Unit 3: Sampling techniques and descriptive statistics

Research rarely studies every member of a population; it studies a manageable subset and reasons back to the whole. This unit establishes how that subset is chosen (sampling), how the observations are classified (types of data), how they are summarised (central tendency), how two variables move together (correlation), and how conclusions are extended from sample to population (inferential statistics).

• Population: the complete set of units under study, denoted N; e.g. all 40,000 voters in a district.
• Sample: a subset of size n drawn from the population; e.g. 400 voters actually interviewed.
• Parameter vs statistic: a fixed value describing the population (parameter, e.g. μ) versus a computed value describing the sample (statistic, e.g. x̄).
• Sampling frame: the operational list from which units are drawn, e.g. an electoral roll.
• Sampling error: the discrepancy between statistic and parameter caused by studying only a part.

II. Sampling — Selecting a representative subset

A. Introduction to sampling

Sampling is the process of selecting units from a population so that the sample supports valid inference about the whole.

• Purpose: saves cost and time; a census of 40,000 voters is replaced by 400, cutting fieldwork ~99%.
• Feasibility: enables study where a census is impossible, e.g. destructive testing of matchboxes.
• Precondition — representativeness: the sample must mirror the population's structure; a survey of only urban voters cannot describe a mixed district.
• Trade-off: larger n lowers sampling error but raises cost; the researcher fixes n against a tolerable margin of error.

B. Random sampling

In random (probability) sampling every unit has a known, non-zero probability of selection, so results are generalisable.

• Simple random sampling: every unit has equal probability n/N; e.g. drawing 50 roll numbers from a lottery drum or random-number table.
• Systematic sampling: pick every k-th unit after a random start, where k = N/n; for N = 1000, n = 100, k = 10, start at 7 → 7, 17, 27…
• Stratified sampling: divide the population into homogeneous strata (e.g. male/female) and sample each in proportion; guarantees each stratum's presence.
• Cluster sampling: divide into groups (clusters), randomly select whole clusters, then study all their units; e.g. choose 5 villages and survey every household in them.
• Strength: sampling error is measurable and confidence limits can be attached.

C. Non-random sampling

In non-random (non-probability) sampling selection depends on the researcher's judgment or convenience, so selection probabilities are unknown.

• Convenience sampling: units are chosen because they are readily available; e.g. interviewing shoppers at the nearest mall.
• Judgment (purposive) sampling: the expert picks units believed typical; e.g. selecting three "model" schools to study a scheme.
• Quota sampling: fixed numbers per category are filled by any available unit; e.g. 30 men and 30 women with no random draw.
• Snowball sampling: existing respondents recruit further ones; used for hidden populations like migrant workers.
• Limitation: prone to selection bias and no valid estimate of sampling error, so generalisation is weak.

III. Types of data — Classifying the observations

A. Introduction to types of data

Data are the recorded values of a variable; their type dictates which statistical tools are legitimate.

• Variable vs constant: a characteristic that varies across units (age) versus one that does not (a fixed rate).
• Core split: every dataset is either qualitative (categories) or quantitative (numbers), with ordinal data forming an ordered middle case.

B. Qualitative data

Qualitative (categorical) data describe a quality or attribute that is not inherently numeric.

• Nature: values are labels, e.g. blood group {A, B, AB, O} or gender {male, female}.
• Permissible summary: counts and modes only; you can say 60% chose "yes" but cannot average categories.
• No natural order: "red" is not greater than "blue"; arithmetic is meaningless.

C. Quantitative data

Quantitative data record a measurable amount expressed as a number.

• Discrete: whole-number counts, e.g. number of children (0, 1, 2); no intermediate values.
• Continuous: any value in a range, e.g. height 172.4 cm, limited only by instrument precision.
• Permissible summary: full arithmetic — mean, standard deviation, correlation.

D. Ordinal

Ordinal data are categories that carry a rank order but unequal or undefined gaps.

• Definition: order is meaningful, distance is not; e.g. satisfaction {poor < fair < good < excellent}.
• Contrast with interval: the gap between "good" and "excellent" need not equal that between "poor" and "fair".
• Permissible summary: median and rank-based measures are valid; the arithmetic mean is generally not.

IV. Measures of central tendency — Locating the centre of the data

A. Arithmetic mean

The arithmetic mean is the sum of all values divided by their count, the most widely used average.

x̄ = (Σxᵢ) / n

• Symbols: x̄ = mean, xᵢ = i-th observation, Σxᵢ = total of observations, n = number of observations.
• Worked example: for {4, 8, 6, 10, 12}, Σx = 40, n = 5, so x̄ = 40/5 = 8.
• Property: the algebraic sum of deviations about the mean is zero, Σ(xᵢ − x̄) = 0.
• Weakness: distorted by extreme values; incomes {2, 3, 4, 100} give x̄ = 27.25, unrepresentative of the typical value.

B. Average

"Average" is the general term for any single value chosen to represent an entire dataset.

• Function: condenses many observations into one typical figure for comparison.
• Three common forms: mean (arithmetic centre), median (positional centre), mode (most frequent value).
• Selection rule: the mean suits symmetric numeric data, the median suits skewed or ordinal data, the mode suits categorical data.

C. Median and its importance

The median is the value that divides an ordered dataset into two equal halves.

Odd n : median = value of the (n+1)/2-th item
Even n: median = mean of the (n/2)-th and (n/2 + 1)-th items

• Worked example: {3, 5, 7, 9, 11} → (5+1)/2 = 3rd item = 7.
• Importance — robustness: unaffected by extreme values, so {2, 3, 4, 100} has median 3.5, a fairer "typical" figure than the mean 27.25.
• Importance — ordinal use: it is the highest valid average for ranked data like satisfaction scores.
• Importance — open-ended classes: computable even when a top class is "above ₹1,00,000".

D. Characteristics of an ideal average

An ideal average is one that satisfies the properties statisticians expect of a representative value.

• Rigidly defined: yields one unambiguous value; the mean qualifies, the mode may not (data can be bimodal).
• Based on all observations: uses every value; the mean does, the median and mode do not.
• Easy to understand and compute: interpretable by a layperson.
• Capable of further algebraic treatment: combinable across groups; combined means can be found, medians cannot.
• Least affected by sampling fluctuation: stable across different samples; the mean is generally the most stable.
• Not unduly affected by extreme values: here the median outperforms the mean.

V. Correlation — Measuring how two variables move together

A. Karl Pearson coefficient of correlation

Pearson's coefficient r measures the strength and direction of the linear relationship between two quantitative variables.

r = Σ(x − x̄)(y − ȳ) / √[ Σ(x − x̄)² · Σ(y − ȳ)² ]

• Symbols: x, y = paired observations; x̄, ȳ = their means; r ranges from −1 to +1.
• Interpretation:
  • r = +1: perfect positive linear relation, e.g. hours studied and marks rising together.
  • r = −1: perfect negative relation, e.g. price and quantity demanded.
  • r = 0: no linear relation.
• Worked example (concept): advertising spend and sales giving r = 0.85 indicates a strong positive linear association.
• Assumptions: both variables are quantitative, the relation is linear, and values are roughly normally distributed.
• Caution — causation: a high r shows association, not cause; ice-cream sales and drowning correlate through a hidden variable (summer heat).

VI. Introduction to inferential statistics — Generalising beyond the sample

A. Introduction to inferential statistics

Inferential statistics use sample data to draw conclusions about the population, quantifying the uncertainty involved.

• Contrast with descriptive statistics:
  • Descriptive: summarises the data in hand, e.g. "this sample's mean income is ₹28,000".
  • Inferential: projects to the population, e.g. "the population mean lies in ₹27,000–₹29,000 with 95% confidence".
• Estimation: using x̄ to estimate μ, reported as a point estimate or an interval estimate (confidence interval).
• Hypothesis testing: framing a null hypothesis H₀ (no effect) and an alternative H₁, then using a test statistic and p-value to accept or reject H₀.
• Role of sampling: valid inference requires probability sampling; the measurable sampling error of Section II is what makes confidence levels possible.
• Significance level: the risk α of wrongly rejecting a true H₀, conventionally set at 0.05, linking the whole chain from sample selection back to population conclusions.