“Overall,” “pooled,” or “combined”
Use the actual group sizes. Each person, cell, or other denominator unit counts equally.
(n₁ × p₁ + n₂ × p₂) / (n₁ + n₂)Here p is a decimal proportion. Multiply the result by 100 for a percentage.
SKILLS · QUANTITATIVE REASONING
Weighted percentages
Before calculating, ask: “A percentage of what?” The denominator tells you how much each group should count.
01 · THE IDEA
Count people before averaging percentages.
If 60% of a small group and 20% of a large group succeed, the overall percentage is closer to 20%. More people sit behind that rate.
Pooled percentage = total successes ÷ total people × 100
For separate, nonoverlapping groups measuring the same outcome: add the successes, add the denominators, then divide. If only rates are given, first use successes = decimal rate × group size.
MOVE THE GROUP SIZES
Same rates. Different combined result.
Group A stays at 60% success. Group B stays at 20%. Move people between the groups while keeping 100 people in total.
A: 12 successes / 20 peopleB: 16 successes / 80 people
Actual pooled rate28%(12 + 16) / 100
Equal-group average40%(60% + 20%) / 2
The pooled rate is closer to 20% because group B has more people. The equal-group average stays at 40% because its weights stay 50:50.
“Standardized” or “equal weight”
Use the reference weights the question specifies. Those may differ from the observed group sizes.
w₁ × p₁ + w₂ × p₂Weights sum to 1. Equal weights for two groups means ½ and ½; standardization does not always use equal weights.
A simple average is valid when the denominators are equal, or when the question explicitly asks for equal weight per group. It can also happen to match when rates are identical; that coincidence does not make it a general pooling rule.
02 · WORKED EXAMPLES
Same arithmetic, different questions
1. Pool the actual people
A task is completed by 60% of group A’s 20 participants and 20% of group B’s 80 participants. What percentage completes it overall?
- A contributes 0.60 × 20 = 12 completions.
- B contributes 0.20 × 80 = 16 completions.
- Combine: (12 + 16) / (20 + 80) = 28%.
Why not 40%? (60% + 20%) / 2 gives the small group the same influence as the large group. The actual weights are 20% and 80%.
2. Now the question changes the weights
Using those same rates, what would completion be if the two groups were represented equally?
- Use reference weights of 0.50 and 0.50.
- 0.50 × 60% + 0.50 × 20% = 40%.
This describes a population with equal group representation. It does not replace the 28% observed pooled rate.
If the reference population were 75% A and 25% B, the standardized rate would instead be 0.75 × 60% + 0.25 × 20% = 50%.
3. Read the comparison within each subgroup
| Task difficulty | Group X: successes / total | Group Y: successes / total |
|---|---|---|
| Easier | 8 / 10 (80%) | 63 / 90 (70%) |
| Harder | 18 / 90 (20%) | 1 / 10 (10%) |
X is 10 percentage points higher within each difficulty. Yet its pooled rate is (8 + 18) / 100 = 26%, while Y’s is (63 + 1) / 100 = 64%.
Why the reversal? X has mostly harder tasks; Y has mostly easier tasks. The group composition changes the pooled comparison. This is a form of Simpson’s paradox.
Give both difficulties equal weight and X is (80% + 20%) / 2 = 50%; Y is (70% + 10%) / 2 = 40%. That comparison answers an equal-composition question. Neither comparison by itself proves a causal effect.
03 · TRY IT IN DIFFERENT FORMS
Eight ways this can be asked
For each question, name the denominator and write the setup before calculating. Use a hint if needed, then choose an answer to see the explanation.
Untimed learning practice. These responses stay on this page and reset when you leave or reload; they do not change your test results.
QUESTION 1 / 8 · P/S
Counts in a table
Two nonoverlapping groups complete the same memory task. What percentage of all participants recalled the target word?
| Group | Recalled the word | Participants |
|---|---|---|
| A | 18 | 30 |
| B | 14 | 70 |
1 of 8
TAKE INTO YOUR NEXT PASSAGE
A four-step setup check
- Name the whole. All participants? Responders only? Total cells? Final volume?
- Choose the weights. Actual sizes for pooling; specified reference weights for standardization.
- Build the numerator. Convert rates to counts or amounts when needed, then add.
- Check the range. With positive weights, a weighted mean lies between its component values and moves toward the more heavily weighted one.
Explain your setup in one sentence: “I’m dividing ___ by ___ because the question asks for ___.”
Concept references
All worked examples and practice questions here are original learning exercises. For the underlying definitions, see NIST’s weighted mean formula and CDC’s explanation of standardization using reference weights.