Measures of Dispersion — MHT-CET Maths

Formulas (4)

• σ² = Σx²/n − x̄²: Recover Any One Quantity From the Others · The identity
  $$\sigma^2 = \frac{\sum x_i^2}{n} - \bar x^2 \qquad \sum x_i^2 = n\left(\sigma^2 + \bar x^2\right)$$
• Deviations From an Assumed Mean: Σ(x − a) and Σ(x − a)² · Assumed-mean formulas
  $$\bar x = a + \frac{\sum d_i}{n},\qquad \sigma^2 = \frac{\sum d_i^2}{n} - \left(\frac{\sum d_i}{n}\right)^2,\qquad d_i = x_i - a$$
• A Wrong Observation Replaced, or Observations Added: Fix the Sums First · Correcting sums
  $$\sum x_{\text{new}} = \sum x - x_{\text{wrong}} + x_{\text{right}},\qquad \sum x^2_{\text{new}} = \sum x^2 - x_{\text{wrong}}^2 + x_{\text{right}}^2$$
• Grouped Data: Midpoints, Σfx and Σfx² · Weighted sums
  $$\bar x = \frac{\sum f_i x_i}{N},\qquad \sigma^2 = \frac{\sum f_i x_i^2}{N} - \bar x^2,\qquad N = \sum f_i$$

Watch out for (4)

• Forgetting to add the mean squared
  → σ² = Σx²/n − x̄²: Recover Any One Quantity From the Others
• Reading Σ(x − 2)²/n as the variance
  → Deviations From an Assumed Mean: Σ(x − a) and Σ(x − a)²
• Keeping the old mean after correcting the sum
  → A Wrong Observation Replaced, or Observations Added: Fix the Sums First
• Using class limits instead of midpoints
  → Grouped Data: Midpoints, Σfx and Σfx²

Formulas (4)

• Adding a Constant: Mean Shifts, Variance Stays · Shift rule
  $$\overline{x + c} = \bar x + c,\qquad \operatorname{Var}(x + c) = \operatorname{Var}(x)$$
• Multiplying by λ: Mean × λ, SD × |λ|, Variance × λ² · Scale rule
  $$\overline{\lambda x} = \lambda\bar x,\qquad \sigma_{\lambda x} = |\lambda|\,\sigma_x,\qquad \operatorname{Var}(\lambda x) = \lambda^2\operatorname{Var}(x)$$
• y = px − q With Target Mean and SD: Two Equations, Watch the Sign of p · Combined rule
  $$y = px - q:\quad \bar y = p\bar x - q,\qquad \sigma_y = |p|\,\sigma_x$$
• Mean of (x − k)² and the b = a + c Identity: Expand, Then Use σ² + x̄² · Shifted squares
  $$\frac{\sum (x_i - k)^2}{n} = \sigma^2 + (\bar x - k)^2$$

Watch out for (4)

• Adding the constant to the variance
  → Adding a Constant: Mean Shifts, Variance Stays
• Multiplying the variance by λ, not λ²
  → Multiplying by λ: Mean × λ, SD × |λ|, Variance × λ²
• Taking p positive by default
  → y = px − q With Target Mean and SD: Two Equations, Watch the Sign of p
• Using Σx²/n = σ² in the expansion
  → Mean of (x − k)² and the b = a + c Identity: Expand, Then Use σ² + x̄²

Formulas (3)

• Variance of the First n Natural Numbers, and of the Evens by Scaling · Standard results
  $$\operatorname{Var}(1..n) = \frac{n^2 - 1}{12},\qquad \operatorname{Var}(2, 4, \dots, 2n) = \frac{n^2 - 1}{3},\qquad \operatorname{Var}(1..2n) = \frac{4n^2 - 1}{12}$$
• Two Missing Observations: x + y From the Mean, x² + y² From the Variance · Two unknowns
  $$x + y = S,\ x^2 + y^2 = Q \ \Rightarrow\ xy = \frac{S^2 - Q}{2},\quad (x - y)^2 = 2Q - S^2$$
• One Unknown Value: Write the Variance in Terms of It and Solve · One unknown
  $$\frac{\sum x_i^2(k)}{n} - \bar x(k)^2 = \sigma^2 \ \Rightarrow\ \text{solve for } k$$

Watch out for (3)

• Doubling instead of quadrupling for the evens
  → Variance of the First n Natural Numbers, and of the Evens by Scaling
• Dividing by n − 1
  → Two Missing Observations: x + y From the Mean, x² + y² From the Variance
• Forgetting the mean also contains k
  → One Unknown Value: Write the Variance in Terms of It and Solve