Differences & the Big Picture of Variability

You’ll be able to

Find the mean and standard deviation of a difference between two independent statistics
Explain how variability of a difference combines the two individual variabilities
Connect sampling distributions to the logic of inference

Sampling distributions of differences

Much of inference compares two groups, so we need the sampling distribution of a difference — p-hat₁ − p-hat₂ or x-bar₁ − x-bar₂. The mean of the difference is the difference of the means (p₁ − p₂ or μ₁ − μ₂). Because the two samples are independent, the variances add: the standard deviation of the difference is the square root of the sum of the two individual variances.

Why variances add

When you subtract two independent random quantities, uncertainty accumulates — each estimate carries its own error, and combining them cannot cancel that uncertainty. That is why we add variances (never standard deviations, and never subtract). This single idea produces the standard-error formulas for two-sample intervals and tests in Units 6 and 7.

Difference of two independent sample proportions

μ = p₁ − p₂ · σ = sqrt( p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂ )

The mean subtracts, but the variances (the terms under the root) add. The same add-the-variances logic gives σ = sqrt(σ₁²/n₁ + σ₂²/n₂) for a difference of means.

Worked example

Two independent samples give p-hat₁ = 0.6 (n₁ = 100) and p-hat₂ = 0.5 (n₂ = 100). Estimate the standard deviation of p-hat₁ − p-hat₂.

1.Write the two variance contributions: p-hat₁(1−p-hat₁)/n₁ = 0.6·0.4/100 = 0.0024 and p-hat₂(1−p-hat₂)/n₂ = 0.5·0.5/100 = 0.0025.
2.Add the variances (independence): 0.0024 + 0.0025 = 0.0049.
3.Take the square root: sqrt(0.0049) = 0.07.

Answer: The standard deviation of the difference is sqrt(0.0024 + 0.0025) = sqrt(0.0049) = 0.07. Note the variances were added before the square root, not the standard deviations.

Checkpoint

Two independent sample means have standard deviations (standard errors) 3 and 4. What is the standard deviation of their difference?

On the exam

Every two-sample standard error you will meet is built by the same move: compute each group’s variance contribution, add them, then square-root. Memorize the pattern once and it powers two-proportion and two-mean intervals and tests alike.

Checkpoint

Why is the sampling distribution the key idea that makes statistical inference possible?

Answer the 2 checkpoints as you read.