Session 1 — Definitions and Concepts
Population: the entire collection of units about which information is wanted. Finite if all members are known/countable (e.g. GPD Year 2 students); infinite if members are large and not all known (e.g. stars in the sky). Also accessible (can be reached) or inaccessible.
Census: collects data on every unit in the population — a complete enumeration.
Sample: a portion of a population selected to meet specific objectives.
Representative sample: one whose key characteristics reflect the target population — essential whenever you want to generalize results back to the population.
Sampling frame: the actual list of units in the population from which the sample is drawn (e.g. an electoral register).
Unit / Sampling unit: a unit is one element of the population; a sampling unit is a clearly defined, identifiable, observable group of units used for sampling purposes.
Why sample instead of census: saves money, time and effort — and a carefully drawn sample can sometimes give more accurate results than an exhausting attempt to study everyone.
Session 2 — Probability Sampling
Probability sampling: every unit has a known, non-zero (definite) probability of being selected, based on randomization/chance. It is more time-consuming and costly than non-probability sampling, but it enables unbiased estimates and calculable sampling error.
1. Simple Random Sampling (SRS)
Every possible sample of size $n$ has an equal chance of selection. Inclusion probability $= n/N$. Selected via raffle/lottery or a table of random numbers.
+ Simplest; no auxiliary info needed; well-established formulas.
− Ignores auxiliary info; can be costly if spread out; can by chance yield a "bad" sample.
2. Stratified Sampling
Population split into homogeneous, mutually exclusive strata; independent sample drawn from each. Best when units within a stratum are similar and strata differ from each other.
+ More precise; guarantees subgroup representation; protects against bad samples.
− Needs good auxiliary info for the whole frame; more complex estimation.
3. Systematic Sampling (SYS)
Units chosen at regular intervals $k = N/n$ after a random start $r$ between 1 and $k$. Used when no list exists or the list is randomly ordered.
+ Simple (one random number); can be better dispersed than SRS.
− Risk of a "bad" sample if $k$ matches hidden periodicity; no unbiased variance estimator.
4. Cluster Sampling
Population divided into clusters; a sample of whole clusters is drawn and everyone within selected clusters is surveyed. Best when units within a cluster are heterogeneous (opposite of stratification).
+ Cuts data-collection cost; no need for a full individual list.
− Usually less efficient than SRS since neighbouring units tend to be similar.
5. Multi-Stage Sampling
Sample selected in 2+ successive stages: Primary Sampling Units (PSUs) at stage 1, Secondary Sampling Units (SSUs) at stage 2, etc. Only some units within each selected PSU are sub-sampled.
+ More efficient than one-stage cluster sampling; only needs a frame at each stage, not the whole population.
− Complex organisation and variance formulas; final sample size not always known in advance.
Quick comparison
| Method | Selection basis | Best when… |
|---|---|---|
| SRS | Every unit equal chance | Simple population, no need to stratify |
| Stratified | Random within each homogeneous group | Sub-group estimates needed |
| Systematic | Every $k^{th}$ unit after random start | No frame available / randomly ordered list |
| Cluster | Random whole groups, survey everyone inside | Naturally grouped population, cost priority |
| Multi-stage | Random at 2+ successive levels | Very large / dispersed population |
Session 3 — Non-Probability Sampling
Units are selected subjectively — by availability or judgment, not chance. It's fast and cheap, but because inclusion probability can't be calculated, there's no way to compute reliable sampling error, and generalizing to the population requires risky assumptions.
When it's still useful: exploratory/idea-generating studies, preliminary steps before a real probability survey, follow-up work to interpret probability-survey results, questionnaire/focus-group testing, or when a frame is genuinely impossible to build.
| Type | How it works | Example / Note |
|---|---|---|
| Haphazard | Units chosen aimlessly, no plan | "Man on the street" interviews; assumes population is homogeneous |
| Volunteer | Respondents self-select | Call-in radio polls — big selection bias, only motivated people respond |
| Judgement | An expert deliberately picks "representative" units | Subject to the expert's own bias |
| Quota | Sampling continues until fixed quotas per sub-group are filled | Like stratified sampling, but selection within groups is non-random |
| Modified probability | Early stages use probability sampling, last stage is non-probability | e.g. probability-select regions, then quota-sample individuals within |
| Snowball | Existing sample members refer future subjects | Good for hidden/hard-to-locate populations, e.g. migrants |
Disadvantages: requires strong (often unjustified) representativeness assumptions; can't compute inclusion probabilities or reliable error estimates.
Session 4 — Essential Steps in a Sample Survey
A survey isn't just "ask questions, compile answers" — it follows precise, interconnected steps:
- Statement of Objectives — precisely define what's measured, the target population, operational definitions and scope (vague goals like "housing conditions of the poor" must be made specific).
- Literature search — check what's already known/done so you don't duplicate work.
- Questionnaire design — clear, well-sequenced, respondent-friendly questions tied to the objectives. Guidelines: keep it simple, ensure applicability, be specific, soften sensitive questions.
- Pilot survey — test the questionnaire on a small sample to catch flaws and refine open-ended questions into closed ones.
- Sample design & sample size determination — choose the sampling method and size, balancing precision against cost.
- Total cost estimation — estimate cost from sample size, manpower, materials; design may loop back if too expensive.
- Manpower training — recruit and train interviewers for uniform data collection; instruction manuals for every stage.
- Main survey — actual data collection; supervisors check and certify interviewer returns.
- Data processing — coding, data entry, tabulation/summary, then subject (cross-tabulation) analysis.
- Post-enumeration survey (PES) — a small follow-up survey after the main one to check accuracy (e.g. wrong entries) and investigate non-response; some units are deliberately shared between the main sample and the PES for a consistency check.
Mnemonic: Objectives → Literature → Questionnaire → Pilot → Sample design → Cost → Manpower → Main survey → Data processing → Post-enumeration.
Session 5 — Simple Random Sampling: Estimation
Notation: $\bar Y$ = population mean, $N$ = population size, $S^2$ = population variance; $\bar y$ = sample mean, $s^2$ = sample variance. Capital letters = population, lowercase = sample.
Estimator: any function of the sample used to estimate a population value (e.g. $\bar y$ estimates $\bar Y$). Unbiased estimator: $E(\text{estimator}) = $ true value, averaged over all possible samples of the same size.
$(1-f)$ is the finite population correction (fpc). If $f \approx 0$ (huge population, tiny sample), it vanishes and $\text{Var}(\bar y) = S^2/n$.
z-values to memorise: 90% CI → 1.645; 95% CI → 1.96; 99% CI → 2.58.
Session 6 — Proportions under SRS
For a yes/no attribute: $P$ = population proportion, $p$ = sample proportion, $Q = 1-P$, $q = 1-p$. Proportions are simply sample means of a 0/1 (dichotomous) variable — that's why the formulas mirror the SRS mean formulas.
Worked pattern: given $N$, $n$, count with attribute → $p = \text{count}/n$ → estimated total $= Np$ → variance of total $= N^2\cdot\widehat{\text{var}}(p)$ → $SE = \sqrt{\text{variance}}$ → CI $=$ estimate $\pm\, z\cdot SE$.
Know the logic even if you don't memorise the derivation: smaller desired margin of error $d$ → larger required $n$.
Session 7 — Stratified Sampling: Estimation & Allocation
Setup: $L$ strata; stratum $h$ has size $N_h$, sample size $n_h$, mean $\bar Y_h$, sample mean $\bar y_h$, variance $S_h^2$. $W_h = N_h/N$ is the stratum weight (proportion of total population in stratum $h$).
Why stratify (recap): precision gain when strata are internally homogeneous but different from each other; guarantees sub-group representation; protects against a "bad" sample.
Allocation methods — splitting total $n$ among strata
Equal allocation
Same $n_h$ in every stratum. Simple, but ignores stratum size and variability.
Proportional allocation (Bowley, 1926)
$$n_h = n\cdot W_h = n\cdot\frac{N_h}{N}$$ Bigger strata get proportionally more sample.
Optimum / Neyman allocation (1934)
$$n_h = n\cdot\frac{W_h S_h}{\sum_{h=1}^{L} W_h S_h}$$ Strata that are larger and more variable get more sample. Minimises $\text{Var}(\bar y_{st})$ for a fixed total $n$, so $V_{opt} \le V_{prop}$ always.
Formula & Derivation Reference (from your uploaded formula sheet)
Standard definitions and notation
$N$ = population size, $n$ = sample size, $Y_i$/$y_i$ = population/sample observation for unit $i$, $T_Y$/$\hat T_Y$ = population total and its sample-based estimate, $\bar Y$/$\bar y$ = population/sample mean, $S^2$/$s^2$ = population/sample variance.
Simple Random Sampling
Proportions in SRS
Show derivation — variance of a proportion
Let $y_i = 1$ if unit $i$ has the attribute, $0$ otherwise, so $p=\frac1n\sum y_i$ and $P=\frac1N\sum Y_i$. Since $Y_i$ is dichotomous, $Y_i^2=Y_i$, so $\sum Y_i^2 = NP$.
$$S^2 = \frac{\sum Y_i^2 - NP^2}{N-1} = \frac{NP-NP^2}{N-1} = \frac{NP(1-P)}{N-1} = \frac{NPQ}{N-1}$$ $$\text{Var}(p) = (1-f)\frac{S^2}{n} = \left(\frac{N-n}{Nn}\right)\frac{NPQ}{N-1} = \frac{N-n}{N-1}\cdot\frac{PQ}{n}$$Show derivation — sample size for proportions
Require the margin of error $d \ge z_{\alpha/2}\cdot SE(p)$. Squaring and substituting $\text{Var}(p)$:
$$d^2 \ge z_{\alpha/2}^2(1-f)\frac{PQ}{n} \ \Rightarrow\ \frac{d^2}{z_{\alpha/2}^2}\ge\frac{PQ}{n}-\frac{PQ}{N}$$ $$n \ge \frac{N\,P\,Q\,z_{\alpha/2}^2}{d^2(N-1)+P\,Q\,z_{\alpha/2}^2}$$Stratified sampling
Show derivation — expectation & variance of the stratified mean
$$E(\bar y_{st})=E\!\left(\sum W_h\bar y_h\right)=\sum W_hE(\bar y_h)=\sum W_h\bar Y_h=\bar Y$$Because sampling is independent across strata, all covariances between strata vanish, so:
$$\text{Var}(\bar y_{st})=\text{Var}\!\left(\sum W_h\bar y_h\right)=\sum W_h^2\,\text{Var}(\bar y_h)=\sum_{h=1}^{L}W_h^2(1-f_h)\frac{S_h^2}{n_h}$$Allocation derivations
Show derivation — proportional allocation
Set $n_h \propto N_h$, i.e. $n_h = KN_h$. Summing over all strata: $n=K\sum N_h = KN \Rightarrow K = n/N$, so:
$$n_h = N_h\cdot\frac nN = nW_h$$Substituting into the general stratified variance and letting $N$ be large (second term vanishes):
$$\text{Var}(\bar y_{prop}) \approx \frac{\sum_{h=1}^{L}W_hS_h^2}{n}$$Show derivation — optimum (Neyman) allocation
Minimise $V=\sum \dfrac{W_h^2S_h^2}{n_h}$ subject to $\sum n_h = n$ using a Lagrange multiplier $\lambda$:
$$\frac{\partial V}{\partial n_h}=-\frac{W_h^2S_h^2}{n_h^2}+\lambda=0 \ \Rightarrow\ n_h=\frac{W_hS_h}{\sqrt\lambda}$$Summing across strata forces $\sqrt\lambda = \dfrac{\sum W_hS_h}{n}$, giving the final allocation and minimum variance:
$$n_h = n\cdot\frac{W_hS_h}{\sum_{h=1}^{L}W_hS_h} \qquad\qquad V_{opt}(\bar y_{st}) = \frac{\left(\sum_{h=1}^{L}W_hS_h\right)^2}{n}$$High-Yield Theory Questions — Be Ready For These
- Distinguish population vs sample; census vs sample survey.
- Explain probability sampling vs non-probability sampling — definitions, and why only one supports valid statistical inference.
- Describe each probability sampling method (SRS, stratified, systematic, cluster, multi-stage): how it works, one advantage, one disadvantage.
- Explain the three reasons for stratification.
- Compare stratified vs cluster sampling (stratify = homogeneous within/heterogeneous between → increases precision; cluster = the opposite → decreases precision but cuts cost).
- Describe each non-probability method (haphazard, volunteer, judgement, quota, modified, snowball) with an example.
- List and explain the ten steps in conducting a sample survey.
- Define estimator and unbiased estimator; explain the finite population correction.
- Explain margin of error / confidence interval in plain language.
- Explain proportional vs optimum (Neyman) allocation, and why optimum allocation is always at least as efficient.