Module 5 Article: Who Do You Ask? — Sampling and Sample Size

The big idea

You almost never study everyone; you study a sample and infer to the population. The whole chain of trust — from a sample mean to a population claim — rests on who you picked and how many. Pick wrong, and even a perfectly analysed dataset is useless.

Population, frame, and sample

Term	Meaning	Trap
Population (target)	Everyone you want to generalise to	Too broad (e.g., "students")
Sampling frame	The list you actually sample from	Frame ≠ target (1936 Gallup used car-owners)
Sample	The units actually studied	Selection bias

For our stress study

• Population: full-time MCA students in India.
• Frame: the enrolment list of three colleges you can access.
• So you must state the limits of your frame — your conclusions apply to Indian full-time MCA students at comparable institutions, not all students.

Probability vs non-probability sampling

Probability (everyone has a known, non-zero chance)

• Simple random: lottery / random numbers.
• Systematic: every kth person from a list (k = N/n).
• Stratified: split the frame into strata, sample within each (e.g., by year of study). More precise if strata differ.
• Cluster: sample whole groups (e.g., randomly pick colleges, then test all their students). Cheaper when individuals are clustered.
• Multi-stage: cluster, then sub-sample within clusters (national surveys).

Non-probability (chance is unknown or zero)

• Convenience: whoever is easy to reach (students in your class).
• Purposive: "information-rich" cases chosen on purpose.
• Quota: fill quotas by trait (no random selection).
• Snowball: referrals from initial participants (hidden populations).

Rule of thumb: probability sampling for quantitative generalisation; non-probability for qualitative exploration.

Sample-size determination

For estimating a mean

n = z² · σ² / d²

• z = 1.96 for 95% confidence
• σ = planning value (from pilot or prior studies)
• d = desired margin of error

For estimating a proportion

n = z² · p(1−p) / d²

• Use p = 0.5 if you have no idea (it maximises n, a conservative choice).

Example

Goal: estimate the proportion of stressed MCA students, 95% confidence, ±5%. n = 1.96² · 0.5·0.5 / 0.05² = 3.84·0.25/0.0025 = 384.

Finite population correction (FPC)

If your sample is a large fraction of the population (n/N > 5%):

n_adj = n / (1 + (n−1)/N)

For our study: power, not just precision

For hypothesis testing, you also want power (1 − β, usually 0.80) to detect a meaningful effect size (from the pilot SD and the minimum effect you care about). Power analysis usually gives a larger n than the precision formula — use the larger.

Sampling errors and bias

• Sampling error: natural variation between a sample statistic and the population parameter. It shrinks with √n (halve the error → 4× the sample). It is random and shown by the confidence interval.
• Sampling bias: systematic over/under-representation (e.g., only surveying students in the gym). It is a design flaw and does not shrink with n.

Minimising bias

• Use a proper, complete frame.
• Randomise selection.
• Follow up non-responders (non-response is a bias, not just noise).
• Weight post-stratification if your sample is off on known margins.

Quantitative vs qualitative sampling

Aspect	Quantitative	Qualitative
Method	Probability sampling	Purposive/snowball
Size driver	Power + precision	Saturation (no new themes)
Goal	Generalise with CI	Understand meaning in depth

Worked example: stress-study sample plan

• Power analysis (two-sample t, d = 0.5, power 0.8, α 0.05) → ~64 per group = 128.
• With 20% expected loss → ~160 total.
• Stratify by year (1st/2nd/3rd) and sex to keep groups balanced.
• Offer a small incentive (₹200) to reduce non-response.

Exam angle

For "sampling design":

1. Define population, frame, sample; explain the frame ≠ target problem.
2. Compare two probability and two non-probability methods and when to use each.
3. Show n = z²σ²/d² (means) or n = z²p(1−p)/d² (proportions); state the conservative p = 0.5 choice.
4. Distinguish sampling error (random, ↓ with √n) from sampling bias (systematic, ↓ with design).

Common mistake

Reporting a sample mean and saying "this is the population mean." A sample mean estimates the population mean with a confidence interval and sampling error; it is almost certainly not exactly equal.

See also

• Module 6 (analysis) and Module 7/8 (probability/statistics underpin inference).
• Glossary.md (sampling bias, sampling error, frame, power).