Sampling & Sampling Distributions: Methods, Bias, Errors & Central Limit Theorem

TAKEAWAYS:

• Sampling vs. Census: Sampling is faster, cheaper, and practical for large populations (e.g., NTC surveys), while census is exhaustive but costly (e.g., NEPSE’s annual reports).
• Probability Sampling: Methods like simple random, stratified, and cluster sampling reduce bias; non-probability methods (e.g., convenience sampling) risk skewing results.
• Sampling Distribution: The distribution of a statistic (e.g., sample mean) across repeated samples follows a normal distribution (CLT), even if the population isn’t normal.
• Central Limit Theorem (CLT): For large samples (n ≥ 30), the sampling distribution of the mean is normal, regardless of the population distribution. This justifies using z-tests and confidence intervals.
• Sampling Error: The difference between a sample statistic (e.g., sample mean) and the population parameter. Reduce it by increasing sample size or using better sampling methods.
• Real-World Tie: Khalti’s fraud detection uses stratified sampling to analyze transaction patterns across regions, while Pathao’s driver allocation relies on cluster sampling for efficient route planning.

1. Why Sample? Sampling vs. Census

Definitions

• Population: The entire group being studied (e.g., all Nepalese IT students, all Daraz customers).
• Sample: A subset of the population used to estimate population parameters.
• Census: Collecting data from every member of the population (e.g., Nepal’s national census).

Advantages of Sampling

graph TD
    A["Sampling"] --> B["Faster"]
    A --> C["Cheaper"]
    A --> D["Practical for large populations"]
    A --> E["Less invasive"]
    A --> F["Allows statistical inference"]

Disadvantages of Sampling

• Sampling Error: The difference between the sample statistic and the true population parameter.
• Non-response Bias: If sampled individuals refuse to participate (e.g., low response rates in NTC’s customer satisfaction surveys).
• Selection Bias: When the sample isn’t representative (e.g., surveying only Kathmandu residents for national trends).

When to Use Census?

• Small populations (e.g., employees in a single company).
• Critical decisions (e.g., NEPSE’s annual audits).

Visual: Two Venn diagrams showing how populations are divided into strata (homogeneous groups) vs. clusters (heterogeneous groups).

2. Types of Sampling Methods

A. Probability Sampling (Unbiased)

Every member has a known chance of being selected. Four key methods:

B. Non-Probability Sampling (Biased but Practical)

Used when probability sampling is infeasible. Three common methods:

WORKED EXAMPLE 1: Stratified Sampling for Ncell’s Customer Satisfaction Ncell wants to survey 500 customers across 3 regions (East, West, Central) with populations of 2M, 1.5M, and 2.5M respectively. Design a stratified sample.

Steps:

1. Calculate stratum sizes:
   • East: $\frac{2,000,000}{6,000,000}\times 500=167$
   • West: $\frac{1,500,000}{6,000,000}\times 500=125$
   • Central: $\frac{2,500,000}{6,000,000}\times 500=208$
2. Randomly select 167 East, 125 West, and 208 Central customers using simple random sampling within each stratum.

Why Stratified?

• Ensures representation from all regions.
• Reduces sampling error compared to simple random sampling.

3. Sampling Distribution and the Central Limit Theorem (CLT)

Key Concepts

• Sampling Distribution: The distribution of a statistic (e.g., sample mean, $\bar{X}$) across all possible samples of size n.
• CLT: For large n (≥30), the sampling distribution of $\bar{X}$ is approximately normal, regardless of the population distribution.

Visualizing the CLT

Key Observations:

1. The sampling distribution is normal, even if the population is skewed (e.g., income data).
2. The mean of the sampling distribution (${\mu }_{\bar{X}}$) equals the population mean ($\mu$).
3. The standard deviation of the sampling distribution (${\sigma }_{\bar{X}}$ or SE) is: $${\sigma }_{\bar{X}}=\frac{\sigma }{\sqrt{n}}$$ where $\sigma$ = population standard deviation.

WORKED EXAMPLE 2: CLT in Action – Daraz’s Order Processing Daraz processes orders with a population mean delivery time $\mu =5$ days and $\sigma =2$ days. If a sample of n=40 orders is taken:

1. What is the probability that the sample mean delivery time is >5.5 days?
2. How does the sampling distribution change if n=100?

Solution:

1. Calculate ${\sigma }_{\bar{X}}$: $${\sigma }_{\bar{X}}=\frac{2}{\sqrt{40}}=0.32$$
2. Standardize the value: $$Z=\frac{5.5-5}{0.32}=1.56$$
3. Find P(Z > 1.56) from z-table: 0.0586 (5.86%).
4. For n=100: $${\sigma }_{\bar{X}}=\frac{2}{\sqrt{100}}=0.2$$ The distribution becomes narrower, reducing sampling error.

Real-World Tie: Daraz uses the CLT to predict delivery delays by analyzing sample means from different regions. A sample mean >5.5 days triggers alerts for logistics teams.

4. Sampling Error and How to Reduce It

Types of Sampling Error

1. Random Error: Due to chance (e.g., selecting a non-representative sample).
2. Systematic Error: Due to flawed methods (e.g., biased sampling frame).

How to Reduce Sampling Error

WORKED EXAMPLE 3: Sample Size Calculation for NEPSE NEPSE wants to estimate the average stock price with a margin of error (E) of Rs. 500 and 95% confidence. The population standard deviation ($\sigma$) is Rs. 2,000. Calculate the required sample size.

Formula: $$n={\left(\frac{Z\times \sigma }{E}\right)}^2$$ where $Z=1.96$ for 95% confidence.

Calculation: $$n={\left(\frac{1.96\times 2000}{500}\right)}^2=(7.84{)}^2=61.5\approx 62$$

Interpretation: NEPSE needs 62 stocks in the sample to ensure the estimate is within Rs. 500 of the true mean with 95% confidence.

5. Common Pitfalls in Sampling

graph TD
    A["Poor Sampling Frame"] --> B["Excludes key groups"]
    A --> C["Biased results"]
    D["Small Sample Size"] --> E["High sampling error"]
    D --> F["Unreliable estimates"]
    G["Non-Response Bias"] --> H["Overrepresents respondents"]
    G --> I["Underrepresents hard-to-reach groups"]

Example:

• NTC’s 2022 survey used convenience sampling (only landline users), leading to underrepresentation of mobile-only customers.

In the Real World

1. Khalti’s Fraud Detection

   • Idea Used: Stratified sampling by transaction type (e.g., peer-to-peer, merchant payments) and region.
   • How: Khalti divides transactions into strata (e.g., high-value vs. low-value) and samples proportionally to detect anomalies. For example, if 10% of high-value transactions are flagged in Kathmandu, they investigate further.

2. Pathao’s Driver Allocation

   • Idea Used: Cluster sampling by city zones (e.g., Thapathali, Lakshmi Marg).
   • How: Pathao clusters drivers by location and samples entire clusters during peak hours to optimize ride matching. This reduces the need for real-time GPS data for every driver.

3. NTC’s Customer Satisfaction Surveys

   • Idea Used: Systematic sampling of phone numbers from a random dialing frame.
   • Challenge: Non-response bias (e.g., elderly users may not answer). NTC mitigates this by offering incentives (e.g., free internet minutes).

4. Daraz’s Inventory Management

   • Idea Used: Central Limit Theorem to predict demand.
   • How: Daraz analyzes sample means of daily orders across product categories. For example, if the sample mean for laptops is 500 units/day (±50), they stock accordingly, reducing over/under-stocking.

5. NEPSE’s Index Calculation

   • Idea Used: Random sampling of listed companies to calculate the NEPSE index.
   • How: NEPSE randomly selects 50–100 stocks from its 200+ listings to compute the index, ensuring representativeness.

Exam Tip

What Examiners Look For

1. Distinguish between sampling methods:

   • Probability vs. non-probability: Always state whether the method is random or biased.
   • Stratified vs. cluster: Explain how the population is divided and why (e.g., "stratified by income to ensure representation").

2. CLT applications:

   • When to use: Always check if n ≥ 30 before applying the CLT.
   • Formulas: Know ${\mu }_{\bar{X}}=\mu$ and ${\sigma }_{\bar{X}}=\frac{\sigma }{\sqrt{n}}$.
   • Real-world link: Connect to confidence intervals (e.g., "Since n=100 > 30, we can use the CLT to construct a 95% CI for the mean").

3. Sampling error calculations:

   • Margin of error (E): $E=Z\times {\sigma }_{\bar{X}}$.
   • Sample size (n): Derive from $n={\left(\frac{Z\times \sigma }{E}\right)}^2$.

4. Common mistakes to avoid:

   • Ignoring non-response bias: Always mention how it affects results.
   • Assuming normality without CLT: Only use z-tests if n ≥ 30 or the population is normal.
   • Confusing stratified and cluster sampling: Stratified = homogeneous groups; cluster = heterogeneous groups.

High-Score Strategies

• Draw diagrams: Sketch sampling distributions (e.g., population vs. sampling distribution) for CLT questions.

• Link to real data: Even in theoretical questions, relate to Nepalese examples (e.g., "Like NTC’s surveys, stratified sampling ensures...").

• Show calculations step-by-step: Examiners reward clarity. For example:

  "Given $\mu =50$, $\sigma =10$, n=25, find P($\bar{X}>52$). Step 1: ${\sigma }_{\bar{X}}=\frac{10}{\sqrt{25}}=2$. Step 2: $Z=\frac{52-50}{2}=1$. Step 3: P(Z > 1) = 0.1587."

• Interpret results: Always explain what the probability/sampling error means in context (e.g., "There’s a 15.87% chance the sample mean exceeds 52, indicating potential delays in Pathao’s delivery times").