# Combined Variance and Standard Deviation

> Combined mean and variance for JEE Mathematics: n1(s1 squared plus d1 squared) formula, worked two-group examples and dispersion shortcut facts.

- Canonical URL: https://prepelephant.com/topics/jee/mathematics/variance-standard-deviation-combined
- Exam / course: JEE · Subject: Mathematics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Combined Variance and Standard Deviation", PrepElephant, https://prepelephant.com/topics/jee/mathematics/variance-standard-deviation-combined

## Direct answer

Combining the statistics of two groups demands two steps: first the combined mean, x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂), a weighted average; then the combined variance, σ² = [n₁(s₁² + d₁²) + n₂(s₂² + d₂²)]/(n₁ + n₂), where d₁ = x̄₁ − x̄ and d₂ = x̄₂ − x̄ measure how far each group's mean sits from the combined mean. Variance is never the plain average of variances — the spread between the group means contributes through the d² terms, so merging two tightly clustered groups with distant means can yield a large combined variance. An equivalent one-line form: σ² = (n₁s₁² + n₂s₂²)/(n₁ + n₂) + n₁n₂(x̄₁ − x̄₂)²/(n₁ + n₂)².

## What you must remember

- **Combined mean:** (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂) — weights are group sizes, always.
- **Combined variance:** [n₁(s₁² + d₁²) + n₂(s₂² + d₂²)]/(n₁ + n₂) with dᵢ = x̄ᵢ − x̄; the same formula extends to k groups by summing nᵢ(sᵢ² + dᵢ²) over all groups.
- **Two-group shortcut:** σ² = (n₁s₁² + n₂s₂²)/(n₁ + n₂) + [n₁n₂/(n₁ + n₂)²]·(x̄₁ − x̄₂)² — the second term is the price of the means' separation.
- **Equal groups simplification:** for n₁ = n₂, σ² = (s₁² + s₂²)/2 + (x̄₁ − x̄₂)²/4 — the most common numerical case.
- **Identical means:** if x̄₁ = x̄₂, combined variance is the size-weighted average of variances and the combined mean equals the common mean.
- **SD bookkeeping:** standard deviations combine by taking the square root of combined variance; SDs never add directly.
- **Shortcut facts:** variance is unchanged by shifting all data (adding a constant), scales by a² when data multiply by a; range-based bound SD ≤ (max − min)/2; for a binomial-type proportion data set these identities recur constantly.

## Two batches, one variance

Group A: 30 items, mean 20, variance 4. Group B: 20 items, mean 10, variance 1. Combined mean: (30×20 + 20×10)/50 = 800/50 = 16. Deviations: d₁ = 20 − 16 = 4, d₂ = 10 − 16 = −6. Combined variance: [30(4 + 16) + 20(1 + 36)]/50 = [600 + 740]/50 = 1340/50 = 26.8, so the combined SD is √26.8 ≈ 5.18. Notice the anatomy: the within-group spreads contributed 30(4) + 20(1) = 140, but the between-group separation contributed 30(16) + 20(36) = 1200 — nearly 90% of the total. Two individually tight groups whose means differ widely behave, after merging, like one very spread-out population.

The equal-size sanity check is worth running: with n₁ = n₂ = 25 and the same statistics, the formula gives σ² = (4 + 1)/2 + (20 − 10)²/4 = 2.5 + 25 = 27.5, confirming that the second term alone can dominate. In JEE numerical questions the numbers are engineered so that d₁ and d₂ are integers; if yours are not, recompute the combined mean before proceeding — a fractional x̄ almost always signals an arithmetic slip earlier.

## Why squaring the deviation matters

The recurring error is averaging variances like means: (4 + 1)/2 = 2.5 is the answer only when the means coincide, and options in the paper always include the unweighted or plainly averaged variance as bait. The second error is computing dᵢ as group mean minus the wrong reference — the deviations must be from the combined mean, not from one group's mean or from zero. Third, the units trap in SD questions: variance carries squared units, SD does not; a question asking for the combined standard deviation requires the final square root, and leaving √26.8 unevaluated when the answer is demanded as a decimal loses the mark. Main-level items feed the formula directly with two groups; Advanced composes — three groups, or a group split and re-merged, or variance asked after a transformation of variables (each score increased by 5: variance unchanged; each doubled: variance quadruples) — testing whether the d² logic and the transformation rules are genuinely understood rather than memorised.

## Frequently asked questions

### What is the formula for combined variance of two groups?

σ² = [n₁(s₁² + d₁²) + n₂(s₂² + d₂²)]/(n₁ + n₂), where d₁ and d₂ are the deviations of the group means from the combined mean.

### Why can't you simply average the two variances?

Because merging shifts each group's data relative to the new combined mean; the squared distance of each group mean from that combined mean adds the between-group term n₁n₂(x̄₁ − x̄₂)²/(n₁ + n₂)².

### What happens to combined variance when both groups have the same mean?

It reduces to the size-weighted average of the variances: (n₁s₁² + n₂s₂²)/(n₁ + n₂), since the deviation terms d₁ = d₂ = 0 vanish.

### Two groups of 30 and 20 items have means 20 and 10 and variances 4 and 1 — what is the combined variance?

Combined mean 16, deviations 4 and −6, so σ² = [30(20) + 20(37)]/50 = 26.8.

### How does adding or multiplying all observations change variance?

Adding a constant changes nothing (spread is shift-invariant); multiplying by a scales variance by a² and standard deviation by |a|.