Essential Mathematics and Statistics Formula Sheet
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Quick revision table for Mathematics and Statistics exams.
Unit I: Integration and Data Presentation
| Topic | Formula | Use |
|---|---|---|
| Integration Algebraic | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n≠-1 ∫1/x dx = ln|x| + C ∫k dx = kx + C ∫x dx = x²/2 + C | Finding area, antiderivative |
| Integration Trigonometric | ∫sin x dx = -cos x + C ∫cos x dx = sin x + C ∫sec²x dx = tan x + C ∫tan x dx = ln|sec x| + C ∫cot x dx = ln|sin x| + C | Trig integration |
| Integration Exponential | ∫eˣ dx = eˣ + C ∫aˣ dx = aˣ/ln a + C | Growth, decay problems |
| Pie Diagram Angle | Angle of sector = (Component value / Total value) × 360° | Constructing pie chart |
| Median from Ogive | Median = X-coordinate of intersection point of "less than" and "more than" ogive | Graphical method |
Unit II: Central Tendency and Dispersion
| Topic | Formula | Note |
|---|---|---|
| Arithmetic Mean Ungrouped | x̄ = Σx / n | n = number of observations |
| Arithmetic Mean Grouped | x̄ = Σfx / Σf | f = frequency, x = mid-value |
| Median Ungrouped (Odd n) | Median = [(n+1)/2]th term | Data must be sorted |
| Median Ungrouped (Even n) | Median = [n/2 th term + (n/2+1)th term] / 2 | Average of 2 middle terms |
| Median Grouped | Median = L + [(N/2 - cf)/f] × h | L=lower limit, N=Σf, cf=cum freq before, f=freq of median class, h=class width |
| Mode Grouped | Mode = L + [(f1-f0)/(2f1-f0-f2)] × h | f1=freq of modal class, f0=freq before, f2=freq after |
| Geometric Mean | GM = (x₁·x₂·...·xₙ)¹/ⁿ or log GM = (Σlog x) / n | Used for average growth rate |
| Harmonic Mean | HM = n / Σ(1/xᵢ) | Used for average speed |
| Range | Range = Maximum value - Minimum value | Simplest measure |
| Mean Deviation | M.D. = Σ|x - x̄| / n; Coefficient of M.D. = M.D. / x̄ | x̄ = mean used |
| Variance Population | σ² = Σ(x - x̄)² / n | n = total frequency |
| Standard Deviation | σ = √[Σ(x - x̄)² / n] | Most reliable measure |
| Coefficient of Variation | C.V. = (σ / x̄) × 100% | Used to compare 2 datasets |
Unit III: Correlation
| Topic | Formula | Range/Note |
|---|---|---|
| Karl Pearson's Coefficient r | r = Σ[(X-x̄)(Y-ȳ)] / √[Σ(X-x̄)² × Σ(Y-ȳ)²] Shortcut: r = [ΣXY - n·x̄·ȳ] / √[ΣX²-n·x̄²][ΣY²-n·ȳ²] | -1 ≤ r ≤ +1; r=+1 perfect positive, r=-1 perfect negative, r=0 no correlation |
| Spearman's Rank Correlation | rₛ = 1 - [6ΣD² / n(n²-1)] | D = difference in ranks; Used for ranked/qualitative data |
Unit IV: Linear Regression
| Topic | Formula | Use |
|---|---|---|
| Regression Equation Y on X | Y = a + b_yx·X b_yx = r·(σy/σx) = Σ[(X-x̄)(Y-ȳ)]/Σ(X-x̄)² a = ȳ - b_yx·x̄ | Predict value of Y for given X |
| Regression Equation X on Y | X = a' + b_xy·Y b_xy = r·(σx/σy) = Σ[(X-x̄)(Y-ȳ)]/Σ(Y-ȳ)² a' = x̄ - b_xy·ȳ | Predict value of X for given Y |
| Relation between r and b | r = ±√(b_yx × b_xy) | r is geometric mean of regression coefficients |
| Principle of Least Squares | Minimize Σ(Y - Ŷ)² | Condition for line of best fit |