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Combining Random Variables: Means Always Add, Variances Do Not

You’ll be able to

Means add without conditions

For any two random variables, μ(X + Y) = μX + μY and μ(X − Y) = μX − μY. This holds whether or not X and Y are independent, which makes expected value unusually well behaved. Variances are not so accommodating, and the difference between the two rules is where most errors in this topic live.

Combining independent random variables
μ(X ± Y) = μX ± μY σ²(X + Y) = σ²X + σ²Y σ²(X − Y) = σ²X + σ²Y σ(X ± Y) = √(σ²X + σ²Y)
Variances ADD in both cases, and only if X and Y are independent. Standard deviations never add — you must square, add, and take the root.

Why variance adds even for a difference

This looks wrong and is not. Variance measures spread, and subtracting one random quantity from another makes the result more variable, not less: both sources of randomness contribute to how much the difference bounces around. Formally the coefficient is squared, so (−1)² = 1 and the second variance enters positively. Concretely, if two independent measurements each vary by a few units, their difference varies by more than either — because a high value of one can coincide with a low value of the other.

When independence fails

The addition rule for variances requires independence, and it is the condition worth checking explicitly. Heights of two randomly chosen strangers are plausibly independent; heights of a randomly chosen parent and their child are not. Drawing two cards without replacement makes the second draw depend on the first. Where variables are dependent the variance of the sum includes a covariance term, which AP Statistics does not develop — so the exam simply requires you to notice when the rule does not apply and say so.

Worked example

X has mean 10 and standard deviation 3; Y has mean 4 and standard deviation 4, and X and Y are independent. Find the mean and standard deviation of X − Y.

  1. 1.Mean: μ(X − Y) = 10 − 4 = 6.
  2. 2.Variances: σ²X = 9 and σ²Y = 16.
  3. 3.Add them, even for a difference: σ²(X − Y) = 9 + 16 = 25.
  4. 4.Standard deviation: σ = √25 = 5.
Answer: Mean 6, standard deviation 5. Note that the standard deviation of the difference, 5, exceeds either individual standard deviation — subtracting a random quantity increases spread rather than reducing it.
Checkpoint

X and Y are independent with standard deviations 6 and 8. The standard deviation of X + Y is —

Tip

A quick sanity check: the standard deviation of a sum or difference of two independent variables is always larger than either one alone but smaller than their sum. If your answer falls outside that window you have made an arithmetic error.

Checkpoint

The rule σ²(X − Y) = σ²X + σ²Y requires that —

Answer the 2 checkpoints as you read.

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