Sampling & Central Limit Theorem

Master the bridge between samples and populations - from sampling methods to the powerful CLT that underpins all inferential statistics.

Inferential Statistics Foundation Advanced Level 50 min

🎯 Population vs Sample

Population

Complete set of all items/individuals of interest.

Parameter: True value (μ, σ)
Often impractical to measure
Example: All voters in country

Sample

Subset selected from population.

Statistic: Estimated value (x̄, s)
Practical to measure
Example: 1000 voters surveyed

📊 Population → Sample → Inference

Population

All elements
Parameters: μ, σ

↓

Sample

Selected subset
Statistics: x̄, s

→

📈

Inference

Make conclusions
about population

🏢 Real World Example: E-commerce

Population: All 1M customers

True conversion rate: μ = 3.2%

Sample: 1000 customers surveyed

Sample rate: x̄ = 3.5%

Inference: Estimate true rate

95% CI: [3.1%, 3.9%]

🎯 Sampling Methods

🎯

Simple Random

Every member has equal chance

Example: Random number generator

✅ No bias
❌ May miss subgroups

📊

Stratified

Divide into strata, sample from each

Example: Age groups 18-25, 26-40, 41+

✅ Represents all groups
❌ Need prior knowledge

🏙️

Cluster

Randomly select clusters, sample all in cluster

Example: Random cities, survey all schools

✅ Cost-effective
❌ Less precise

🔢

Systematic

Select every kth element

Example: Every 10th customer

✅ Simple to implement
❌ Periodic bias risk

📊 Sampling Methods Comparison

Simple Random

Random selection

Stratified

By subgroups

🎯 When to Use Which Method?

🎯

Simple Random
No prior info, homogeneous

📊

Stratified
Important subgroups exist

🏙️

Cluster
Geographic/cost constraints

🔢

Systematic
Assembly line, queues

⚠️ Bias in Sampling

🎯 Selection Bias

Sample not representative of population

Example: Phone survey misses people without phones

Solution: Use appropriate sampling frame

🗣️ Response Bias

Answers not truthful or accurate

Example: Social desirability bias in surveys

Solution: Anonymous surveys, neutral questions

📭 Non-response Bias

Participants differ from non-participants

Example: Online survey only reaches tech-savvy

Solution: Follow-ups, incentives

📊 Bias vs Random Error

Unbiased, Precise

Centered on target, low spread

Biased, Precise

Consistently off-target

Key Insight: Bias = Systematic error | Random error = Natural variation

🛡️ How to Avoid Bias

🎯

Random Sampling
Equal chance for all

📋

Clear Questions
Neutral, unambiguous

📊

Adequate Response Rate
Follow up non-respondents

🔍

Pilot Testing
Test survey first

📊 Sampling Error

📖 What is Sampling Error?

Error = x̄ - μ

Sample mean - Population mean

Definition: Difference between sample statistic and population parameter due to random chance.

✅ Sampling Error

Due to random chance
Reduced by larger samples
Quantifiable with SE
Always present
Example: Different samples give different x̄

❌ Bias

Systematic error
Not reduced by larger samples
Hard to quantify
Can be eliminated
Example: Poor sampling method

📐 Standard Error Formulas

                                SE = σ / √n
                            

Population σ known
σ = population SD
n = sample size

                                SE ≈ s / √n
                            

Population σ unknown
s = sample SD
n = sample size

SE measures precision: Smaller SE → More precise estimate

🎯 Example: Customer Satisfaction

Scenario: Survey customer satisfaction (scale 1-10)

True population mean

7.2

Population SD

1.5

Sample size

100

Standard Error: SE = 1.5 / √100 = 1.5 / 10 = 0.15

Interpretation: Sample means will typically vary by about 0.15 from true population mean.

📉 Reducing Sampling Error

📈

Larger n
SE ∝ 1/√n

🎯

Stratified Sampling
Reduces variance

📊

Efficient Design
Better sampling methods

🔍

Pilot Studies
Estimate variance

📊 Sampling Distribution

📖 What is Sampling Distribution?

x̄'s

Distribution of sample means

Definition: Probability distribution of a statistic (like sample mean) obtained from all possible samples of size n from population.

🎯 Visual Example: Multiple Samples

Population: Mean = 50, SD = 10 | Sample size: n = 30

Population Distribution μ = 50, σ = 10

μ = 50

↓ Take multiple samples (n=30)

Sampling Distribution of x̄ μ_x̄ = 50, σ_x̄ = 10/√30 ≈ 1.83

μ_x̄ = 50

Narrower spread!

📊 Properties of Sampling Distribution of x̄

μ_x̄ = μ

Mean

Mean of sample means equals population mean

σ_x̄ = σ/√n

Standard Error

Spread decreases with larger n

Shape

Distribution

Normal if population normal or n ≥ 30

📈 Effect of Sample Size on SE

n = 10

SE = σ/√10
≈ 0.316σ

n = 30

SE = σ/√30
≈ 0.183σ

n = 100

SE = σ/√100
= 0.1σ

💡 Key Insight: To halve SE, need to quadruple n! (SE ∝ 1/√n)

🌟 Central Limit Theorem (CLT)

📖 The Magic of CLT

CLT

Statistical Superpower

Theorem: For large enough sample size (n ≥ 30), sampling distribution of sample mean approaches normal distribution, regardless of population distribution.

📊 CLT in Action: Any Population → Normal

Skewed Population

Right-skewed distribution

→

Take samples
n ≥ 30

Sampling Distribution of x̄

Normal distribution!

Magic: Even from skewed population, sample means become normally distributed!

✅ CLT Conditions

🎯

Random Sampling

Samples must be random

📊

Independence

Observations independent (n ≤ 10% population)

🔢

Sample Size

n ≥ 30 or population normal

📐 CLT Formulas & Implications

                            x̄ ~ N(μ, σ/√n)
                        

Sample mean distribution
Normal with mean μ
Standard error σ/√n

                            Z = (x̄ - μ) / (σ/√n)
                        

Z-score for sample mean
Standard normal distribution
For confidence intervals & testing

🎯 CLT Example: Website Load Times

Scenario: Website load times are exponentially distributed with mean 2 seconds.

Population mean

2 seconds

Population SD

2 seconds

Sample size

CLT Application: Even though load times are exponential, sample mean of 50 measurements ~ Normal(μ=2, SE=2/√50≈0.283)

Question: P(average load time > 2.5 seconds)?
Answer: Z = (2.5-2)/0.283 = 1.77 → P ≈ 0.038 (3.8%)

🔢 The n ≥ 30 Rule

🤔 Why n = 30 Specifically?

Magic Number

Reason: At n=30, Student's t-distribution closely approximates normal distribution (difference < 5%). Provides good balance between practicality and accuracy.

📈 Sample Size Effect on Normality

n = 5

Still skewed
Not normal

n = 15

Getting closer
Almost normal

n = 30

Good approximation
CLT works well

n = 50

Excellent
Very normal

⚠️ When n ≥ 30 is NOT Enough

📊 Highly Skewed Data

For extremely skewed distributions (e.g., income), need n > 100 for CLT to work well.

📉 Heavy Tails

Distributions with extreme outliers (e.g., financial returns) need larger samples.

🎯 Small Population

If sampling >10% of population, need finite population correction.

🎯 Practical Sample Size Guidelines

📊

Normal Population
Any n works

🎯

Moderate Skew
n ≥ 30

⚠️

Extreme Skew
n ≥ 100

📈

Binary Data
np ≥ 10, n(1-p) ≥ 10

🧠 Data Science Applications

📊

A/B Testing

CLT Application: Compare conversion rates between groups

Process:

Collect sample data (n ≥ 30 per group)
CLT ensures normal distribution of means
Calculate p-value using normal approximation
Make decision with confidence

Without CLT: Need complex non-parametric tests

🎯

Confidence Intervals

CLT Application: Estimate population parameters

Formula (95% CI):

                            x̄ ± 1.96 × (s/√n)
                        

Why it works: CLT guarantees x̄ ~ Normal, allowing Z-scores

🏭

Quality Control

Sampling Application: Monitor production processes

Control Charts:

Take samples of n items periodically
Plot sample means over time
CLT: Means normally distributed
Set control limits using SE

Example: Detect machine malfunction early

🏢 Case Study: E-commerce A/B Test

🎯 Problem

Test new checkout button color (red vs blue)

📊 Sampling

Randomly assign 1000 users to each group (n=1000)

📈 CLT Application

Conversion rates ~ Normal by CLT
SE = √[p(1-p)/n]

Results:
• Red: 120/1000 = 12% conversion
• Blue: 150/1000 = 15% conversion
• Difference: 3% (p-value = 0.02) → Statistically significant!

🎯 Interview Questions Preview

Q: Why is n=30 considered the magic number for CLT?

A: At n=30, t-distribution approximates normal within 5%. Balance between practicality and statistical accuracy.

Q: When would you NOT use CLT even with n=30?

A: Extremely skewed distributions, heavy-tailed data, or when sampling >10% of population.

✅ Chapter Summary

🎯

Sampling

Population → Sample → Inference

Bias vs Random Error

📊

Sampling Distribution

Distribution of sample statistics

                        μ_x̄ = μ, σ_x̄ = σ/√n
                    

🌟

Central Limit Theorem

n ≥ 30 → Normal distribution

                        x̄ ~ N(μ, σ/√n)
                    

⚡ Key Formulas

SE = σ/√n μ_x̄ = μ 95% CI: x̄ ± 1.96×SE Z = (x̄-μ)/SE n ≥ 30 rule

💡 Practical Tips

🎯

Always check for bias

📊

n=30 for CLT

⚠️

Watch for extreme skew

📈

SE decreases with √n

Sampling & Central Limit Theorem

🎯 Population vs Sample

Population

Sample

📊 Population → Sample → Inference

Population

Sample

Inference

🏢 Real World Example: E-commerce

🎯 Sampling Methods

Simple Random

Stratified

Cluster

Systematic

📊 Sampling Methods Comparison

Simple Random

Stratified

🎯 When to Use Which Method?

⚠️ Bias in Sampling

🎯 Selection Bias

🗣️ Response Bias

📭 Non-response Bias

📊 Bias vs Random Error

Unbiased, Precise

Biased, Precise

🛡️ How to Avoid Bias

📊 Sampling Error

📖 What is Sampling Error?

✅ Sampling Error

❌ Bias

📐 Standard Error Formulas

🎯 Example: Customer Satisfaction

📉 Reducing Sampling Error

📊 Sampling Distribution

📖 What is Sampling Distribution?

🎯 Visual Example: Multiple Samples

📊 Properties of Sampling Distribution of x̄

Mean

Standard Error

Distribution

📈 Effect of Sample Size on SE

🌟 Central Limit Theorem (CLT)

📖 The Magic of CLT

📊 CLT in Action: Any Population → Normal

Skewed Population

Sampling Distribution of x̄

✅ CLT Conditions

Random Sampling

Independence

Sample Size

📐 CLT Formulas & Implications

🎯 CLT Example: Website Load Times

🔢 The n ≥ 30 Rule

🤔 Why n = 30 Specifically?

📈 Sample Size Effect on Normality

⚠️ When n ≥ 30 is NOT Enough

📊 Highly Skewed Data

📉 Heavy Tails

🎯 Small Population

🎯 Practical Sample Size Guidelines

🧠 Data Science Applications

A/B Testing

Confidence Intervals

Quality Control

🏢 Case Study: E-commerce A/B Test

🎯 Problem

📊 Sampling

📈 CLT Application

🎯 Interview Questions Preview

✅ Chapter Summary

Sampling

Sampling Distribution

Central Limit Theorem

⚡ Key Formulas

💡 Practical Tips

Sampling & CLT Practice

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