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.
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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.
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Kaart 1
Küsimus
In a weighted arithmetic mean, what does a weight describe?
Vastus
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.
Kaart 2
Küsimus
For values xᵢ with nonnegative weights wᵢ and a positive total weight, what is the weighted arithmetic mean?
Vastus
Divide the sum of weight × value by the sum of weights.
Kaart 3
Küsimus
A list contains the value 4 twice and the value 9 three times. What frequency weights reproduce the mean of all five entries?
Vastus
Weights 2 for 4 and 3 for 9.
Each weight counts how often its value occurs.
Kaart 4
Küsimus
Two assessments have scores 9/12 and 36/40. No averaging policy is stated. Is the overall percentage uniquely determined?
Vastus
No. The weighting policy is missing.
Giving each assessment equal weight and pooling earned/possible points answer different questions.
Kaart 5
Küsimus
Relative weights are 2, 3, and 5. What are their normalized weights?
Vastus
0.2, 0.3, and 0.5.
Divide each weight by the total, 10. Normalized weights sum to 1.
Kaart 6
Küsimus
With nonnegative weights and positive total weight, where must a weighted arithmetic mean lie?
Vastus
Between the smallest and largest values that have positive weight, inclusive.
Zero-weight values do not set the relevant endpoints.
Kaart 7
Küsimus
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?
Vastus
The group sizes.
Group size × group mean reconstructs each group total. Add those totals and divide by the total number of observations.
Kaart 8
Küsimus
What does a weighted arithmetic mean become when all weights are equal and positive?
Vastus
The ordinary arithmetic mean.
The common weight cancels from numerator and denominator.
Kaart 9
Küsimus
Values 14, 20, and 200 have respective weights 2, 3, and 0. What is their weighted mean?
Vastus
17.6.
(2 × 14 + 3 × 20) / (2 + 3) = 88/5. The value 200 contributes nothing.
Kaart 10
Küsimus
A category score is 90% and its course weight is 20%. Which number describes performance, and which describes relative influence?
Vastus
90% is performance; 20% is relative influence.
Its full-course contribution is 0.20 × 90 = 18 percentage points.
Kaart 11
Küsimus
Scores are 8/10 and 21/30. A policy gives each assessment equal weight after converting to percentages. What is the overall percentage?
Vastus
75%.
The assessment percentages are 80% and 70%; their equal-weight mean is (80 + 70) / 2.
Kaart 12
Küsimus
A recorded value is 6 on four occasions and 11 on two occasions. What is the mean over all six occasions?
Vastus
23/3, or 7⅔.
(4 × 6 + 2 × 11) / 6 = 46/6.
Kaart 13
Küsimus
All weights in a weighted mean are multiplied by 7. The original total weight is positive. What happens to the mean?
Vastus
It stays the same.
Both numerator and denominator gain the same factor, which cancels.
Kaart 14
Küsimus
A group has 15 observations and an exact mean of 12 units per observation. What is the total of its observations?
Vastus
180 units.
15 × 12 = 180. Reconstruct totals before combining groups.
Kaart 15
Küsimus
Scores are 8/10 and 21/30. A policy pools all earned points and all possible points. What is the overall percentage?
Vastus
72.5%.
(8 + 21) / (10 + 30) × 100 = 29/40 × 100.
Kaart 16
Küsimus
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?
Vastus
The numbers of trials in the batches.
Success percentage × trial count reconstructs successes, after converting percentages to proportions.
Kaart 17
Küsimus
Values 18 and 30 have relative weights 1 and 3. What is their weighted mean?
Vastus
27.
(1 × 18 + 3 × 30) / 4 = 108/4.
Kaart 18
Küsimus
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?
Vastus
15.
(10 × 12 + 15 × 17) / 25 = 375/25.
Kaart 19
Küsimus
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?
Vastus
71%.
0.25 × 80 + 0.75 × 68 = 20 + 51.
Kaart 20
Küsimus
Values 10 and 22 have weights 2 and 4. A solution reports the numerator 108 as the weighted mean. What single operation is missing?
Vastus
Divide by the total weight, 6.
108/6 = 18. A weighted sum becomes a weighted mean only after the correct division.
Kaart 21
Küsimus
A course category has a missing score and a positive weight. Is that equivalent to giving the category zero weight?
Vastus
No.
A missing score is unknown; zero weight explicitly removes influence. A policy must say how missing work is treated.
Kaart 22
Küsimus
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?
Vastus
8.9.
(4 × 5 + 6 × 8 + 10 × 11) / (4 + 6 + 10) = 178/20.
Kaart 23
Küsimus
Completed work has score 84% and counts for 40% of a course. What contribution has it earned toward the full-course percentage?
Vastus
33.6 percentage points.
0.40 × 84 = 33.6. The remaining 60% has not yet supplied a contribution.
Kaart 24
Küsimus
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?
Vastus
75%.
(9 + 21) / (15 + 25) × 100 = 30/40 × 100.
Kaart 25
Küsimus
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?
Vastus
No.
87 exceeds every positively weighted value. Check the products, scales, and denominator.
Kaart 26
Küsimus
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?
Vastus
62.5.
Use the counts: (6 × 40 + 18 × 70) / 24 = 1500/24. Equal weighting would give each group equal influence.
Kaart 27
Küsimus
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?
Vastus
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.
Kaart 28
Küsimus
Values 24 and 36 have normalized weights 0.25 and 0.75. What is their weighted mean?
Vastus
33.
0.25 × 24 + 0.75 × 36 = 6 + 27. No further division is needed because the weights sum to 1.
Kaart 29
Küsimus
All weights in a proposed weighted arithmetic mean are zero. Is the mean defined?
Vastus
No.
The total weight is zero, so the formula would divide by zero.
Kaart 30
Küsimus
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?
Vastus
20.
The second total is 20 × 16 − 8 × 10 = 240; divide by 12. Check: (80 + 240)/20 = 16.
Kaart 31
Küsimus
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%?
Vastus
86%.
0.60 × 76 + 0.40 × x = 80, so x = (80 − 45.6)/0.40 = 86.
Kaart 32
Küsimus
Two reported group means use some of the same observations. Can weighting them by group sizes automatically recover the mean of the distinct observations?
Vastus
No. Overlapping observations would be counted more than once.
You need information that removes the duplicate contribution before reconstructing the target total and count.
Kaart 33
Küsimus
Values 20 and 80 have positive weights 4 and 1. Without calculating the mean, which value is it closer to?
Vastus
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.
Kaart 34
Küsimus
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?
Vastus
24 observations.
(216 + 30n)/(12 + n) = 26 gives 4n = 96. Check: (216 + 720)/36 = 26.
Kaart 35
Küsimus
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?
Vastus
23 percentage points.
0.25 × 92 = 23. The 92% score describes performance on completed work.
Kaart 36
Küsimus
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?
Vastus
80%.
(0.50 × 10 + 0.90 × 30)/40 × 100 = 32/40 × 100. Weight the rates by their trial counts.
Kaart 37
Küsimus
Two categories have course weights 30% and 70%. Only the first score, 90%, is known. Is the final course percentage determined?
Vastus
No. The second score is still needed.
The known contribution is 27 percentage points; the final result is 27 + 0.70 × the second score.
Kaart 38
Küsimus
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?
Vastus
No; it gives an estimate of 11 units.
(10 × 8 + 30 × 12)/40 = 11, but rounding discarded information about both true group totals.
Kaart 39
Küsimus
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?
Vastus
No. The maximum is 65%.
0.70 × 50 + 0.30 × 100 = 65. Reaching 80 would require 150% on the remaining assessment.
Kaart 40
Küsimus
Scores 0.8 and 70 represent 80% and 70%. Equal assessment weights are intended. What must be repaired before averaging the two recorded numbers?
Vastus
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.
Kaart 41
Küsimus
Completed categories contribute 36 percentage points and occupy 45% of a course. The remaining 55% receives score 80%. What is the final course percentage?
Vastus
80%.
36 + 0.55 × 80 = 80. Add full-course contributions; do not average the contribution 36 with the score 80.
Kaart 42
Küsimus
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?
Vastus
16.
(10 + 22)/2 = 16. This answers the equal-group policy; it need not equal the mean across individuals.
42 kaarti
Weighted Average Flashcards: Choose Weights & Combine Means
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