Weighted Average Flashcards: Choose Weights & Combine Means
Practise 42 original weighted-average prompts: choose weights, combine group means, compare stated grading policies, and repair common calculation errors.
About this deck
Choose what counts equally before calculating an average. These 42 weighted average flashcards use original short problems to practise relative weights, combined means, pooled success rates, and explicitly stated grading policies.
You’ll choose weights from a situation; turn relative weights into normalized shares; calculate means from frequencies or disjoint groups; and distinguish equal-assessment, pooled-points, and supplied category-weight policies. Selected inverse cards recover a missing group mean, group size, or target assessment score. Local mistake repairs address the denominator, mismatched score scales, equal averaging of unequal groups, and full-course contributions confused with completed-work percentages. Every front supplies the assumptions needed after shuffling.
The sequence introduces the formula and weight meanings before dependent calculations. It then mixes short computations with policy choices, range checks, inverses, and repairs. Related variants are separated by at least three other cards; the review scheduler handles long-term spacing. All cards use the weighted-average tag.
You should know ordinary means, percentages, and simple algebra. Weights are nonnegative and their total must be positive. Group-combination problems state their target set and use exact means unless rounding is explicitly mentioned. Grading examples define fictional policies; they do not reproduce a particular school or gradebook.
This evergreen Mathematics deck is not aligned to an exam or curriculum. It excludes negative and inverse-variance weights, survey weighting, investment returns, institutional GPA rules, missing-work assumptions, dropped work, extra credit, grouped-interval estimates, average-speed problems, and statistical inference. It includes basic pooling but no Simpson’s-paradox reversal datasets. Mechanical reversals are omitted when the inputs would not be uniquely determined; selected inverses give enough information to solve them.
The formula was checked against NIST’s weighted mean reference. Normalization, bounds, reconstructed totals, and inverse answers were independently derived from that formula. The grading calculations follow only the policies stated on their own fronts. No source exercises, competitor cards, textbook examples, exam questions, or source diagrams were copied or adapted.
AI tools assisted with drafting and the original decorative cover. Every final front and back was manually read in author review, and numerical answers were checked with exact arithmetic or substitution. This does not claim learner testing or guaranteed results. The CC0 dedication covers original wording, invented examples, organization, metadata, and generated cover to the extent applicable rights exist. Mathematical facts and cited materials retain their own status. This is independent study practice, with no affiliation to or endorsement by the cited organizations.
Cards in this deck
Card 1
Question
In a weighted arithmetic mean, what does a weight describe?
Answer
How much a value counts relative to the other values.
The weight might represent a frequency, a group size, or a stated policy. Choose its meaning before calculating.
Card 2
Question
For values xᵢ with nonnegative weights wᵢ and a positive total weight, what is the weighted arithmetic mean?
Answer
Divide the sum of weight × value by the sum of weights.
Card 3
Question
A list contains the value 4 twice and the value 9 three times. What frequency weights reproduce the mean of all five entries?
Answer
Weights 2 for 4 and 3 for 9.
Each weight counts how often its value occurs.
Card 4
Question
Two assessments have scores 9/12 and 36/40. No averaging policy is stated. Is the overall percentage uniquely determined?
Answer
No. The weighting policy is missing.
Giving each assessment equal weight and pooling earned/possible points answer different questions.
Card 5
Question
Relative weights are 2, 3, and 5. What are their normalized weights?
Answer
0.2, 0.3, and 0.5.
Divide each weight by the total, 10. Normalized weights sum to 1.
Card 6
Question
With nonnegative weights and positive total weight, where must a weighted arithmetic mean lie?
Answer
Between the smallest and largest values that have positive weight, inclusive.
Zero-weight values do not set the relevant endpoints.
Card 7
Question
Disjoint groups cover the whole target set and measure the same quantity on the same scale. Each group mean is exact. What weights give the mean of all individual observations?
Answer
The group sizes.
Group size × group mean reconstructs each group total. Add those totals and divide by the total number of observations.
Card 8
Question
What does a weighted arithmetic mean become when all weights are equal and positive?
Answer
The ordinary arithmetic mean.
The common weight cancels from numerator and denominator.
Card 9
Question
Values 14, 20, and 200 have respective weights 2, 3, and 0. What is their weighted mean?
Answer
17.6.
(2 × 14 + 3 × 20) / (2 + 3) = 88/5. The value 200 contributes nothing.
Card 10
Question
A category score is 90% and its course weight is 20%. Which number describes performance, and which describes relative influence?
Answer
90% is performance; 20% is relative influence.
Its full-course contribution is 0.20 × 90 = 18 percentage points.
Card 11
Question
Scores are 8/10 and 21/30. A policy gives each assessment equal weight after converting to percentages. What is the overall percentage?
Answer
75%.
The assessment percentages are 80% and 70%; their equal-weight mean is (80 + 70) / 2.
Card 12
Question
A recorded value is 6 on four occasions and 11 on two occasions. What is the mean over all six occasions?
Answer
23/3, or 7⅔.
(4 × 6 + 2 × 11) / 6 = 46/6.
Card 13
Question
All weights in a weighted mean are multiplied by 7. The original total weight is positive. What happens to the mean?
Answer
It stays the same.
Both numerator and denominator gain the same factor, which cancels.
Card 14
Question
A group has 15 observations and an exact mean of 12 units per observation. What is the total of its observations?
Answer
180 units.
15 × 12 = 180. Reconstruct totals before combining groups.
Card 15
Question
Scores are 8/10 and 21/30. A policy pools all earned points and all possible points. What is the overall percentage?
Answer
72.5%.
(8 + 21) / (10 + 30) × 100 = 29/40 × 100.
Card 16
Question
Two disjoint batches cover all trials in a target set. Each trial has one success/failure outcome. To pool exact batch success percentages, what should their weights represent?
Answer
The numbers of trials in the batches.
Success percentage × trial count reconstructs successes, after converting percentages to proportions.
Card 17
Question
Values 18 and 30 have relative weights 1 and 3. What is their weighted mean?
Answer
27.
(1 × 18 + 3 × 30) / 4 = 108/4.
Card 18
Question
Two disjoint groups cover all 25 observations of one quantity on the same scale. Their exact means are 12 for 10 observations and 17 for 15 observations. What is the combined mean?
Answer
15.
(10 × 12 + 15 × 17) / 25 = 375/25.
Card 19
Question
A course uses an 80% project score with weight 25% and a 68% exam score with weight 75%. These are all categories. What is the course percentage?
Answer
71%.
0.25 × 80 + 0.75 × 68 = 20 + 51.
Card 20
Question
Values 10 and 22 have weights 2 and 4. A solution reports the numerator 108 as the weighted mean. What single operation is missing?
Answer
Divide by the total weight, 6.
108/6 = 18. A weighted sum becomes a weighted mean only after the correct division.
Card 21
Question
A course category has a missing score and a positive weight. Is that equivalent to giving the category zero weight?
Answer
No.
A missing score is unknown; zero weight explicitly removes influence. A policy must say how missing work is treated.
Card 22
Question
Three disjoint groups cover all observations of one quantity on the same scale. Their (size, exact mean) pairs are (4, 5), (6, 8), and (10, 11). What is the combined mean?
Answer
8.9.
(4 × 5 + 6 × 8 + 10 × 11) / (4 + 6 + 10) = 178/20.
Card 23
Question
Completed work has score 84% and counts for 40% of a course. What contribution has it earned toward the full-course percentage?
Answer
33.6 percentage points.
0.40 × 84 = 33.6. The remaining 60% has not yet supplied a contribution.
Card 24
Question
Two disjoint batches cover all trials in a target set. Exactly 9/15 trials succeed in one batch and 21/25 in the other. What is the pooled success percentage?
Answer
75%.
(9 + 21) / (15 + 25) × 100 = 30/40 × 100.
Card 25
Question
Values with positive weights range from 30 to 80. A calculation gives a weighted arithmetic mean of 87. Can that result be correct under nonnegative weighting?
Answer
No.
87 exceeds every positively weighted value. Check the products, scales, and denominator.
Card 26
Question
Two disjoint groups cover a target set on one scale. Their exact means are 40 and 70, with sizes 6 and 18. A solution averages the means as (40 + 70)/2. What is the correct combined mean?
Answer
62.5.
Use the counts: (6 × 40 + 18 × 70) / 24 = 1500/24. Equal weighting would give each group equal influence.
Card 27
Question
Two completed course categories have scores 70% and 90%, with course weights 20% and 30%. The other 50% is unfinished. When completed weights are renormalized, what is the completed-work percentage?
Answer
82%.
(0.20 × 70 + 0.30 × 90) / 0.50 = 41/0.50. This is the completed-work result, not the contribution to the whole course.
Card 28
Question
Values 24 and 36 have normalized weights 0.25 and 0.75. What is their weighted mean?
Answer
33.
0.25 × 24 + 0.75 × 36 = 6 + 27. No further division is needed because the weights sum to 1.
Card 29
Question
All weights in a proposed weighted arithmetic mean are zero. Is the mean defined?
Answer
No.
The total weight is zero, so the formula would divide by zero.
Card 30
Question
Two disjoint groups cover all 20 observations of one quantity on one scale. The combined mean is exactly 16. One group has 8 observations with exact mean 10; the other has 12. What is the second group’s mean?
Answer
20.
The second total is 20 × 16 − 8 × 10 = 240; divide by 12. Check: (80 + 240)/20 = 16.
Card 31
Question
Completed work has score 76% and weight 60%. A final exam has weight 40%. With no other categories, what final-exam percentage gives a course result of 80%?
Answer
86%.
0.60 × 76 + 0.40 × x = 80, so x = (80 − 45.6)/0.40 = 86.
Card 32
Question
Two reported group means use some of the same observations. Can weighting them by group sizes automatically recover the mean of the distinct observations?
Answer
No. Overlapping observations would be counted more than once.
You need information that removes the duplicate contribution before reconstructing the target total and count.
Card 33
Question
Values 20 and 80 have positive weights 4 and 1. Without calculating the mean, which value is it closer to?
Answer
20.
With two distinct values, the mean is closer to the one with the larger weight. Its distance from 20 is one fifth of the 60-unit gap.
Card 34
Question
Two disjoint groups cover a target set on one scale. One has 12 observations with exact mean 18; the other has exact mean 30. The combined mean is exactly 26. How many observations are in the second group?
Answer
24 observations.
(216 + 30n)/(12 + n) = 26 gives 4n = 96. Check: (216 + 720)/36 = 26.
Card 35
Question
Completed work worth 25% of a course has score 92%. A solution calls 92 the percentage points already earned toward the whole course. What is the correct contribution?
Answer
23 percentage points.
0.25 × 92 = 23. The 92% score describes performance on completed work.
Card 36
Question
Two disjoint batches cover all trials. Their exact success rates are 50% of 10 trials and 90% of 30 trials. A solution reports their equal mean, 70%, as the pooled rate. What is the pooled rate?
Answer
80%.
(0.50 × 10 + 0.90 × 30)/40 × 100 = 32/40 × 100. Weight the rates by their trial counts.
Card 37
Question
Two categories have course weights 30% and 70%. Only the first score, 90%, is known. Is the final course percentage determined?
Answer
No. The second score is still needed.
The known contribution is 27 percentage points; the final result is 27 + 0.70 × the second score.
Card 38
Question
Two disjoint groups cover a set on one scale, with sizes 10 and 30. Their means, rounded to the nearest whole unit, are reported as 8 and 12. Does weighting these reports give the exact combined mean?
Answer
No; it gives an estimate of 11 units.
(10 × 8 + 30 × 12)/40 = 11, but rounding discarded information about both true group totals.
Card 39
Question
Completed work has score 50% and weight 70%. The remaining assessment has weight 30% and a maximum score of 100%. Is a final course result of 80% attainable?
Answer
No. The maximum is 65%.
0.70 × 50 + 0.30 × 100 = 65. Reaching 80 would require 150% on the remaining assessment.
Card 40
Question
Scores 0.8 and 70 represent 80% and 70%. Equal assessment weights are intended. What must be repaired before averaging the two recorded numbers?
Answer
Put both scores on the same scale.
Use 80 and 70 for a percentage result of 75%, or 0.8 and 0.7 for a proportion of 0.75.
Card 41
Question
Completed categories contribute 36 percentage points and occupy 45% of a course. The remaining 55% receives score 80%. What is the final course percentage?
Answer
80%.
36 + 0.55 × 80 = 80. Add full-course contributions; do not average the contribution 36 with the score 80.
Card 42
Question
A policy gives two groups equal influence, regardless of size. Their exact means on one scale are 10 and 22. What is the policy-weighted mean?
Answer
16.
(10 + 22)/2 = 16. This answers the equal-group policy; it need not equal the mean across individuals.
42 cards
Weighted Average Flashcards: Choose Weights & Combine Means
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