Day 13 — Stratified Sampling: The Smart Way to Sample 🎯📊
Divide and conquer your sampling strategy for maximum precision.
Stratified sampling guarantees coverage of important subgroups while reducing variance by 50-95% compared to simple random sampling.
💡 Note: This article uses technical terms and abbreviations. For definitions, check out the Key Terms & Glossary page.
🎲 The Random Sampling Trap
Imagine you're conducting a health survey in a company of 1,000 employees:
-
900 office workers (90%)
-
100 executives (10%)
You randomly sample 100 people. Here's what can go wrong:
Unlucky Sample #1:
Office workers: 95 people ✅
Executives: 5 people 😬
Problem: Only 5 executives - can't say much about this group!
Unlucky Sample #2:
Office workers: 87 people 🤔
Executives: 13 people 🤷
Different from reality (90/10 split)!
Unlucky Sample #3:
Office workers: 100 people 😱
Executives: 0 people 💀
Complete miss on executive health!
The problem: Simple Random Sampling (SRS) is... well, random! 🎲
The solution: Stratified Sampling - sample smartly within groups! 🧠✨
🎯 What is Stratified Sampling?
Stratified Sampling means:
-
Divide population into non-overlapping groups (strata)
-
Sample from each stratum separately
-
Combine results with proper weighting
Visual Comparison
Simple Random Sampling (SRS):
Show code (10 lines)
Population: 🔵🔵🔵🔵🔵🔵🔵🔵🔵 (office workers)
🔴 (executive)
Random sample of 10:
Picked: 🔵🔵🔵🔵🔵🔵🔵🔵🔵🔵
Result: All office workers! 😱
Stratified Sampling:
Show code (14 lines)
Population:
Stratum 1: 🔵🔵🔵🔵🔵🔵🔵🔵🔵 (90 office workers)
Stratum 2: 🔴 (10 executives)
Stratified sample of 10:
From Stratum 1: 🔵🔵🔵🔵🔵🔵🔵🔵🔵 (9 people)
From Stratum 2: 🔴 (1 person)
Result: Proper representation! ✅
🎯 Why Stratify? Three Big Reasons
1. Guaranteed Coverage 🛡️
Problem with SRS: Might miss rare but important groups
Example:
Show code (16 lines)
City population:
- Urban: 70%
- Suburban: 20%
- Rural: 10%
SRS of 100 might give:
Urban: 65, Suburban: 25, Rural: 10
OR
Urban: 75, Suburban: 22, Rural: 3 ← Rural underrepresented!
Stratified solution:
Show code (10 lines)
Explicitly sample from each:
Urban: 70 people (guaranteed)
Suburban: 20 people (guaranteed)
Rural: 10 people (guaranteed)
Coverage ensured! ✅
2. Variance Reduction 📉
The Math Intuition:
Variance comes from differences:
-
Between-stratum variance: How different are the groups?
-
Within-stratum variance: How different are people within each group?
Key insight: If strata are homogeneous (similar within), stratified sampling has lower variance than SRS!
Visual:
Show code (22 lines)
POPULATION (high variance):
Health scores: 45, 48, 50, 52, 85, 87, 88, 90, 91, 92
↑_________↑ ↑___________________↑
Office Executives
(lower) (higher)
Within-stratum variance:
Office: σ² = 6.5 (people similar)
Executive: σ² = 7.8 (people similar)
But between-stratum difference is HUGE (50 vs 90)!
SRS estimates affected by this big gap.
Stratified sampling accounts for it separately! ✅
3. Domain Insights 🔍
SRS result:
"Average health score: 75"
Okay... but tells us nothing about groups!
Stratified result:
"Average health scores:
Office workers: 52 (95% CI: 50-54)
Executives: 88 (95% CI: 86-90)"
Rich insights about each segment! 🎯
📊 The Math: How Much Better Is It?
Variance Formula
Simple Random Sampling variance:
Show code (12 lines)
Var(ȳ_SRS) = σ²/n × (N-n)/N
Where:
- σ² = overall population variance
- n = sample size
- N = population size
- (N-n)/N = finite population correction
Stratified Sampling variance:
Show code (12 lines)
Var(ȳ_strat) = Σ(Wₕ² × σₕ²/nₕ × (Nₕ-nₕ)/Nₕ)
Where:
- Wₕ = stratum h weight (Nₕ/N)
- σₕ² = variance within stratum h
- nₕ = sample size in stratum h
- Nₕ = population size in stratum h
The Variance Reduction:
Var(ȳ_SRS) - Var(ȳ_strat) = Σ Wₕ(μₕ - μ)²
This is the between-stratum variance!
Translation: The more different your strata are, the bigger the variance reduction! 🎉
Example Calculation 🧮
Population:
-
Stratum 1 (Office): N₁ = 900, μ₁ = 50, σ₁² = 100
-
Stratum 2 (Executive): N₂ = 100, μ₂ = 90, σ₂² = 64
-
Total: N = 1000
Sample: n = 100
Proportional allocation:
-
n₁ = 90 (90% of sample)
-
n₂ = 10 (10% of sample)
SRS Variance:
First, calculate overall variance:
Show code (12 lines)
μ = 0.9(50) + 0.1(90) = 45 + 9 = 54
σ² = 0.9(100 + (50-54)²) + 0.1(64 + (90-54)²)
= 0.9(100 + 16) + 0.1(64 + 1296)
= 0.9(116) + 0.1(1360)
= 104.4 + 136
= 240.4
Var(ȳ_SRS) = 240.4/100 × (1000-100)/1000
= 2.404 × 0.9
= 2.16
Standard error: √2.16 = 1.47 📏
Stratified Variance:
Show code (20 lines)
W₁ = 900/1000 = 0.9
W₂ = 100/1000 = 0.1
Var(ȳ_strat) = 0.9² × (100/90) × (900-90)/900
+ 0.1² × (64/10) × (100-10)/100
= 0.81 × 1.11 × 0.9
+ 0.01 × 6.4 × 0.9
= 0.81 + 0.058
= 0.87
Standard error: √0.87 = 0.93 📏
The Improvement:
Show code (10 lines)
Variance reduction: 2.16 - 0.87 = 1.29 (60% reduction! 🎉)
Standard error:
SRS: 1.47
Stratified: 0.93
Stratified is 58% more precise! ✨
Translation: To get the same precision with SRS, you'd need 2.5× more samples!
🎚️ Allocation Strategies: How Many Per Stratum?
Once you decide to stratify, how do you divide your sample across strata?
1. Proportional Allocation ⚖️ (Most Common)
Rule: Sample proportionally to stratum size
nₕ = n × (Nₕ/N)
Example:
Population: 900 office, 100 executive (1000 total)
Sample size: n = 100
Office sample: 100 × (900/1000) = 90
Executive sample: 100 × (100/1000) = 10
Pros:
-
✅ Simple, intuitive
-
✅ Self-weighting (no complex weights needed)
-
✅ Represents population structure
Cons:
- ⚠️ Small strata get small samples (might be imprecise)
2. Equal Allocation 🟰
Rule: Same sample size for each stratum
nₕ = n / H
Where H = number of strata
Example:
Show code (10 lines)
Population: 900 office, 100 executive
Sample size: n = 100
Strata: H = 2
Office sample: 100/2 = 50
Executive sample: 100/2 = 50
Pros:
-
✅ Good for comparing strata (equal precision)
-
✅ Ensures small strata have enough data
Cons:
-
⚠️ Oversamples small strata (need complex weights)
-
⚠️ Less efficient for overall mean estimation
3. Neyman Allocation 🎯 (Optimal)
Rule: Allocate proportional to stratum size AND variance
nₕ = n × (Nₕ × σₕ) / Σ(Nₖ × σₖ)
Intuition: Sample more from:
-
Large strata (more people → more important)
-
High-variance strata (more diverse → need more samples)
Example:
Show code (14 lines)
Stratum 1: N₁ = 900, σ₁ = 10
Stratum 2: N₂ = 100, σ₂ = 8
Stratum 1 weight: 900 × 10 = 9,000
Stratum 2 weight: 100 × 8 = 800
Total weight: 9,800
Office sample: 100 × (9000/9800) = 91.8 ≈ 92
Executive sample: 100 × (800/9800) = 8.2 ≈ 8
Pros:
-
✅ Mathematically optimal (minimizes variance!)
-
✅ Accounts for both size and heterogeneity
Cons:
-
⚠️ Requires knowing σₕ in advance (often unknown!)
-
⚠️ Might still undersample important small strata
4. Optimal Allocation with Cost 💰
Rule: Account for different sampling costs per stratum
nₕ = n × (Nₕ × σₕ / √cₕ) / Σ(Nₖ × σₖ / √cₖ)
Where cₕ = cost to sample one unit from stratum h
Example:
Executives cost 5× more to survey (busy, need incentives)
c₁ = $10 (office worker)
c₂ = $50 (executive)
This would reduce executive sample further!
Use when: Budget constrained, different costs per stratum
📈 Visual: Variance vs Allocation
Let's see how variance changes with different allocations:
Show code (34 lines)
Variance (SE²)
│
3.0 │
│ • SRS
2.5 │
│
2.0 │ • Equal
│
1.5 │
│
1.0 │ • Proportional
│
0.5 │ • Neyman
│ (Optimal!)
0.0 └────────────────────────────────
Different Allocation Strategies
Lower is better! ✅
Takeaway: Neyman always wins (if you know the variances)!
🎨🔬 Defining Strata: The Art and Science
Good strata are:
1. Mutually Exclusive 🚫
Each unit belongs to exactly one stratum
❌ Bad: "Young", "Students"
(Young students counted twice!)
✅ Good: "Student", "Non-Student"
2. Exhaustive 📦
Every unit belongs to some stratum
❌ Bad: "<30", "40-60", ">60"
(Missing 30-40 age range!)
✅ Good: "<30", "30-40", "40-60", ">60"
3. Homogeneous Within 🟰
Units within stratum are similar
❌ Bad stratum: "People" (too diverse!)
✅ Good stratum: "Female doctors aged 40-50"
4. Heterogeneous Between 🎭
Strata are different from each other
❌ Bad: "Age 30-40", "Age 31-41"
(Too much overlap, not distinct!)
✅ Good: "Age 18-30", "Age 31-50", "Age 51+"
5. Meaningful 💡
Based on domain knowledge, not arbitrary
❌ Bad: "First 500 rows", "Last 500 rows"
(Arbitrary split!)
✅ Good: "Urban", "Suburban", "Rural"
(Meaningful demographic divisions)
Common Stratification Variables:
Demographics:
-
Age groups
-
Gender
-
Education level
-
Income brackets
-
Geographic region
Business:
-
Customer segments (high/medium/low value)
-
Product categories
-
Time periods (Q1, Q2, Q3, Q4)
Medical:
-
Disease severity (mild/moderate/severe)
-
Treatment type
-
Risk factors present/absent
🎓 Exercise: SRS vs Stratified Variance
Let's work through a complete example!
The Setup
Population of 20 people:
Stratum 1 (Group A) - 15 people:
Values: [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
Mean (μ₁) = 17
Variance (σ₁²) = 20
SD (σ₁) = 4.47
Stratum 2 (Group B) - 5 people:
Values: [50, 52, 54, 56, 58]
Mean (μ₂) = 54
Variance (σ₂²) = 8
SD (σ₂) = 2.83
Overall population:
N = 20
μ = (15×17 + 5×54)/20 = (255 + 270)/20 = 26.25
We want to sample n = 8 people.
Approach 1: Simple Random Sampling
Variance calculation:
First, overall variance:
Show code (28 lines)
σ² = Σ(xᵢ - μ)² / N
For stratum 1 contribution:
= (15/20) × [σ₁² + (μ₁ - μ)²]
= 0.75 × [20 + (17 - 26.25)²]
= 0.75 × [20 + 85.56]
= 0.75 × 105.56
= 79.17
For stratum 2 contribution:
= (5/20) × [σ₂² + (μ₂ - μ)²]
= 0.25 × [8 + (54 - 26.25)²]
= 0.25 × [8 + 770.06]
= 0.25 × 778.06
= 194.52
Total: σ² = 79.17 + 194.52 = 273.69
Var(ȳ_SRS) = σ²/n × (N-n)/N
= 273.69/8 × (20-8)/20
= 34.21 × 0.6
= 20.53
Standard Error: √20.53 = 4.53 📏
Approach 2: Proportional Stratified Sampling
Sample allocation:
n₁ = 8 × (15/20) = 6
n₂ = 8 × (5/20) = 2
Variance calculation:
Show code (26 lines)
W₁ = 15/20 = 0.75
W₂ = 5/20 = 0.25
Var(ȳ_strat) = W₁² × (σ₁²/n₁) × (N₁-n₁)/N₁
+ W₂² × (σ₂²/n₂) × (N₂-n₂)/N₂
= 0.75² × (20/6) × (15-6)/15
+ 0.25² × (8/2) × (5-2)/5
= 0.5625 × 3.33 × 0.6
+ 0.0625 × 4 × 0.6
= 1.125 + 0.15
= 1.275
Standard Error: √1.275 = 1.13 📏
Approach 3: Neyman Allocation
Optimal allocation:
Show code (10 lines)
Stratum 1: N₁ × σ₁ = 15 × 4.47 = 67.05
Stratum 2: N₂ × σ₂ = 5 × 2.83 = 14.15
Total: 81.20
n₁ = 8 × (67.05/81.20) = 6.6 ≈ 7
n₂ = 8 × (14.15/81.20) = 1.4 ≈ 1
Variance calculation:
Show code (16 lines)
Var(ȳ_Neyman) = 0.75² × (20/7) × 0.53
+ 0.25² × (8/1) × 0.8
= 0.5625 × 2.86 × 0.53
+ 0.0625 × 8 × 0.8
= 0.85 + 0.40
= 1.25
Standard Error: √1.25 = 1.12 📏
The Comparison Table 📊
| Method | n₁ | n₂ | Variance | SE | Efficiency |
|--------|----|----|----------|-----|------------|
| SRS | varies | varies | 20.53 | 4.53 | 1.00× |
| Proportional | 6 | 2 | 1.275 | 1.13 | 16.1× 🎉 |
| Neyman | 7 | 1 | 1.25 | 1.12 | 16.4× 🏆 |
Key Insights:
✅ Stratified sampling reduces variance by 94%! (From 20.53 to 1.27)
✅ Why? The two groups are VERY different (means of 17 vs 54), but within each group, people are similar
✅ Neyman only slightly better than proportional (1.25 vs 1.275) - proportional is good enough!
✅ To match stratified precision with SRS, you'd need 129 samples instead of 8! (16× more)
Visual Representation 📊
Show code (10 lines)
Standard Error Comparison
SRS: ████████████████████████████████ (4.53)
Proportional: ███ (1.13)
Neyman: ███ (1.12)
Shorter bars = Better (more precise)! ✅
🐍 Python Implementation
perform_stratification Function
Show code (100 lines)
import pandas as pd
import numpy as np
def perform_stratification(df, stratum_col, target_col,
sample_size, allocation='proportional'):
"""
Perform stratified sampling
Parameters:
- df: DataFrame
- stratum_col: Column defining strata
- target_col: Variable of interest
- sample_size: Total sample size
- allocation: 'proportional', 'equal', or 'neyman'
Returns:
- sample_df: Stratified sample
- summary: Allocation summary
"""
# Calculate stratum statistics
stratum_stats = df.groupby(stratum_col).agg({
target_col: ['count', 'mean', 'std']
}).reset_index()
stratum_stats.columns = [stratum_col, 'N', 'mean', 'std']
stratum_stats['weight'] = stratum_stats['N'] / len(df)
# Calculate sample sizes per stratum
if allocation == 'proportional':
stratum_stats['n'] = (
sample_size * stratum_stats['weight']
).round().astype(int)
elif allocation == 'equal':
n_strata = len(stratum_stats)
stratum_stats['n'] = sample_size // n_strata
elif allocation == 'neyman':
# Optimal allocation
stratum_stats['allocation_weight'] = (
stratum_stats['N'] * stratum_stats['std']
)
total_weight = stratum_stats['allocation_weight'].sum()
stratum_stats['n'] = (
sample_size * stratum_stats['allocation_weight'] / total_weight
).round().astype(int)
# Adjust for rounding errors
total_allocated = stratum_stats['n'].sum()
if total_allocated != sample_size:
diff = sample_size - total_allocated
# Add/subtract from largest stratum
largest_idx = stratum_stats['N'].idxmax()
stratum_stats.loc[largest_idx, 'n'] += diff
# Perform stratified sampling
sample_dfs = []
for _, row in stratum_stats.iterrows():
stratum_data = df[df[stratum_col] == row[stratum_col]]
stratum_sample = stratum_data.sample(
n=int(row['n']),
replace=False,
random_state=42
)
sample_dfs.append(stratum_sample)
sample_df = pd.concat(sample_dfs, ignore_index=True)
return sample_df, stratum_stats
def show_stratification_summary(stratum_stats, allocation_type):
"""
Display stratification summary with variance estimates
"""
print(f"\n{'='*60}")
print(f"Stratification Summary - {allocation_type.title()} Allocation")
print(f"{'='*60}\n")
print(stratum_stats.to_string(index=False))
# Calculate overall variance
total_n = stratum_stats['n'].sum()
variance_components = (
stratum_stats['weight']**2 *
stratum_stats['std']**2 / stratum_stats['n'] *
(stratum_stats['N'] - stratum_stats['n']) / stratum_stats['N']
)
total_variance = variance_components.sum()
se = np.sqrt(total_variance)
print(f"\n{'='*60}")
print(f"Overall Variance: {total_variance:.4f}")
print(f"Standard Error: {se:.4f}")
print(f"{'='*60}\n")
return total_variance, se
Usage Example
Show code (36 lines)
# Create toy dataset
np.random.seed(42)
df = pd.DataFrame({
'group': ['A']*150 + ['B']*50,
'value': np.concatenate([
np.random.normal(50, 10, 150), # Group A
np.random.normal(90, 8, 50) # Group B
])
})
# Proportional allocation
sample_prop, stats_prop = perform_stratification(
df, 'group', 'value',
sample_size=100,
allocation='proportional'
)
var_prop, se_prop = show_stratification_summary(stats_prop, 'proportional')
# Neyman allocation
sample_neyman, stats_neyman = perform_stratification(
df, 'group', 'value',
sample_size=100,
allocation='neyman'
)
var_neyman, se_neyman = show_stratification_summary(stats_neyman, 'neyman')
# Compare with SRS
overall_std = df['value'].std()
var_srs = overall_std**2 / 100 * (200-100)/200
se_srs = np.sqrt(var_srs)
print(f"\nComparison:")
print(f"SRS SE: {se_srs:.4f}")
print(f"Proportional SE: {se_prop:.4f} ({(1-se_prop/se_srs)*100:.1f}% reduction)")
print(f"Neyman SE: {se_neyman:.4f} ({(1-se_neyman/se_srs)*100:.1f}% reduction)")
🎯 When to Use Stratified Sampling
Perfect For:
✅ Known subgroups that differ meaningfully
-
Demographics (age, gender, region)
-
Business segments (customer tiers)
-
Risk categories (low/medium/high)
✅ Small important groups you must include
-
Rare diseases
-
Executive opinions
-
Minority populations
✅ Variance reduction is critical
-
Limited budget (need maximum precision)
-
Policy decisions (small errors matter)
-
Clinical trials (safety critical)
✅ Domain insights needed per group
-
Compare regions
-
Track segments over time
-
Identify disparities
Don't Use When:
❌ No clear strata exist
-
Homogeneous population
-
Unknown groupings
-
Exploratory research (don't know what matters)
❌ Stratum information unavailable at sampling time
-
Can't identify strata before sampling
-
Post-hoc stratification (use weighting instead)
❌ Very small sample sizes
-
n < 30: Stratification overhead not worth it
-
Not enough samples per stratum
❌ Cost prohibitive
-
Different strata require vastly different effort
-
Geographic dispersion makes stratified sampling impractical
⚠️ Common Pitfalls
1. Too Many Strata 🚫
❌ Bad: 20 strata with n=100 samples
→ 5 samples per stratum (unreliable!)
✅ Good: 4-5 strata with n=100 samples
→ 20-25 per stratum (stable estimates)
Rule of thumb: At least 10-15 samples per stratum
2. Ignoring Weights 🚫
With non-proportional allocation, you MUST weight results:
# Wrong (unweighted)
overall_mean = sample_df['value'].mean()
# Right (weighted)
stratum_means = sample_df.groupby('stratum')['value'].mean()
stratum_weights = population_stratum_sizes / population_total
overall_mean = (stratum_means * stratum_weights).sum()
3. Defining Overlapping Strata 🚫
Show code (18 lines)
❌ Bad:
Stratum A: "Students"
Stratum B: "Age < 25"
→ Young students belong to both!
✅ Good:
Stratum A: "Student, Age < 25"
Stratum B: "Student, Age ≥ 25"
Stratum C: "Non-student, Age < 25"
Stratum D: "Non-student, Age ≥ 25"
4. Forgetting Finite Population Correction 🚫
When sampling a large fraction of the stratum:
# Include FPC
var_stratum = (sigma²/n) * (N-n)/N
# Not just
var_stratum = sigma²/n # Wrong!
🌟 Takeaway
Stratified sampling is the "divide and conquer" of sampling:
Key Concepts:
✅ Strata = non-overlapping, exhaustive groups
✅ Proportional allocation = sample proportionally (simple, self-weighting)
✅ Neyman allocation = optimal (proportional to Nₕ × σₕ)
✅ Variance reduction = can be 50-95% lower than SRS!
✅ Coverage guarantee = ensures rare groups included
✅ Domain insights = separate estimates per stratum
The Math Win:
Variance reduction = Σ Wₕ(μₕ - μ)²
Translation: The more different your strata,
the bigger the improvement! 📈
Allocation Decision Tree:
Show code (18 lines)
Do you know σₕ for each stratum?
│
├─ Yes → Use Neyman (optimal!) 🏆
│
└─ No → Do you need equal precision per stratum?
│
├─ Yes → Use Equal allocation ⚖️
│
└─ No → Use Proportional (simplest) ✅
Real Impact:
In our exercise, stratified sampling gave 16× more precision than SRS with the same sample size. That's like getting 129 samples for the price of 8! 💰✨
📚 References
-
Cochran, W. G. (1977). Sampling Techniques (3rd ed.). John Wiley & Sons.
-
Lohr, S. L. (2019). Sampling: Design and Analysis (3rd ed.). Chapman and Hall/CRC.
-
Kish, L. (1965). Survey Sampling. John Wiley & Sons.
-
Neyman, J. (1934). On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection. Journal of the Royal Statistical Society, 97(4), 558-625.
-
Särndal, C. E., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. Springer-Verlag.
-
Thompson, S. K. (2012). Sampling (3rd ed.). John Wiley & Sons.
-
Valliant, R., Dever, J. A., & Kreuter, F. (2018). Practical Tools for Designing and Weighting Survey Samples (2nd ed.). Springer.
-
Groves, R. M., Fowler, F. J., Couper, M. P., Lepkowski, J. M., Singer, E., & Tourangeau, R. (2009). Survey Methodology (2nd ed.). John Wiley & Sons.
-
Little, R. J., & Rubin, D. B. (2019). Statistical Analysis with Missing Data (3rd ed.). John Wiley & Sons.
-
Lumley, T. (2010). Complex Surveys: A Guide to Analysis Using R. John Wiley & Sons.
Day 13 Complete! 🎉
This is Day 13 of my 30-day challenge documenting my Data Science journey at Oracle! Stay tuned for more insights and mathematical foundations of data science. 🚀
Next: Day 14 - Power Analysis and Sample Size Calculation, where we'll figure out how many samples you REALLY need to detect an effect! 📊🔬
💡 Note: This article uses technical terms like stratified sampling, variance reduction, allocation, Neyman allocation, and finite population correction. For definitions, check out the Key Terms & Glossary page.
