A weighted mean is the mean of a combined dataset, found by multiplying each group's mean by its weight (usually the group's size), adding those products, and dividing by the total weight. In AP Stats it shows up when you merge groups of different sizes, where simply averaging the means gives the wrong answer.
A weighted mean is what you get when you combine groups of data that don't all count equally. Instead of adding up raw values, you multiply each group's mean by its weight (almost always the number of values in that group), add those products together, and divide by the total weight. It works because mean × count gives you back the group's total. So the weighted mean is really just the regular mean formula from learning objective 1.7.A (sum of all values divided by n), computed in a shortcut way when you only know group summaries.
Here's the move in action. A downtown branch has 50 employees with a mean salary of $60,000 and a suburban branch has 150 employees with a mean of $40,000. The combined mean is (50 × 60,000 + 150 × 40,000) / 200 = $45,000. Notice that's not $50,000, the simple average of the two means. The suburban branch has three times the people, so it pulls the combined mean toward its value. Bigger group, bigger pull. That's the whole idea of weighting.
Weighted mean lives in Topic 1.7 (Summary Statistics for a Quantitative Variable) in Unit 1: Exploring One-Variable Data, and it directly supports learning objective 1.7.A, calculating measures of center for quantitative data. The CED defines the mean as the sum of all values divided by n, and the weighted mean is the only way to honor that definition when your data arrive pre-summarized as group means and group sizes. It also feeds 1.7.C, choosing and justifying a measure of center, because understanding how each value (or group) pulls on the mean is exactly why the mean is nonresistant. The exam loves this concept because it punishes autopilot. Students who reflexively average two means lose easy points to students who stop and ask how big each group is.
Keep studying AP® Statistics Unit 1
Mean (Unit 1)
The weighted mean isn't a new statistic. It's the ordinary mean, sum of values over n, computed from group summaries instead of raw data. Mean × count recovers each group's total, which is the trick that makes the shortcut work.
Sensitivity to extreme values (Unit 1)
Weighting explains why the mean is nonresistant. Every value pulls on the mean in proportion to how far out it sits, so one extreme value acts like a heavy weight dragging the mean toward it. The median ignores that pull entirely.
Median (Unit 1)
There's no clean 'weighted median' shortcut for combining groups. Knowing two groups' medians and sizes does not let you compute the combined median, which is one practical reason combined-group problems are always about means.
Expected value of a random variable (Unit 4)
Expected value is a weighted mean in disguise. You multiply each outcome by its probability (the weight) and sum. If weighted means click for you in Unit 1, the expected value formula in Unit 4 will feel familiar instead of new.
This is almost always a multiple-choice calculation, and the wrong answer choices are built from the most tempting mistake, averaging the means without weighting. Typical stems give you two groups with different sizes and different means, like 10 values averaging 80 combined with 20 values averaging 95 (answer: (10×80 + 20×95)/30 = 90, not 87.5). A frequency version is also common, like a quiz where two students score 5, three score 8, and five score 10, giving (2×5 + 3×8 + 5×10)/10 = 8.4. Some questions test the concept instead of the arithmetic by asking what additional information you'd need to combine two means. The answer is the size of each group. No released FRQ has used the phrase 'weighted mean' verbatim, but combining group summaries correctly is fair game anywhere descriptive statistics appear.
Averaging the means (like saying the district mean is (78 + 84)/2 = 81) only works if both groups are exactly the same size. The weighted mean fixes this by letting each group count in proportion to how many values it has. In the salary example, averaging $60,000 and $40,000 gives $50,000, but the correct weighted mean is $45,000 because the $40,000 branch has three times as many employees. On the AP exam, the unweighted average is almost always sitting there as a trap answer choice.
The weighted mean of combined groups is the sum of (each group's mean × its size) divided by the total number of values.
You can only average two group means directly when the groups are exactly the same size; otherwise you must weight by group size.
The shortcut works because mean × count gives back each group's total, so you're really just applying the standard mean formula from LO 1.7.A.
The combined mean always lands closer to the mean of the larger group, which is a quick way to sanity-check your answer.
If a question asks what you need to combine two means, the answer is the number of values in each group.
Expected value in Unit 4 uses the same logic, with probabilities playing the role of weights.
It's the mean of a combined dataset found by multiplying each group's mean by its size (the weight), summing those products, and dividing by the total count. It appears in Topic 1.7 under learning objective 1.7.A.
No, unless the groups are the same size. If 10 values average 80 and 20 values average 95, the combined mean is (10×80 + 20×95)/30 = 90, not the simple average of 87.5.
It's the same statistic computed differently. The regular mean sums raw values and divides by n; the weighted mean reaches the same number using group means and group sizes when you don't have the raw data.
You need each group's mean and each group's size. A district can't combine School A's mean of 78 with School B's mean of 84 without knowing how many students attend each school.
Yes, it's tested under Topic 1.7 (LO 1.7.A), usually as a multiple-choice question combining two groups of different sizes. The unweighted average of the means is the classic trap answer.
Connect this key term to the AP exam workflow: review the course, practice questions, and check related study tools.
Review units, study guides, and course resources.
Check this vocabulary in multiple-choice context.
Apply key concepts in written AP responses.
Estimate the exam score you are working toward.
Review the highest-yield facts before practice.
Put the full course together before test day.