MHT-CET Maths · Teaching notes

Measures of Dispersion — MHT-CET Maths

Measures of Dispersion was dropped from the MHT-CET Maths paper after 2024: it ran one question a paper across the 2023 and 2024 shifts and then scored zero across every 2025 paper. The notes are here because the bank has its past-year questions and because the same two formulas — the variance as Σx²/n − x̄², and the shift-and-scale rules — are exactly what the Probability Distribution chapter's mean-and-variance questions use. Fewer than one in ten of these questions is HARD. The three pages below run from computing a variance out of sums, through what happens to mean and variance when every observation is shifted or scaled, to the standard-series results and the two-missing-observations stem that was set five times. Every PYQ is tagged. Treat the chapter as a formula rehearsal for Probability Distribution, not as a paper topic.

Every subtopic, worked example, formula and trap in one printable document — answers shown, ready to share.

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Subtopic notes

PYQ weightage by concept

11 concepts · 32 PYQs — where the marks actually sit, so you know what to drill first

Mean and Variance From Sums — Σx, Σx² and Deviations From an Assumed Mean9 PYQs · 28%

Concept	PYQs	Share
Deviations From an Assumed Mean: Σ(x − a) and Σ(x − a)²	3	9%
σ² = Σx²/n − x̄²: Recover Any One Quantity From the Others	2	6%
A Wrong Observation Replaced, or Observations Added: Fix the Sums First	2	6%
Grouped Data: Midpoints, Σfx and Σfx²	2	6%

Shift and Scale — How Adding and Multiplying Change Mean, Variance and SD10 PYQs · 31%

Concept	PYQs	Share
Multiplying by λ: Mean × λ, SD × \|λ\|, Variance × λ²	5	16%
Mean of (x − k)² and the b = a + c Identity: Expand, Then Use σ² + x̄²	3	9%
Adding a Constant: Mean Shifts, Variance Stays	1	3%
y = px − q With Target Mean and SD: Two Equations, Watch the Sign of p	1	3%

Standard Series and Missing Observations — First n Naturals, Evens, Primes and Two Unknowns13 PYQs · 41%

Concept	PYQs	Share
Two Missing Observations: x + y From the Mean, x² + y² From the Variance	6	19%
Variance of the First n Natural Numbers, and of the Evens by Scaling	5	16%
One Unknown Value: Write the Variance in Terms of It and Solve	2	6%

Formula & revision sheet

11 formulas · 11 gotchas across all subtopics — the exam-eve cheat-sheet

Mean and Variance From Sums — Σx, Σx² and Deviations From an Assumed Mean

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

Shift and Scale — How Adding and Multiplying Change Mean, Variance and SD

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

Standard Series and Missing Observations — First n Naturals, Evens, Primes and Two Unknowns

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