Worked examples: VCE General Mathematics Formula Sheets

Formula index with worked examples

1. Data analysis · standardised score

   Formula or reference: z = (x − x̄)/sₓ

   Units and conditions: z has no units. x, x̄ and sₓ share the original units.

   Worked example: An observation of 18 with mean 12 and standard deviation 4 has z = 1.5.

2. Data analysis · boxplot fences

   Formula or reference: Lower = Q₁ − 1.5 × IQR. Upper = Q₃ + 1.5 × IQR.

   Units and conditions: Fences use the observations’ units.

   Worked example: Q₁ = 8 and Q₃ = 12 give an IQR of 4 and fences of 2 and 18.

3. Data analysis · least-squares regression

   Formula or reference: ŷ = a + bx, where b = r sᵧ/sₓ and a = ȳ − b x̄.

   Units and conditions: b uses y units per x unit. a and ŷ use y units.

   Worked example: With r = 0.8, sᵧ = 10, sₓ = 4, x̄ = 5 and ȳ = 18, b = 2 and a = 8. At x = 6 the prediction is 20.

4. Data analysis · residual

   Formula or reference: Residual = observed value − predicted value

   Units and conditions: Use the response variable’s units.

   Worked example: A value of 22 with a prediction of 20 gives a residual of +2.

5. Data analysis · seasonal index

   Formula or reference: Seasonal index = observed figure ÷ deseasonalised figure

   Units and conditions: The index is a unitless ratio.

   Worked example: An observed 150 sales against a deseasonalised 120 gives a seasonal index of 1.25.

6. Finance · first-order recurrence

   Formula or reference: u₀ = a. uₙ₊₁ = Ruₙ + d.

   Units and conditions: R is unitless. a, u and d use the same units.

   Worked example: For u₀ = $100, R = 1.05 and d = −$10, u₁ = $95 and u₂ = $89.75.

7. Finance · effective annual rate

   Formula or reference: Effective rate = [(1 + r/(100n))ⁿ − 1] × 100%.

   Units and conditions: r is the quoted annual percentage number. n counts compounding periods per year.

   Worked example: For a nominal 8% compounded quarterly, (1.02⁴ − 1) × 100% ≈ 8.24%.

8. Matrices · 2 × 2 determinant

   Formula or reference: For A = [[a,b],[c,d]], det A = ad − bc.

   Units and conditions: Use the units implied by the entries, when applicable.

   Worked example: For A = [[3,1],[2,4]], det A = 12 − 2 = 10.

9. Matrices · 2 × 2 inverse

   Formula or reference: A⁻¹ = [[d,−b],[−c,a]] ÷ det A when det A ≠ 0.

   Units and conditions: Divide every entry by the determinant.

   Worked example: For A = [[3,1],[2,4]], the inverse is [[4,−1],[−2,3]]/10.

10. Matrices · recurrence with a constant input

    Formula or reference: S₀ is the initial state. Sₙ₊₁ = TSₙ + B.

    Units and conditions: S and B use the same units and dimensions. T is the transition matrix.

    Worked example: Let T = [[0.9,0.2],[0.1,0.8]], S₀ = (100,50)ᵀ and B = (4,0)ᵀ. Then S₁ = (104,50)ᵀ.

11. Matrices · Leslie recurrence

    Formula or reference: S₀ is the initial population. Sₙ₊₁ = LSₙ.

    Units and conditions: S is a population vector. L contains fertility and survival factors.

    Worked example: With L = [[0,2],[0.5,0]] and S₀ = (20,10)ᵀ, S₁ = (20,10)ᵀ.

12. Networks · Euler’s relation

    Formula or reference: v + f = e + 2 for a connected planar drawing.

    Units and conditions: v, e and f count vertices, edges and faces, including the outer face.

    Worked example: A planar drawing with 5 vertices and 7 edges has 4 faces.