Kerala Plus Two Statistics (DHSE) cheat sheet

Primary data are collected first-hand by the investigator; secondary data are already collected by someone else and reused.

Classification groups raw data by one attribute; tabulation arranges the classified data in rows and columns with a stub, caption, and body.

Frequency distribution: exclusive classes (0–10, 10–20) have no ambiguity at boundaries; inclusive classes (0–9, 10–19) need a correction factor of half the gap.

Class width = upper limit − lower limit; class mark (mid-value) = (lower + upper)/2; relative frequency = class frequency / total frequency.

Less-than and more-than ogives intersect at the median; the ogive is a plot of cumulative frequency against class boundary.

Histogram uses class boundaries on the X-axis; the area of each bar is proportional to class frequency when widths are unequal.

Pie chart: angle of a sector = (component / total) × 360°; useful for parts of a whole, not for comparison over time.

Skewness sign rule: mean > median > mode indicates positive (right) skew; mean < median < mode indicates negative (left) skew.

Coefficient of variation = (SD / mean) × 100; it compares variability of two series measured in different units.

Arithmetic mean of grouped data with frequencies: x̄ = Σfx / Σf, where x is the class mark.

Combined mean of two groups: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂); this is a weighted mean with weights n₁ and n₂.

Median of grouped data: median = L + [(N/2 − cf)/f] × h, with L the lower boundary of the median class, cf the cumulative frequency before it, f its frequency, h the width.

Mode of grouped data: mode = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the modal frequency, f₀ and f₂ the frequencies of adjacent classes.

Empirical relation for moderately skewed data: mode = 3 median − 2 mean.

Quartile deviation = (Q₃ − Q₁)/2; coefficient = (Q₃ − Q₁)/(Q₃ + Q₁).

Mean deviation about a value A = Σf|x − A|/Σf; it is least when A is the median.

Variance of grouped data: σ² = (Σfx²/Σf) − (Σfx/Σf)²; standard deviation σ is its positive square root.

Change of origin and scale: adding a constant leaves SD unchanged; multiplying every observation by k multiplies SD by |k|.

Karl Pearson coefficient: r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²]; it lies between −1 and +1 and is unit-free.

Direct computation form: r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

Spearman rank correlation: ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference of ranks; for tied ranks use mean ranks.

Regression of y on x: ŷ = a + bx with b = r(σy/σx) = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² and a = ȳ − bx̄; the line passes through (x̄, ȳ).

Two regression lines are y on x and x on y; they intersect at (x̄, ȳ) and coincide only when |r| = 1.

r² = bxy × byx (product of the two regression coefficients); the sign of r equals the sign of each regression coefficient.

Regression coefficients are unaffected by change of origin but change under change of scale; r is unaffected by both.

Regression predicts one variable from another and is asymmetric; correlation measures only the strength and direction of linear association.

Sample space S is the set of all outcomes; an event is a subset of S; P(S) = 1 and P(∅) = 0.

Classical probability: P(A) = number of favourable outcomes / total equally likely outcomes.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.

Complement rule: P(A′) = 1 − P(A); also P(A ∩ B′) = P(A) − P(A ∩ B).

De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Mutually exclusive events cannot occur together; independent events satisfy P(A ∩ B) = P(A)P(B) and can occur together.

Multiplication rule: P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B).

Odds in favour of A are P(A) : P(A′); odds against A are P(A′) : P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), defined only when P(B) > 0.

Independence test: if P(A|B) = P(A), then P(A ∩ B) = P(A)P(B) and the events are independent.

Total probability theorem: for a partition B₁, B₂, …, Bₙ, P(A) = Σ P(Bᵢ) P(A|Bᵢ).

Bayes' theorem: P(Bᵢ|A) = P(Bᵢ) P(A|Bᵢ) / Σⱼ P(Bⱼ) P(A|Bⱼ); the denominator is the total probability of A.

The prior P(Bᵢ) is updated to the posterior P(Bᵢ|A) after observing A; the likelihood is P(A|Bᵢ).

Multiplication theorem for n events: P(A₁A₂…Aₙ) = P(A₁) P(A₂|A₁) P(A₃|A₁A₂) …

For independent events, conditional probability reduces to the unconditional probability: P(A|B) = P(A).

A random variable is a real-valued function on the sample space; it is discrete if it takes countable values, continuous otherwise.

Probability mass function p(x) must satisfy p(x) ≥ 0 and Σp(x) = 1 over the support.

Distribution function F(x) = P(X ≤ x) is non-decreasing, right-continuous, with limits 0 at −∞ and 1 at +∞.

For a discrete variable, P(a < X ≤ b) = F(b) − F(a); P(X = x) is the jump of F at x.

Expectation E(X) = Σx p(x); E(aX + b) = aE(X) + b.

Variance Var(X) = E(X²) − [E(X)]²; Var(aX + b) = a² Var(X).

If Y = g(X), then E(Y) = Σg(x) p(x); in general E[g(X)] ≠ g[E(X)].

Mean of a symmetric distribution equals its centre of symmetry; the median splits probability equally.

E(X + Y) = E(X) + E(Y) always; Var(X + Y) = Var(X) + Var(Y) only when X and Y are independent.

If X and Y are independent, E(XY) = E(X)E(Y); the converse does not hold in general.

Cov(X, Y) = E(XY) − E(X)E(Y); Cov(aX, bY) = ab Cov(X, Y).

Correlation of variables: ρ = Cov(X, Y)/[σx σy], lying between −1 and 1.

Var(X − Y) = Var(X) + Var(Y) − 2Cov(X, Y); the covariance term vanishes under independence.

E(X − μ)² is the variance; E|X − μ| is the mean deviation and is minimised about the median.

Moment generating function M(t) = E(e^{tX}); its derivatives at t = 0 give the raw moments.

For independent variables, the mgf of the sum is the product of the individual mgfs.

Binomial: n independent Bernoulli trials, constant success probability p; P(X = r) = C(n, r) p^r q^(n−r), q = 1 − p.

Binomial mean = np, variance = npq; variance is less than the mean unless p = 1.

Binomial mgf = (q + pe^t)^n; the sum of independent binomials with the same p is binomial with added n.

Poisson: P(X = r) = e^(−λ) λ^r / r!, r = 0, 1, 2, …; mean = variance = λ.

Poisson is the limit of binomial when n → ∞, p → 0 with np = λ fixed; use it for rare events.

Poisson recurrence: P(r + 1) = [λ/(r + 1)] P(r), useful for computing successive probabilities.

Binomial is right-skewed for p < 0.5, symmetric for p = 0.5, left-skewed for p > 0.5; Poisson is right-skewed.

Sum of independent Poisson variables is Poisson with parameter equal to the sum of the parameters.

Simple random sampling without replacement: number of samples of size n from N units is C(N, n).

Parameter describes the population (μ, σ); statistic describes the sample (x̄, s) and is a random variable.

Standard error of the mean with replacement: SE = σ/√n; without replacement multiply by √[(N − n)/(N − 1)], the finite population correction.

For large n, the sampling distribution of x̄ is approximately normal with mean μ and variance σ²/n (central limit theorem).

Sample proportion p̂ has mean P and standard error √[P(1 − P)/n]; the finite population correction applies without replacement.

Sampling error arises from observing only part of the population; non-sampling errors include bias in frame, non-response, and measurement error.

Stratified sampling divides the population into homogeneous strata and samples each; it usually reduces variance relative to simple random sampling.

Systematic sampling selects every k-th unit after a random start; it is convenient when a list of the population exists.

An estimator is unbiased if E(θ̂) = θ; the sample mean is unbiased for μ and s² = Σ(x − x̄)²/(n − 1) is unbiased for σ².

Consistency means θ̂ converges to θ in probability as n increases; efficiency compares variances of unbiased estimators.

Confidence interval for μ with known σ: x̄ ± z(α/2) σ/√n; use t with n − 1 df when σ is unknown and n is small.

Null hypothesis H₀ states no effect; the alternative H₁ may be one-sided or two-sided and determines the rejection region.

Type I error rejects a true H₀ with probability α; Type II error accepts a false H₀ with probability β; power = 1 − β.

One-sample z statistic for the mean: z = (x̄ − μ₀)/(σ/√n); for a proportion: z = (p̂ − P₀)/√[P₀(1 − P₀)/n].

t statistic for small samples: t = (x̄ − μ₀)/(s/√n) with n − 1 degrees of freedom.

Reject H₀ when the computed statistic falls in the critical region; a p-value below α gives the same decision.

The level of significance α is fixed before sampling; it is not the probability that H₀ is true.