normsum() — Adding two normals

Adds or subtracts two bell-curve things into one new bell curve.

X ± Y ~ N(μ₁±μ₂, σ₁²+σ₂²)

Key idea

• A normal (bell curve) is a hump-shaped spread of values; adding or subtracting two independent ones gives another bell curve.
• Variance — how far values scatter from the average — ADDS, whether you add or subtract the variables. (e.g. 9 + 16 = 25)
• To get the standard deviation (typical spread) of the result, take the square root of the summed variances. (e.g. sqrt(9 + 16) = 5)
• You cannot simply add the standard deviations directly. (e.g. 3 + 4 = 7 is wrong)
• The means (the centres) add for a sum and subtract for a difference.

Definition

$$aX\pm bY\sim N(a{\mu }_X\pm b{\mu }_Y,a^2{\sigma }_X^2+b^2{\sigma }_Y^2)$$

$X,Y$

two independent normal variables

$a,b$

constants you scale by

${\mu }_X,{\mu }_Y$

their means

${\sigma }_X^2,{\sigma }_Y^2$

their variances (always add)

In plain words

Pack a lunchbox and a water bottle, each a little uneven in weight. The total weight is still bell-shaped — that hump-in-the-middle pattern — just wobblier than either piece alone.

Where you'd use it

• Two parcels weigh Normal(2 kg, sd 0.3) and Normal(3 kg, sd 0.4). Their combined weight is Normal with mean 5 kg and sd √(0.3² + 0.4²) = 0.5 kg, wider than either parcel alone.
• Two race legs each take Normal(30 s, sd 3). The total time is Normal with mean 60 s and sd √(3² + 3²) ≈ 4.2 s, since the two wobbles pile up.
• You subtract two measurements, each Normal(0, sd 4). The difference is Normal with mean 0 but sd √(4² + 4²) ≈ 5.7 — note the spread still grows even though you subtracted.

Worked example 9

Add N(0, 3²) and N(0, 4²) — bells with sd 3 and sd 4. The result is N(0, 5²): the variances add (9 + 16 = 25), so the new sd is √(3² + 4²) = 5.

Frequently asked questions

How do I combine two normal variables?

Means add; variances add (if independent). Combine SDs via √(σ₁² + σ₂²), never by adding them.