# Sums of normal variables

If X and Y are independent normals, X + Y is also normal. Means add, and variances add: mean = mu1 + mu2, variance = sigma1 squared + sigma2 squared. Standard deviations do not add.

Statistics (university) · Part 04: Continuous distributions and the normal curve · Card 8 of 8 · Last updated 2026-10-19

## Analogy

Two noisy measurements combined: each adds its own wobble, and the wobbles partly cancel.

## Example

**Two parcels**

Each parcel: mean 5 kg, SD 0.4
Total mean = 10 kg
Total variance = 0.16 + 0.16 = 0.32
Total SD = 0.566 kg
P(total > 11): z = 1.77, about 0.038

## In practice

Used for total packing weight, combined journey times and measurement error budgets.

## Practice

### Flashcards

- **Total variance of two independent normals?** Add their variances.

## Questions people ask

### Why do variances add but not standard deviations?

For independent variables, the squared spreads combine like the sides of a right-angled triangle. The total spread is the square root of the sum of squares, which is less than adding the two SDs. Two parcels each with SD 0.4 give 0.566, not 0.8.

### Does the sum of two normal variables stay normal?

Yes, if they are independent. The sum is exactly normal with the added means and variances. This closure property is rare. Most distributions change shape when added, which is part of why the normal is so convenient to work with.

## Related concepts

- [Normal distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-4)
- [Standard normal and z-scores](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-5)
- [Normal probabilities and percentiles](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-6)
- [Normal approximation to the binomial](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-7)

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