Sampling Distributions for the Difference Between Two Sample Means

AP Statistics· Unit 7.2, 7.3· 20 min read

1. Core Definition and Parameters of the Sampling Distribution★★☆☆☆⏱ 5 min

📘 Definition

Sampling Distribution of $ar{x}_1 - ar{x}_2$

$ar{x}_1 - ar{x}_2$

The distribution of all possible values of the difference between sample means from two independent random samples of fixed sizes $n_{1}$ and $n_{2}$ drawn from two distinct populations.

Example:

If you take 50 samples of size 10 from Population A and 50 samples of size 12 from Population B, compute $ar{x}_A - ar{x}_B$ for each pair, the resulting histogram approximates this distribution.

\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2

This result holds for any two independent samples, no matter the population distribution shape, as long as the samples are randomly selected from their respective populations.

\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}

📐 Worked Example

Population 1 has mean 22 and standard deviation 5. Population 2 has mean 18 and standard deviation 3. For independent samples of size $n_{1} = 16$ and $n_{2} = 25$ , calculate the mean and standard deviation of $ar{x}_1 - ar{x}_2$ .

1
Calculate the mean of the sampling distribution:
2
$\mu_{\bar{x}_1 - \bar{x}_2} = 22 - 18 = 4$
3
Calculate the standard deviation of the sampling distribution:
4
$\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{5^2}{16} + \frac{3^2}{25}} = \sqrt{1.5625 + 0.36} \approx 1.39$

Exam tip:

AP graders will deduct points if you incorrectly write the spread formula as $r a c σ_{1} - σ_{2} n_{1} + n_{2}$ — always square each population standard deviation first.

2. Shape Conditions for the Normal Approximation★★★☆☆⏱ 6 min

For the sampling distribution of $ar{x}_1 - ar{x}_2$ to be approximately Normal, you must satisfy three core conditions, none of which can be skipped for full AP exam credit.

Random: Both samples are selected via simple random sampling from their respective populations
Independence: Observations within each sample are independent, and the two samples are independent of each other; verify the 10% condition if sampling without replacement
Normal/Large Sample: Either both populations are Normally distributed, or both sample sizes are ≥ 30, with no extreme outliers in small samples for roughly symmetric populations

✓ Quick check

Test your understanding of condition checks:

We take a sample of size 15 from a right-skewed population and a sample of size 18 from a left-skewed population. Can we use a Normal approximation for $ar{x}_1 - ar{x}_2$ ?
- Yes, because the total sample size is 33 which is over 30
- No, neither sample size is ≥ 30, and the populations are skewed
- Yes, the skews cancel out so the distribution is Normal
Reveal answer
No, neither sample size is ≥ 30, and the populations are skewed —
The CLT applies to each sample individually, not the sum of sample sizes, so you cannot use the Normal approximation here.

📐 Worked Example

We have two independent SRS samples: $n_{1} = 40$ from Population 1 with $σ_{1} = 8$ , $n_{2} = 35$ from Population 2 with $σ_{2} = 6$ . Both population distributions are unknown. Confirm if the sampling distribution of $ar{x}_1 - ar{x}_2$ is approximately Normal.

1
Check the Random condition: The problem explicitly states both are independent simple random samples, so this is satisfied.
2
Check the 10% condition: We can assume the two populations are at least 400 and 350 members respectively, standard for AP problems unless stated otherwise.
3
Check the Normal/Large Sample condition: $n_{1} = 40 \geq 30$ , $n_{2} = 35 \geq 30$ , so the CLT applies to both individual sampling distributions, making the difference distribution approximately Normal.

Exam tip:

You must explicitly name all three conditions in AP free response to earn full credit for two-sample inference procedures.

3. Calculating Probabilities for $ar{x}_1 - ar{x}_2$★★★☆☆⏱ 5 min

Once you confirm the sampling distribution is approximately Normal, you can standardize any observed difference $ar{x}_1 - ar{x}_2$ to a z-score, then use standard Normal tables or a calculator to find the corresponding probability.

z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

📐 Worked Example

Given the sampling distribution from the earlier example where $\mu_{ar{x}_1 - ar{x}_2}=4$ and $\sigma_{ar{x}_1 - ar{x}_2}=1.39$ , find the probability that $ar{x}_1 - ar{x}_2$ is greater than 6.

1
Compute the z-score for the value 6:
2
$z = \frac{6 - 4}{1.39} \approx 1.44$
3
Find the area to the right of z=1.44 on the standard Normal curve.
4
This area equals $1 - Φ (1.44) \approx 1 - 0.9251 = 0.0749$ , so the probability is ~7.5%.

4. Common Pitfalls

Wrong move:

Using the sum of standard deviations instead of the sum of variances under the square root for the spread of the difference

Why:

Variances of independent variables add, but standard deviations do not, so this will overestimate the spread.

Correct move:

Square each population standard deviation, divide by its sample size, sum those values, then take the square root of the total.

Wrong move:

Applying the Central Limit Theorem to the total combined sample size $n_{1} + n_{2}$ instead of checking each sample size individually

Why:

The CLT applies to each individual sampling distribution of $ar{x}_1$ and $ar{x}_2$ , not their sum.

Correct move:

Confirm $n_{1} \geq 30$ AND $n_{2} \geq 30$ for the Normal approximation to hold for the difference.

Wrong move:

Using sampling distribution rules for paired data on two independent samples

Why:

Paired data reduces to a single sample of differences, which uses a completely different sampling distribution.

Correct move:

Always confirm if samples are independent or paired before selecting the correct sampling distribution framework.

Wrong move:

Calculating the mean of the difference as $μ_{1} + μ_{2}$ instead of $μ_{1} - μ_{2}$

Why:

The difference of two random variables has a mean equal to the difference of their individual means, not the sum.

Correct move:

Subtract the second population mean from the first to get the center of the sampling distribution.

Wrong move:

Forgetting to apply the 10% condition when sampling without replacement for two groups

Why:

The standard deviation formula assumes observations are independent, which fails if you sample more than 10% of a small population.

Correct move:

Explicitly state that both populations are at least 10 times their respective sample sizes to satisfy independence.

5. Quick Reference Cheatsheet

Property	Formula / Rule
Mean of sampling distribution	$μ_{\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}} = μ_{1} - μ_{2}$
Standard deviation (independent samples)	$σ_{\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}} = \frac{σ _{1}^{2}}{n _{1}} + \frac{σ _{2}^{2}}{n _{2}}$
Normal condition 1	Both populations are Normally distributed
Normal condition 2	$n_{1} \geq 30$ AND $n_{2} \geq 30$
Z-score for difference	$z = \frac{( x ˉ _{1} - x ˉ _{2} ) - ( μ _{1} - μ _{2} )}{σ _{\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}}}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 1
FRQ 2 on two mean difference properties
2022 · Paper 2
MCQ set on sampling distribution shape
2021 · Paper 1
FRQ 4 on condition verification

What's Next

Mastering this sampling distribution is the critical foundation for all formal two-sample t inference procedures you will encounter on the AP Statistics exam. The rules you learned here directly extend to two-sample t confidence intervals and two-sample t significance tests, which make up over 15% of the points on the AP Stats free response section most years. You will also use this framework to distinguish between paired t procedures and two-sample t procedures, one of the most commonly tested high-difficulty discrimination points on the exam. Ensure you can quickly state all three conditions for the Normal approximation, as graders will deduct points for missing condition checks even if your final numerical answer is correct.

1. Core Definition and Parameters of the Sampling Distribution★★☆☆☆⏱ 5 min

Sampling Distribution of $ar{x}_1 - ar{x}_2$

2. Shape Conditions for the Normal Approximation★★★☆☆⏱ 6 min

3. Calculating Probabilities for $ar{x}_1 - ar{x}_2$★★★☆☆⏱ 5 min

4. Common Pitfalls

5. Quick Reference Cheatsheet

When this came up on past exams

What's Next

Sampling Distribution of $ar{x}_1 - ar{x}_2$

3. Calculating Probabilities for $ar{x}_1 - ar{x}_2$★★★☆☆⏱ 5 min