Sampling Distributions for the Difference Between Sample Proportions

AP Statistics· 12 min read

1. Core Definition and Key Parameters★★☆☆☆⏱ 3 min

When you take independent random samples from two separate populations, the difference between their sample proportions will vary across repeated sampling. This set of all possible differences forms a unique sampling distribution with predictable, fixed parameters.

📘 Definition

Sampling Distribution of $\hat{p}_1 - \hat{p}_2$

$\overset{p}{^}_{1} - \overset{p}{^}_{2}$

The full set of values for the difference between sample proportions from two independent groups, calculated across all possible random samples of size $n_{1}$ and $n_{2}$

Example:

Comparing the proportion of 18-24 year old voters who support a policy to the proportion of 25-40 year old voters who support the same policy

\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2

The mean of the difference distribution is exactly equal to the difference between the two population proportions, no approximation required. The standard deviation uses the property that variances of independent variables add, even for differences.

\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}

📐 Worked Example

Population 1 has true proportion $p_{1} = 0.6$ , sample size $n_{1} = 100$ . Population 2 has true proportion $p_{2} = 0.45$ , sample size $n_{2} = 80$ . Calculate the mean and standard deviation of the sampling distribution of $\overset{p}{^}_{1} - \overset{p}{^}_{2}$ .

1
Step 1: Calculate the mean of the distribution directly using the population proportions
2
$\mu_{\hat{p}_1 - \hat{p}_2} = 0.6 - 0.45 = 0.15$
3
Step 2: Compute the individual variances for each sample proportion first
4
$\text{Var}(\hat{p}_1) = \frac{0.6(0.4)}{100} = 0.0024, \quad \text{Var}(\hat{p}_2) = \frac{0.45(0.55)}{80} = 0.00309$
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Step 3: Sum the variances and take the square root to get the combined standard deviation
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$\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{0.0024 + 0.00309} \approx 0.074$

2. Conditions for Normal Approximation★★★☆☆⏱ 4 min

To treat the sampling distribution of $\overset{p}{^}_{1} - \overset{p}{^}_{2}$ as approximately normal, you must verify three non-negotiable conditions for AP exam full credit.

Random: Both samples are selected independently via simple random sampling from their respective populations
Independent (10% Condition): Each sample size is no larger than 10% of its parent population when sampling without replacement
Normal (Large Counts): For both groups, the expected number of successes and failures is at least 10

📐 Worked Example

Check if the sampling distribution of $\overset{p}{^}_{1} - \overset{p}{^}_{2}$ is approximately normal given $n_{1} = 50$ , $p_{1} = 0.2$ , $n_{2} = 40$ , $p_{2} = 0.3$ .

1
Step 1: Confirm both samples are independent random samples as stated in the problem context
2
Step 2: Verify 10% condition: Both populations are at least 500 and 400 people respectively, which is standard for real survey contexts
3
Step 3: Check all four large count products
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$n_1p_1 = 10, n_1(1-p_1) = 40, n_2p_2 =12, n_2(1-p_2)=28$
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All values are ≥10, so the normal approximation is fully valid.

3. Probability Calculations for $\hat{p}_1 - \hat{p}_2$★★★☆☆⏱ 3 min

Once you have confirmed the distribution is approximately normal, you can use standard z-score calculations to find the probability of observing a difference as extreme or more extreme than a given value.

z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}}

📐 Worked Example

Using the earlier distribution with mean 0.15 and standard deviation 0.074, calculate the probability that $\overset{p}{^}_{1} - \overset{p}{^}_{2} > 0.2$ .

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Step 1: Plug the observed difference, mean, and standard deviation into the z-score formula
2
$z = \frac{0.2 - 0.15}{0.074} \approx 0.676$
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Step 2: Use a standard normal table or calculator to find the area to the right of z=0.676
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Step 3: Final probability ≈ 0.249, meaning roughly 25% of all possible sample differences will exceed 0.2

4. Common Pitfalls

Wrong move:

Adding standard deviations directly instead of adding variances then taking the square root

Why:

Variances of independent variables add, but standard deviations do not, so this will produce an overestimated combined SD

Correct move:

Calculate variance for each sample proportion separately, sum the two variances, then take the square root to get the combined standard deviation

Wrong move:

Using a pooled proportion to calculate the SD of the sampling distribution

Why:

Pooling is only required for two-proportion hypothesis tests, not for describing the theoretical sampling distribution

Correct move:

Use the individual known population proportions $p_{1}$ and $p_{2}$ in the SD formula for sampling distribution calculations

Wrong move:

Checking large counts only for the combined total sample instead of each individual group

Why:

Normality requires both individual single-proportion sampling distributions to be approximately normal, not just the combined difference

Correct move:

Verify all four product terms $n_{1} p_{1}$ , $n_{1} (1 - p_{1})$ , $n_{2} p_{2}$ , $n_{2} (1 - p_{2})$ are ≥10

Wrong move:

Forgetting to confirm the 10% independence condition when sampling without replacement

Why:

If sample size exceeds 10% of the parent population, observations are not independent, and the SD formula will be inaccurate

Correct move:

Explicitly state that both samples are less than 10% of their respective populations for full exam credit

Wrong move:

Using the combined total sample size $n_{1} + n_{2}$ in the SD denominator

Why:

The two samples are fully independent, so their sample sizes cannot be merged for variance calculations

Correct move:

Use $n_{1}$ only for group 1 variance, $n_{2}$ only for group 2 variance

5. Quick Reference Cheatsheet

Parameter	Formula	Required Condition
Mean of $\overset{p}{^}_{1} - \overset{p}{^}_{2}$	$p_{1} - p_{2}$	Independent random samples
Standard Deviation of $\overset{p}{^}_{1} - \overset{p}{^}_{2}$	$\frac{p _{1} ( 1 - p _{1} )}{n _{1}} + \frac{p _{2} ( 1 - p _{2} )}{n _{2}}$	10% condition satisfied for both samples
Z-score for Probability	$z = \frac{( p ^ _{1} - p ^ _{2} ) - ( p _{1} - p _{2} )}{σ _{\overset{p}{^}_{1} - \overset{p}{^}_{2}}}$	Large counts condition satisfied for both samples

What's Next

Mastering sampling distributions for the difference in sample proportions is the critical foundational step before moving to formal inference for two proportions. You will apply these exact properties to construct confidence intervals for the difference between two population proportions, which lets you estimate how large a gap exists between two groups with a specified level of confidence. You will also use the same core framework to run two-proportion z hypothesis tests, to evaluate if there is statistically significant evidence of a difference between two population proportions. This topic is heavily weighted on the AP Stats exam, appearing in roughly 15% of all FRQ points across recent administrations, so fluency with condition checks and parameter calculations will directly boost your score on multi-part inference questions.