The hard problems

Questions in the other exams ask you to apply a formula. A hard DSE question asks you to find the formula. Look at how they are phrased: a mean and a variance and two corrected entries; a sum of squares and a standard deviation, but no mean; the equations of two regression lines; a distribution function that lies to the right of another. Five habits cover almost all of them.

1. Keep sums, not data. A mean is Σx ÷ n and a variance is Σx² ÷ n − x̄². If you are given those, you hold the sums, and any change to the data is a change to the sums.
2. Work in symbols. When the numbers are letters, keep them letters. Orderings, signs and cancellations show up in algebra that arithmetic hides.
3. Look for what cancels. Shared terms, constants in a shifted variable, and a divisor that appears top and bottom.
4. Ask what is invariant. Changing units, shifting, swapping variables: some quantities survive (r, R², a t-ratio) and some do not (a slope, an SSE).
5. Test the extremes. What does the formula give at r = 0, at r = ±1, with one observation, with all values equal? An option that fails an extreme case is wrong.

Each section below takes one family of questions. Every worked question is a real DSE paper question; the full sets are at the end.

These questions test whether you have understood what an average is, not whether you can compute one. Four typical forms:

• Summation algebra. Simplify an expression with Σ signs, using Σc = nc, Σ(x − y) = Σx − Σy and Σx = nx̄.
• An average from symbols. Compare a mean and a median when the values are p, q, p + q and q − p.
• Grouped data. Use class marks for the mean (the Gymkhana Club weights) and the cumulative frequency for the median class (the household expenditure table).
• Logs. The arithmetic mean of the logs is the log of the geometric mean (Lesson 3), which is why wage regressions in logs describe geometric averages.

Typically a question reports a mean and a variance for n values, then says some entries were wrong. You never see the data. Do not try to rebuild it. Convert what you are given to Σx and Σx², patch the few wrong terms and convert back.

The same sums-first move recovers a missing mean. The 2009 question about passengers without tickets gives Σx² = 2,000, n = 20 and σ = 6, and no mean. From σ² = Σx² ÷ n − x̄², 36 = 100 − x̄², so x̄ = 8: the fine is ₹8 each (Lesson 4).

Covariance is linear in each of its two arguments. So the covariance of two sums is the sum of all the cross covariances, and when the underlying variables are uncorrelated, nearly all of them vanish. What is left is exactly the part the two sums share.

The other correlation questions test the converse trap. Independent variables are always uncorrelated, but the reverse fails. If X is symmetric about zero, such as a uniform on [−1, 1] or a variable taking −1, 0, 1 with equal probability, then Cov(X, X²) = E[X³] − E[X]E[X²] = 0 although X² is a function of X. They are uncorrelated but dependent (DSE 2010, DSE 2012).

A simple regression is fixed by five numbers: x̄, ȳ, σₓ, σᵧ and r. Give the examiner any three of the five and they can ask for the rest. So the first move on every regression question is to write down which of the five you have, and then:

The rest of the regression questions are one-line facts if you ask which quantities are invariant:

• Rescaling. If Y* = w₁Y and X* = w₂X, the slope becomes (w₁ ÷ w₂)b, the intercept w₁a, the variance of the slope (w₁ ÷ w₂)² times its old value, and R² does not change (DSE 2012, DSE 2016). In a log-log model, dividing the price by 1,000 changes the intercept by β ln 1,000 and leaves the elasticity β alone (DSE 2013).
• R² equals r² only with an intercept (DSE 2014).
• Reverse regression. Regressing x on y gives a different slope, but the same t-ratio and the same R² (DSE 2014).
• R² is not comparable between models with different dependent variables, such as demand per household and demand per head (DSE 2012).
• Equal variances. If σₓ = σᵧ and 0 < ρ < 1, both OLS slopes equal ρ, so both are below 1 (DSE 2020).
• Conditional mean. The estimate of the mean of y given x is ȳ + b(x − x̄), which depends on x, so “the mean of y” is not the answer (DSE 2018).
• Log-level. In ln Y = 1.2 + 0.5X a one-unit rise in X goes with about a 50% rise in Y. The exact figure is e^0.5 − 1 = 64.9%, and the approximation gets worse as the slope grows (DSE 2011).

Three questions from 2006 and 2008 ask for a Laspeyres index (115.3, worked in Lesson 8), for the properties of an ideal index, and for a chain-base index. The last two take care.

In the exam the hardest step is the first: recognising which of the techniques above the question is inviting. Six situations; pick the opening move.

All 32 DSE questions above the easy level from this module, grouped by where they come from. Moderate questions come first in each set, then the hard ones. Easy DSE questions are in the lessons they belong to.