
         -2.400758277161838e+00, -2.549732539343734e+00,
          4.374664141464968e+00,  2.938163982698783e+00]
    d = [7.784695709041462e-03,  3.224671290700398e-01,
         2.445134137142996e+00,  3.754408661907416e+00]

    p_low  = 0.02425
    p_high = 1 - p_low

    if p < p_low:
        q = math.sqrt(-2 * math.log(p))
        return (((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) / \
               ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1)
    elif p <= p_high:
        q = p - 0.5
        r = q * q
        return (((((a[0]*r+a[1])*r+a[2])*r+a[3])*r+a[4])*r+a[5])*q / \
               (((((b[0]*r+b[1])*r+b[2])*r+b[3])*r+b[4])*r+1)
    else:
        q = math.sqrt(-2 * math.log(1 - p))
        return -(((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) / \
                ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1)


# ---------------------------------------------------------------------------
# Core calculation
# ---------------------------------------------------------------------------

def calculate_sample_size(
    baseline: float,
    mde: float,
    alpha: float = 0.05,
    power: float = 0.80,
) -> dict:
    """
    Two-proportion z-test sample size formula (two-tailed).

    n = (Z_alpha/2 + Z_beta)^2 * (p1*(1-p1) + p2*(1-p2)) / (p2 - p1)^2

    Args:
        baseline  : baseline conversion rate (e.g. 0.05 for 5%)
        mde       : minimum detectable effect as relative lift (e.g. 0.20 for +20%)
        alpha     : significance level (Type I error rate), default 0.05
        power     : statistical power (1 - Type II error rate), default 0.80

    Returns dict with all intermediate values and results.
    """
    p1 = baseline
    p2 = baseline * (1 + mde)          # expected conversion with treatment

    if not (0 < p1 < 1):
        raise ValueError(f"baseline must be in (0,1), got {p1}")
    if not (0 < p2 < 1):
        raise ValueError(
            f"baseline * (1 + mde) = {p2:.4f} is outside (0,1). "
            "Reduce mde or increase baseline."
        )

    z_alpha = _norm_ppf(1 - alpha / 2)   # two-tailed
    z_beta  = _norm_ppf(power)

    pooled_var = p1 * (1 - p1) + p2 * (1 - p2)
    effect_sq  = (p2 - p1) ** 2

    n_raw = ((z_alpha + z_beta) ** 2 * pooled_var) / effect_sq
    n     = math.ceil(n_raw)

    return {
        "inputs": {
            "baseline_conversion_rate": p1,
            "minimum_detectable_effect_relative": mde,
            "expected_variant_conversion_rate": round(p2, 6),
            "significance_level_alpha": alpha,
            "statistical_power": power,
        },
        "z_scores": {
            "z_alpha_2": round(z_alpha, 4),
            "z_beta":    round(z_beta,  4),
        },
        "results": {
            "sample_size_per_variation": n,
            "total_sample_size":         n * 2,
            "absolute_lift":             round(p2 - p1, 6),
            "relative_lift_pct":         round(mde * 100, 2),
        },
        "formula": (
            "n = (Z_α/2 + Z_β)² × (p1(1−p1) + p2(1−p2)) / (p2−p1)²  "
            "[two-proportion z-test, two-tailed]"
        ),
        "assumptions": [
            "Two-tailed test (detecting lift in either direction)",
            "Independent samples (no within-subject correlation)",
            "Fixed horizon (not sequential / always-valid)",
            "Binomial outcome (conversion yes/no)",
            "No novelty effect correction applied",
        ],
    }


def add_duration(result: dict, daily_traffic: int) -> dict:
    """Append estimated test duration given total daily traffic (both variants)."""
    n_total = result["results"]["total_sample_size"]
    days    = math.ceil(n_total / daily_traffic)
    weeks   = round(days / 7, 1)
    result["duration"] = {
        "daily_traffic_both_variants": daily_traffic,
        "estimated_days":  days,
        "estimated_weeks": weeks,
        "note": (
            "Assumes traffic is evenly split 50/50 between control and variant. "
            "Add ~10–20% buffer for weekday/weekend variance."
        ),
    }
    return result


# ---------------------------------------------------------------------------
# Scoring helper (0-100)
# ---------------------------------------------------------------------------

def score_test_design(result: dict) -> dict:
    """Heuristic quality score for the A/B test design."""
    score = 100
    reasons = []
    inputs = result["inputs"]

    # Penalise very low baseline (unreliable estimates)
    if inputs["baseline_conversion_rate"] < 0.01:
        score -= 15
        reasons.append("Baseline <1%: high variance, consider aggregating more data first.")

    # Penalise tiny MDE (will need enormous sample)
    mde = inputs["minimum_detectable_effect_relative"]
    if mde < 0.05:
        score -= 20
        reasons.append("MDE <5%: very small effect, experiment may take months.")
    elif mde < 0.10:
        score -= 10
        reasons.append("MDE <10%: moderately small effect size.")

    # Penalise overly aggressive alpha
    if inputs["significance_level_alpha"] > 0.10:
        score -= 15
        reasons.append("α >10%: high false-positive risk.")

    # Penalise low power
    if inputs["statistical_power"] < 0.80:
        score -= 20
        reasons.append("Power <80%: elevated risk of missing real effects (Type II error).")

    # Duration penalty (if available)
    dur = result.get("duration")
    if dur:
        days = dur["estimated_days"]
        if days > 90:
            score -= 20
            reasons.append(f"Test duration {days}d >90 days: novelty/seasonal effects likely.")
        elif days > 30:
            score -= 10
            reasons.append(f"Test duration {days}d >30 days: monitor for external confounders.")

    score = max(0, score)
    return {
        "design_quality_score": score,
        "score_interpretation": _score_label(score),
        "issues": reasons if reasons else ["No major design issues detected."],
    }


def _score_label(s: int) -> str:
    if s >= 90: return "Excellent"
    if s >= 75: return "Good"
    if s >= 60: return "Fair"
    if s >= 40: return "Poor"
    return "Critical"


# ---------------------------------------------------------------------------
# Pretty-print
# ---------------------------------------------------------------------------

def pretty_print(result: dict, score: dict) -> None:
    inp = result["inputs"]
    res = result["results"]
    zs  = result["z_scores"]

    print("\n" + "=" * 60)
    print("  A/B TEST SAMPLE SIZE CALCULATOR")
    print("=" * 60)

    print("\n📥  INPUTS")
    print(f"  Baseline conversion rate : {inp['baseline_conversion_rate']*100:.2f}%")
    print(f"  Variant conversion rate  : {inp['expected_variant_conversion_rate']*100:.2f}%")
    print(f"  Minimum detectable effect: {inp['minimum_detectable_effect_relative']*100:.1f}% relative "
          f"(+{res['absolute_lift']*100:.3f}pp absolute)")
    print(f"  Significance level (α)   : {inp['significance_level_alpha']}")
    print(f"  Statistical power        : {inp['statistical_power']*100:.0f}%")

    print("\n📐  FORMULA")
    print(f"  {result['formula']}")
    print(f"  Z_α/2 = {zs['z_alpha_2']}   Z_β = {zs['z_beta']}")

    print("\n📊  RESULTS")
    print(f"  ✅ Sample size per variation : {res['sample_size_per_variation']:,}")
    print(f"  ✅ Total sample size (both)  : {res['total_sample_size']:,}")

    if "duration" in result:
        d = result["duration"]
        print(f"\n⏱️   DURATION ESTIMATE  (traffic: {d['daily_traffic_both_variants']:,}/day)")
        print(f"  Estimated test duration : {d['estimated_days']} days  (~{d['estimated_weeks']} weeks)")
        print(f"  Note: {d['note']}")

    print("\n💡  ASSUMPTIONS")
    for a in result["assumptions"]:
        print(f"  • {a}")

    print(f"\n🎯  DESIGN QUALITY SCORE: {score['design_quality_score']}/100  ({score['score_interpretation']})")
    for issue in score["issues"]:
        print(f"  ⚠  {issue}")

    print()


# ---------------------------------------------------------------------------
# CLI
# ---------------------------------------------------------------------------

def parse_args():
    parser = argparse.ArgumentParser(
        description="Calculate required sample size for an A/B test (stdlib only).",
        formatter_class=argparse.RawDescriptionHelpFormatter,
        epilog=__doc__,
    )
    parser.add_argument("--baseline",       type=float, default=None,
                        help="Baseline conversion rate (e.g. 0.05 for 5%%)")
    parser.add_argument("--mde",            type=float, default=None,
                        help="Minimum detectable effect as relative lift (e.g. 0.20 for +20%%)")
    parser.add_argument("--alpha",          type=float, default=0.05,
                        help="Significance level α (default: 0.05)")
    parser.add_argument("--power",          type=float, default=0.80,
                        help="Statistical power 1-β (default: 0.80)")
    parser.add_argument("--daily-traffic",  type=int,   default=None,
                        help="Total daily visitors across both variants (for duration estimate)")
    parser.add_argument("--json",           action="store_true",
                        help="Output results as JSON")
    return parser.parse_args()


DEMO_SCENARIOS = [
    {"label": "E-commerce checkout (low baseline)",
     "baseline": 0.03, "mde": 0.20, "alpha": 0.05, "power": 0.80, "daily_traffic": 800},
    {"label": "SaaS free-trial signup (medium baseline)",
     "baseline": 0.08, "mde": 0.15, "alpha": 0.05, "power": 0.80, "daily_traffic": 2000},
    {"label": "Button CTA (high baseline)",
     "baseline": 0.25, "mde": 0.10, "alpha": 0.05, "power": 0.80, "daily_traffic": 5000},
]


def main():
    args = parse_args()
    demo_mode = (args.baseline is None and args.mde is None)

    if demo_mode:
        print("🔬  DEMO MODE — running 3 sample scenarios\n")
        all_results = []
        for sc in DEMO_SCENARIOS:
            res = calculate_sample_size(sc["baseline"], sc["mde"], sc["alpha"], sc["power"])
            res = add_duration(res, sc["daily_traffic"])
            sc_score = score_test_design(res)
            res["scenario"] = sc["label"]
            res["score"] = sc_score
            all_results.append(res)
            if not args.json:
                print(f"\n{'─'*60}")
                print(f"SCENARIO: {sc['label']}")
                pretty_print(res, sc_score)

        if args.json:
            print(json.dumps(all_results, indent=2))
        return

    # Single calculation mode
    if args.baseline is None or args.mde is None:
        print("Error: --baseline and --mde are required (or omit both for demo mode).", file=sys.stderr)
        sys.exit(1)

    result = calculate_sample_size(args.baseline, args.mde, args.alpha, args.power)
    if args.daily_traffic:
        result = add_duration(result, args.daily_traffic)
    sc_score = score_test_design(result)
    result["score"] = sc_score

    if args.json:
        print(json.dumps(result, indent=2))
    else:
        pretty_print(result, sc_score)


if __name__ == "__main__":
    main()