Weighted Average Flashcards: Choose Weights & Combine Means

Question 1

In a weighted arithmetic mean, what does a weight describe?

Accepted 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.

Question 2

For values xᵢ with nonnegative weights wᵢ and a positive total weight, what is the weighted arithmetic mean?

Accepted Answer

Divide the sum of weight × value by the sum of weights.

\bar{x}_w = \frac{\sum_i w_i x_i}{\sum_i w_i}

Question 3

A list contains the value 4 twice and the value 9 three times. What frequency weights reproduce the mean of all five entries?

Accepted Answer

Weights 2 for 4 and 3 for 9.

Each weight counts how often its value occurs.

Question 4

Two assessments have scores 9/12 and 36/40. No averaging policy is stated. Is the overall percentage uniquely determined?

Accepted Answer

No. The weighting policy is missing.

Giving each assessment equal weight and pooling earned/possible points answer different questions.

Question 5

Relative weights are 2, 3, and 5. What are their normalized weights?

Accepted Answer

0.2, 0.3, and 0.5.

Divide each weight by the total, 10. Normalized weights sum to 1.

Question 6

With nonnegative weights and positive total weight, where must a weighted arithmetic mean lie?

Accepted Answer

Between the smallest and largest values that have positive weight, inclusive.

Zero-weight values do not set the relevant endpoints.

Question 7

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?

Accepted Answer

The group sizes.

Group size × group mean reconstructs each group total. Add those totals and divide by the total number of observations.

Question 8

What does a weighted arithmetic mean become when all weights are equal and positive?

Accepted Answer

The ordinary arithmetic mean.

The common weight cancels from numerator and denominator.

Question 9

Values 14, 20, and 200 have respective weights 2, 3, and 0. What is their weighted mean?

Accepted Answer

17.6.

(2 × 14 + 3 × 20) / (2 + 3) = 88/5. The value 200 contributes nothing.

Question 10

A category score is 90% and its course weight is 20%. Which number describes performance, and which describes relative influence?

Accepted Answer

90% is performance; 20% is relative influence.

Its full-course contribution is 0.20 × 90 = 18 percentage points.

Question 11

Scores are 8/10 and 21/30. A policy gives each assessment equal weight after converting to percentages. What is the overall percentage?

Accepted Answer

75%.

The assessment percentages are 80% and 70%; their equal-weight mean is (80 + 70) / 2.

Question 12

A recorded value is 6 on four occasions and 11 on two occasions. What is the mean over all six occasions?

Accepted Answer

23/3, or 7⅔.

(4 × 6 + 2 × 11) / 6 = 46/6.

Question 13

All weights in a weighted mean are multiplied by 7. The original total weight is positive. What happens to the mean?

Accepted Answer

It stays the same.

Both numerator and denominator gain the same factor, which cancels.

Question 14

A group has 15 observations and an exact mean of 12 units per observation. What is the total of its observations?

Accepted Answer

180 units.

15 × 12 = 180. Reconstruct totals before combining groups.

Question 15

Scores are 8/10 and 21/30. A policy pools all earned points and all possible points. What is the overall percentage?

Accepted Answer

72.5%.

(8 + 21) / (10 + 30) × 100 = 29/40 × 100.

Question 16

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?

Accepted Answer

The numbers of trials in the batches.

Success percentage × trial count reconstructs successes, after converting percentages to proportions.

Question 17

Values 18 and 30 have relative weights 1 and 3. What is their weighted mean?

Accepted Answer

27.

(1 × 18 + 3 × 30) / 4 = 108/4.

Question 18

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?

Accepted Answer

15.

(10 × 12 + 15 × 17) / 25 = 375/25.

Question 19

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?

Accepted Answer

71%.

0.25 × 80 + 0.75 × 68 = 20 + 51.

Question 20

Values 10 and 22 have weights 2 and 4. A solution reports the numerator 108 as the weighted mean. What single operation is missing?

Accepted Answer

Divide by the total weight, 6.

108/6 = 18. A weighted sum becomes a weighted mean only after the correct division.

Question 21

A course category has a missing score and a positive weight. Is that equivalent to giving the category zero weight?

Accepted Answer

No.

A missing score is unknown; zero weight explicitly removes influence. A policy must say how missing work is treated.

Question 22

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?

Accepted Answer

8.9.

(4 × 5 + 6 × 8 + 10 × 11) / (4 + 6 + 10) = 178/20.

Question 23

Completed work has score 84% and counts for 40% of a course. What contribution has it earned toward the full-course percentage?

Accepted Answer

33.6 percentage points.

0.40 × 84 = 33.6. The remaining 60% has not yet supplied a contribution.

Question 24

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?

Accepted Answer

75%.

(9 + 21) / (15 + 25) × 100 = 30/40 × 100.

Question 25

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?

Accepted Answer

No.

87 exceeds every positively weighted value. Check the products, scales, and denominator.

Question 26

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?

Accepted Answer

62.5.

Use the counts: (6 × 40 + 18 × 70) / 24 = 1500/24. Equal weighting would give each group equal influence.

Question 27

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?

Accepted 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.

Question 28

Values 24 and 36 have normalized weights 0.25 and 0.75. What is their weighted mean?

Accepted Answer

33.

0.25 × 24 + 0.75 × 36 = 6 + 27. No further division is needed because the weights sum to 1.

Question 29

All weights in a proposed weighted arithmetic mean are zero. Is the mean defined?

Accepted Answer

No.

The total weight is zero, so the formula would divide by zero.

Question 30

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?

Accepted Answer

20.

The second total is 20 × 16 − 8 × 10 = 240; divide by 12. Check: (80 + 240)/20 = 16.

Question 31

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%?

Accepted Answer

86%.

0.60 × 76 + 0.40 × x = 80, so x = (80 − 45.6)/0.40 = 86.

Question 32

Two reported group means use some of the same observations. Can weighting them by group sizes automatically recover the mean of the distinct observations?

Accepted 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.

Question 33

Values 20 and 80 have positive weights 4 and 1. Without calculating the mean, which value is it closer to?

Accepted 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.

Question 34

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?

Accepted Answer

24 observations.

(216 + 30n)/(12 + n) = 26 gives 4n = 96. Check: (216 + 720)/36 = 26.

Question 35

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?

Accepted Answer

23 percentage points.

0.25 × 92 = 23. The 92% score describes performance on completed work.

Question 36

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?

Accepted Answer

80%.

(0.50 × 10 + 0.90 × 30)/40 × 100 = 32/40 × 100. Weight the rates by their trial counts.

Question 37

Two categories have course weights 30% and 70%. Only the first score, 90%, is known. Is the final course percentage determined?

Accepted 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.

Question 38

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?

Accepted Answer

No; it gives an estimate of 11 units.

(10 × 8 + 30 × 12)/40 = 11, but rounding discarded information about both true group totals.

Question 39

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?

Accepted Answer

No. The maximum is 65%.

0.70 × 50 + 0.30 × 100 = 65. Reaching 80 would require 150% on the remaining assessment.

Question 40

Scores 0.8 and 70 represent 80% and 70%. Equal assessment weights are intended. What must be repaired before averaging the two recorded numbers?

Accepted 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.

Question 41

Completed categories contribute 36 percentage points and occupy 45% of a course. The remaining 55% receives score 80%. What is the final course percentage?

Accepted Answer

80%.

36 + 0.55 × 80 = 80. Add full-course contributions; do not average the contribution 36 with the score 80.

Question 42

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?

Accepted Answer

16.

(10 + 22)/2 = 16. This answers the equal-group policy; it need not equal the mean across individuals.

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Weighted Average Flashcards: Choose Weights & Combine Means