Sampling Distribution of a Sample Mean — AP Statistics
1. What Is the Sampling Distribution of a Sample Mean? ★★☆☆☆ ⏱ 3 min
The sampling distribution of a sample mean (often shortened to sampling distribution of $ar{x}$) is the probability distribution of the sample mean statistic calculated from every possible random sample of the same fixed size $n$ drawn from a given population. This is distinct from the population distribution (distribution of all individual values in the population) and the sample distribution (distribution of values in a single collected sample).
2. Mean and Standard Error of the Sampling Distribution of $ar{x}$ ★★★☆☆ ⏱ 4 min
Two core properties of any sampling distribution are its center (mean) and spread (standard deviation, called standard error here). For any simple random sample of size $n$ drawn from a population with mean $ \mu$ and standard deviation $ \sigma$, the mean of the sampling distribution of $ \bar{x}$ is always equal to the population mean:
\mu_{\bar{x}} = \mu
This means the sample mean $ \bar{x}$ is an unbiased estimator of the population mean $ \mu$: over repeated sampling, the average of all possible sample means equals the true population mean. The spread of the sampling distribution is given by:
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
This formula only holds if two conditions are met: (1) observations in the sample are independent, which for sampling without replacement requires the 10% condition: the sample size $n$ is no more than 10% of the total population size $N$ ($n \leq 0.1N$). If we sample with replacement or the population is infinite, the 10% condition is automatically satisfied. Intuitively, increasing the sample size $n$ reduces the standard error, meaning larger samples produce sample means that are closer on average to the true population mean.
Exam tip: On AP FRQs, you will lose a point if you do not explicitly state and check the 10% condition before using the $\sigma/\sqrt{n}$ formula. Always include this step even if the condition is obviously satisfied.
3. The Central Limit Theorem (CLT) ★★★☆☆ ⏱ 3 min
The Central Limit Theorem (CLT) is the key result that allows us to use normal distribution calculations for sample means even when the underlying population is not normally distributed. Formally, the CLT states that for any population distribution (regardless of its shape: skewed, uniform, bimodal, etc.), the sampling distribution of the sample mean $ \bar{x}$ becomes approximately normally distributed as the sample size $n$ increases. For AP Statistics, we use the rule of thumb that $n \geq 30$ is a large enough sample size for the CLT approximation to hold. If the original population is already normally distributed, the sampling distribution of $ \bar{x}$ is exactly normally distributed for any sample size, no matter how small, so we do not need the CLT in that case.
Intuitively, the CLT works because averaging cancels out extreme values in individual observations. Even if many individual values are very high or very low, the average of multiple values will tend to cluster around the mean, producing a bell-shaped distribution for the average even if the original distribution is not bell-shaped.
Exam tip: Only invoke the Central Limit Theorem when the population is non-normal. If the population is already normal, you do not need CLT to claim normality of the sampling distribution; stating the population is normal is sufficient for any sample size. AP graders will deduct points for mis-citing CLT in this case.
4. Calculating Probabilities for Sample Means ★★★★☆ ⏱ 4 min
The most common application of this topic on the AP exam is calculating probabilities for a sample mean falling in a given range. The step-by-step process for this calculation is:
- Check conditions: (a) 10% condition for independence, (b) normality of the sampling distribution (either population is normal, or $n \geq 30$ and CLT applies).
- Calculate $ \mu_{\bar{x}} = \mu$ and $ \sigma_{\bar{x}} = \sigma/\sqrt{n}$.
- Calculate the z-score for the observed sample mean $ \bar{x}$:
- Use the standard normal distribution to find the desired probability.
z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}
Note that this is different from the z-score for an individual observation, which uses $ \sigma$ (population standard deviation) instead of $ \sigma/\sqrt{n}$ (standard error).
Exam tip: If the question asks for the probability that an individual observation falls in a range, use $ \sigma$; if it asks for the probability that a sample mean falls in a range, always use $ \sigma/\sqrt{n}$. Double-check which one the question asks for before calculating.
Common Pitfalls
Why: Confuses the standard deviation of the population (for individual observations) with the standard deviation of the sampling distribution of the sample mean, a very common mix-up on MCQs.
Why: Confuses the sampling distribution of the statistic with the original population distribution.
Why: Students focus on normality conditions and skip the independence check required for the formula to hold.
Why: Confuses the CLT requirement for non-normal populations with the case of normally distributed populations.
Why: Students memorize 'CLT = normality' and cite it automatically, even when it is unnecessary.
Why: Mixes up the three levels of distribution: population, sample, and sampling distribution.