Tamil Nadu HSE Second Year Statistics common mistakes
Tamil Nadu HSE Second Year Statistics · Understand the traps and how to avoid them.
When finding the inverse of a 3×3 matrix, students compute the adjugate by transposing the original matrix's cofactor matrix incorrectly - they either forget to transpose the cofactor matrix or transpose the original matrix before taking cofactors, then divide by the determinant without checking it is nonzero.
The procedure has two nested operations (cofactor, then transpose) and textbooks write adj(A) = (cofactor matrix)^T, so students lose track of which matrix gets transposed. They also skip verifying |A| ≠ 0 because the formula is applied mechanically.
What to do instead
Write a fixed 5-step checklist and follow it every time: (1) compute |A| first and state it; (2) build the cofactor matrix C entry by entry using C_ij = (−1)^(i+j) M_ij; (3) transpose C to get adj(A); (4) A⁻¹ = adj(A)/|A|; (5) verify A·A⁻¹ = I. Practise at least ten 3×3 inverses and check the final product each time.
Students solve a system of three linear equations by Cramer's rule or matrix inversion without first testing consistency, so they divide by a zero determinant or report a unique solution for a system that is inconsistent or has infinitely many solutions.
Cramer's rule is taught as a formula, and the consistency condition (rank comparison) is treated as a separate chapter. Students do not connect Δ = 0 with the need to examine Δx, Δy, Δz.
What to do instead
Before solving, always compute Δ = |A|. If Δ ≠ 0, state 'unique solution' and proceed. If Δ = 0, compute Δx, Δy, Δz: if all are zero the system has infinitely many solutions (solve by elimination); if any is nonzero it is inconsistent. Also practise the rank form ρ(A) = ρ([A|B]) = number of unknowns.
When computing products, quotients, or powers of complex numbers, students expand everything in Cartesian form and make sign errors, especially writing i² = 1 or mishandling the argument when converting between polar and Cartesian form.
Cartesian expansion is familiar from algebra, so students default to it even for high powers, and the quadrant of the argument is ignored because arg(z) is computed as arctan(y/x) without adjustment.
What to do instead
For products, quotients, and powers above 2, convert to polar form z = r(cos θ + i sin θ) and use De Moivre's theorem. Always locate the quadrant first and compute θ accordingly (add π in quadrants II and III). Memorise i² = −1 and verify by multiplying back. For roots, remember there are n distinct nth roots spaced by 2π/n.
Students apply Descartes' rule of signs and the rational root theorem but then report the number of positive or negative roots as exact rather than as a maximum, and they fail to pair complex roots as conjugates when counting roots.
The rule gives an upper bound differing by an even number, but students treat the sign-change count as the answer. The conjugate-pair property is stated once and forgotten when totalling roots.
What to do instead
State the rule precisely: the number of positive real roots is at most the number of sign changes in f(x) and differs from it by an even number; apply the same to f(−x) for negative roots. After finding real roots by synthetic division, use the conjugate pair theorem to account for remaining complex roots. Always check the total equals the degree.
When evaluating inverse trigonometric expressions such as sin⁻¹(sin 5π/6) or tan⁻¹(tan 3π/4), students substitute the angle directly and ignore the principal-value range, giving 5π/6 instead of π/6.
Students treat sin⁻¹∘sin as an identity for all inputs, forgetting that it holds only when the angle lies in the principal branch. The restricted ranges are memorised as a list without being applied.
What to do instead
Write the principal ranges at the top of every solution: sin⁻¹, tan⁻¹, cosec⁻¹, cot⁻¹ give values in (−π/2, π/2); cos⁻¹, sec⁻¹ in [0, π]. For sin⁻¹(sin x), first reduce x into [−π/2, π/2] using symmetry (sin(π − x) = sin x). Do the same reduction for each function before substituting.
In permutation and combination problems, students cannot decide whether order matters and whether repetition is allowed, so they apply nPr where nCr is needed (or vice versa), and they forget to divide by factorials for repeated identical items.
Word cues like 'arrange' and 'select' are unreliable, and the standard formulas are memorised without the underlying counting principle. Identical-object cases are rarely drilled.
What to do instead
Before computing, answer three questions in writing: does order matter, is repetition allowed, and are any objects identical? Then choose: ordered without repetition nPr; unordered nCr; arrangements of n objects with p, q identical = n!/(p! q!). Verify small cases by listing manually. Practise mixed problems such as forming committees with at least one woman.
When expanding a binomial with a negative or fractional term, students drop the sign on alternate terms or mis-index the general term, and when asked for the term independent of x they set the wrong exponent to zero.
The general term is written as T_(r+1) = nCr a^(n−r) b^r but students substitute loosely, losing the (−1)^r from a negative b, and they confuse the power of x in the general term with the term number.
What to do instead
Write the general term explicitly with signs: T_(r+1) = nCr a^(n−r) (−b)^r, keeping (−1)^r visible. Simplify the power of x in the general term to a single expression in r, then set that exponent to 0 for the term independent of x and solve for r; reject non-integer r. Check by expanding a small case such as (x − 1/x)⁴.
Students confuse arithmetic, geometric, and arithmetico-geometric series, applying the wrong sum formula, and they mishandle infinite geometric series without checking |r| < 1 before using a/(1−r).
The formulas look similar and are memorised as a block; the convergence condition is treated as a footnote rather than a prerequisite.
What to do instead
Classify the series first by testing consecutive differences and ratios on paper. State the condition before using any infinite sum: a/(1−r) only when |r| < 1. For arithmetico-geometric series use the standard multiply-by-r and subtract technique rather than a memorised formula. Keep a one-page formula sheet with the condition written beside each formula.
In probability distribution problems, students use the binomial or Poisson formula without verifying its conditions, confuse P(X = r) with P(X ≥ r), and forget that the total probability must sum to 1 when finding an unknown constant.
Formula recognition is driven by keywords ('trials', 'rare events') rather than by checking independence, fixed n, and constant p. Cumulative probabilities are read carelessly.
What to do instead
Before applying a distribution, write the conditions: binomial needs fixed n, two outcomes, independent trials, constant p; Poisson needs a rare-event rate with mean λ = np. Use P(X ≥ r) = 1 − P(X ≤ r−1). When a constant k is unknown, use Σ P(X = x) = 1 to solve for it, then verify.
Students describe sampling distributions and the standard error of the mean incorrectly - they use the population standard deviation σ instead of σ/√n, or they claim the sampling distribution is normal regardless of sample size.
The standard error formula is recalled without understanding that it measures the spread of sample means, and the central limit theorem is overstated as applying to any n.
What to do instead
Write the definition: for samples of size n from a population with mean μ and SD σ, E(x̄) = μ and SE(x̄) = σ/√n (or s/√n when σ is unknown). State the central limit theorem condition - approximately normal for large n (commonly n ≥ 30) or exactly normal if the population is normal. Compute SE numerically in several worked examples to build intuition for how it shrinks with n.
In estimation problems, students report a point estimate when an interval is required, or they construct a confidence interval using the wrong critical value (z instead of t when σ is unknown and n is small) and misplace the margin of error.
The distinction between point and interval estimation is blurred, and the choice between z and t is memorised as a rule of thumb without the condition being checked.
What to do instead
Read the question for the words 'estimate' versus 'interval'. For intervals write the structure first: estimate ± critical value × standard error. Choose z when σ is known or n is large; choose t with n − 1 degrees of freedom when σ is unknown and n is small. State the confidence level and interpret the interval in context as a sentence.
In hypothesis testing, students state conclusions as 'the null hypothesis is proved true' when failing to reject, set up one-tailed and two-tailed tests incorrectly, and confuse the p-value with the probability that H₀ is true.
The logic of falsification is counterintuitive and the language of 'accept' versus 'fail to reject' is not emphasised. The p-value is often taught only as a number to compare with α.
What to do instead
Write H₀ and H₁ with the claim in H₁, and choose the tail from the direction of H₁ (≠ gives two-tailed, halving α). State the decision rule before computing. Conclude in the form 'at the 5% level there is sufficient/insufficient evidence to reject H₀', never 'H₀ is proved'. Interpret the p-value as the probability of a result at least as extreme as observed, given H₀ is true.
When computing correlation and regression, students mix up the regression of y on x with x on y, use the wrong slope formula, or assume that a high correlation coefficient proves causation or that r = 0 implies no relationship.
The two regression lines and their coefficients b_yx and b_xy are visually similar, and the interpretive caveats are stated briefly in textbooks without being practised.
What to do instead
Label the lines explicitly: y on x uses b_yx = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and is used to predict y; x on y uses b_xy with the denominator Σ(y−ȳ)². Remember b_yx·b_xy = r² and that both lines pass through (x̄, ȳ) - use this to check. Interpret r as measuring only the strength of linear association, and r = 0 as no linear relationship, not no relationship.