📊 Statistics (Advanced)

📌 1. Unit 1 · Collection, Classification and Presentation of Data

• Methods of collecting primary data: direct personal investigation, indirect oral investigation, information from local sources/correspondents, mailed questionnaire (filled by the informant), schedules through enumerators (filled by the enumerator).
• Census = every unit is studied; sample survey = only some units are studied.
• Bases of classification: geographical (place), chronological (time), qualitative (attribute), quantitative (class intervals).
• Class intervals: exclusive (10–20 excludes 20) and inclusive (10–19). Sturges’ rule: number of classes k = 1 + 3.322 log₁₀N.
• Parts of a table: table number, title, head note, column headings (caption), row headings (stub), body, footnote, source note.
• Presentation: histogram, frequency polygon, ogive (the ‘less than’ and ‘more than’ ogives meet at the median), pie chart (angle = value/total × 360°).

Frequency distributions and diagrams

• Discrete distribution: separate values (e.g. number of children in a family); continuous distribution: class intervals (e.g. height 150–155 cm).
• Cumulative frequency (cf): "less than" (add from the top) and "more than" (add from the bottom). Relative frequency = f/N.
• A histogram is only for continuous data; the area of each rectangle is proportional to the frequency. With unequal class widths, height = frequency density = f ÷ class width.
• One-dimensional diagrams: line and bar diagrams; two-dimensional: rectangles, squares, circles (pie). A Lorenz curve shows inequality of income or wealth.

📌 2. Unit 2 · Measures of Central Tendency and Partition Values

Data are classified into class intervals. Class mark (mid-value) = (lower limit + upper limit)/2. Presentation: histogram (locates mode), ogive (locates median and quartiles), frequency polygon, bar and pie diagrams.

Partition values: deciles and percentiles

Deciles (D₁…D₉) split the data into 10 equal parts and percentiles (P₁…P₉₉) into 100 equal parts; quartiles (Q₁, Q₂, Q₃) into 4.

📌 3. Unit 3 · Measures of Dispersion

Dispersion shows how widely values are spread around the average. Absolute measures (range, QD, MD, SD) carry the unit of the data; relative measures (coefficients, CV) are unit-free and are used to compare two series.

📌 4. Unit 4 · Moments, Skewness and Kurtosis

📌 5. Unit 5 · Correlation and Regression

• Scatter diagram: points rising → positive, falling → negative, no pattern → r ≈ 0.
• Coefficient of determination r² = the share of the variance of y explained by x; probable error PE = 0.6745(1 − r²)/√n.
• At r = ±1 the two regression lines coincide; at r = 0 they are perpendicular (x = x̄ and y = ȳ).

Multiple regression and partial/multiple correlation (three variables)

📌 6. Unit 6 · Probability Theory

• Random experiment, sample space S, event (a subset of S). Mutually exclusive: A ∩ B = ∅; exhaustive: A ∪ B ∪ … = S; equally likely: all have equal probability.
• Definitions: classical P(A) = favourable/total equally likely outcomes; relative frequency (empirical) = limit of m/n over a very large number of trials; axiomatic (Kolmogorov): P(A) ≥ 0, P(S) = 1, and P(A ∪ B) = P(A) + P(B) for mutually exclusive events.
• Independent events: one happening does not change the probability of the other; then A and B′, and A′ and B′, are also independent.
• Odds in favour a : b → P = a/(a + b); odds against a : b → P = b/(a + b).

📌 7. Unit 7 · Random Variables and Probability Distributions

• A random variable is a function that gives a number to every outcome of the sample space. Discrete (counts: 0, 1, 2…) → probability mass function p(x), Σp(x) = 1; continuous (measurements) → probability density function f(x), ∫f(x)dx = 1 and P(a < X < b) = area under the curve.
• Distribution function F(x) = P(X ≤ x): non-decreasing, F(−∞) = 0, F(∞) = 1. For a continuous variable P(X = a) = 0.

The four main distributions — which one when

• Binomial: n fixed independent trials, two outcomes each, constant success probability p. Symmetric when p = ½; positively skewed when p < ½. Variance npq ≤ n/4.
• Poisson: rare events (misprints on a page, calls in an hour). Limit of the binomial when n is large, p small and np = λ. P(X = 0) = e⁻λ. The sum of two independent Poisson variables is Poisson (λ₁ + λ₂).
• Normal: bell-shaped, symmetric (β₁ = 0, β₂ = 3); points of inflexion at μ ± σ; QD ≈ 0.6745σ (≈ ⅔σ), MD ≈ 0.7979σ (≈ ⅘σ). Standard normal Z = (X − μ)/σ ~ N(0, 1). Table: P(0 < Z < 1) = 0.3413, P(0 < Z < 1.96) = 0.4750, P(0 < Z < 2) = 0.4772.
• Exponential: waiting time / lifetime. P(X > t) = e⁻λᵗ; mean 1/λ and SD also 1/λ. Memoryless: P(X > s + t | X > s) = P(X > t).
• Continuous uniform U(a, b): mean (a + b)/2, variance (b − a)²/12.

Joint distribution (two discrete random variables)

📌 8. Unit 8 · Sampling Theory

• Parameter = measure of the population (μ, σ); statistic = measure of the sample (x̄, s).
• Probability sampling: simple random, stratified, systematic, cluster, multistage, multiphase.
• Non-probability sampling: purposive, convenience, quota.
• Sampling error arises from taking a sample (falls as n grows); non-sampling error (wrong responses, processing mistakes) occurs even in a census.
• Properties of a good estimator: unbiased, consistent, efficient, sufficient.
• Methods of estimation: method of moments, maximum likelihood (MLE), least squares.

• Population: finite or infinite; sampling frame = the list of units from which the sample is drawn; sampling unit = the unit chosen (person, household, village).
• Advantages of a sample survey: lower cost, less time, more detailed study, and the only way for destructive testing (e.g. bulb life). Census: information on every unit, but costly.
• SRSWR (with replacement): Nⁿ possible samples; SRSWOR (without replacement): ᴺCₙ. Select by the lottery method or a random number table.
• Systematic: interval k = N/n, first unit random from 1 to k, then every k-th. Stratified: proportional allocation nₕ = n × Nₕ/N; Neyman allocation nₕ ∝ Nₕσₕ. Cluster: whole groups (e.g. villages) are chosen. Multistage: first districts, then villages, then households.
• Central limit theorem: for large n (above about 30) the sampling distribution of x̄ is approximately normal N(μ, σ²/n), whatever the population distribution.

📌 9. Unit 9 · Statistical Inference: Estimation and Testing of Hypotheses

Estimation

• Point estimate = a single number (e.g. x̄ = 52); interval estimate = a range that contains the parameter with a stated confidence (e.g. a 95% confidence interval).
• Unbiased: E(T) = θ. x̄ is an unbiased estimator of the population mean μ; Σ(x − x̄)²/n is biased, Σ(x − x̄)²/(n − 1) is unbiased. Consistent: T → θ as n grows. Efficient: least variance among unbiased estimators. Sufficient: keeps all the information in the sample.
• Maximum likelihood estimator (MLE): the value that maximises the likelihood L(θ). Examples: binomial p̂ = x/n; Poisson λ̂ = x̄; normal μ̂ = x̄, σ̂² = Σ(x − x̄)²/n (biased but consistent).

Testing of hypotheses

• Steps: (1) state the null hypothesis H₀ and the alternative H₁; (2) choose the level of significance α; (3) compute the test statistic; (4) compare with the critical value; (5) reject H₀ if |calculated| > critical.
• H₁: μ ≠ μ₀ → two-tailed; H₁: μ > μ₀ or μ < μ₀ → one-tailed. Reject H₀ if the p-value < α.

• Type-I error: rejecting a true H₀ (probability α = level of significance); Type-II error: accepting a false H₀ (β); power of a test = 1 − β.
• Large sample (n > 30) or known σ → Z-test; small sample with unknown σ → t-test; comparing variances → F; independence / goodness of fit → χ².
• At the 5% level the two-tailed critical Z = 1.96; one-tailed 1.645.

📌 10. Unit 10 · Analysis of Variance (ANOVA)

ANOVA (R. A. Fisher) tests the means of three or more groups at once. H₀: μ₁ = μ₂ = … = μₖ. The total variation is split into "between groups" and "within groups (error)" parts and F = MS(between)/MS(within) is computed; a large F means the means are not all equal.

• Assumptions: independent observations, each group from a normal population, equal variances in all groups (homoscedasticity), additive effects. Equal group sizes are not required.
• Link with design of experiments: completely randomised design (CRD) → one-way; randomised block design (RBD) → two-way; in an m × m Latin square (LSD) the error df = (m − 1)(m − 2).

📌 11. Unit 11 · Time Series

• Components of a time series: secular trend, seasonal (within a year), cyclical (business cycle, several years), irregular (floods, wars).
• Measuring trend: free-hand curve, semi-averages, moving averages, least squares.

• Models: additive Y = T + S + C + I; multiplicative Y = T × S × C × I (more common in practice).
• Examples: sales rising at festivals → seasonal; boom and recession → cyclical; steady population growth → secular trend; earthquake/strike → irregular.
• Moving averages: an odd period (3, 5 years) is placed against the middle year; an even period (4 years) must be centred. If the period equals the cycle length, the fluctuations are removed. Some values at the ends are lost.
• Least squares method: Y = a + bX; code X so that ΣX = 0, then a = ΣY/n and b = ΣXY/ΣX².
• Seasonal indices: simple averages, ratio-to-trend, ratio-to-moving-average, link relatives. Quarterly indices add up to 400, monthly ones to 1200. Deseasonalised value = actual ÷ seasonal index × 100.

📌 12. Unit 12 · Index Numbers

• An index number measures the relative change in a group (prices, quantities) over time or place; base year = 100. Price, quantity and value (value = Σp₁q₁/Σp₀q₀ × 100) indices.
• Simple aggregative Σp₁/Σp₀ × 100; simple average of relatives Σ(p₁/p₀ × 100)/n; weighted: Laspeyres, Paasche, Fisher, Marshall–Edgeworth, Dorbish–Bowley (L + P)/2, Kelly (fixed weights).
• Tests: unit test; time reversal P₀₁ × P₁₀ = 1; factor reversal P₀₁ × Q₀₁ = V₀₁; circular test (met by the simple aggregative and fixed-weight indices).
• Chain base: each year uses the previous year = 100 (link relatives); multiplying them gives a fixed-base series.

Both methods give the same answer. A CPI measures the change in the cost of living of a particular group (e.g. industrial workers).

📌 13. Practice MCQs