Descriptive Statistics • Topic 3 of 4

Weighted & Missing-Value Means

A weighted mean applies when values do not all count equally — different group sizes, different credit hours, or different frequencies. You cannot simply average the sub-averages; you must weight each by its count: total of (weight × value), divided by the total weight. A classroom of 30 students averaging 70 combined with 20 students averaging 90 does not average to 80 — it is (30×70 + 20×90)/50 = 78, pulled toward the larger group. The mirror-image skill is the missing-value problem, which the GRE loves as Numeric Entry: if the target mean and the count are known, the required sum is fixed (sum = mean × n), so any single missing value is forced. Think in totals, not averages — nearly every "what must the sixth score be to average 85?" question collapses to one subtraction once you convert the target average into a required total.

✅ Solved examples

1. A class of 30 averages 70 on a test; a class of 20 averages 90. What is the combined average?
Weighted mean = (30×70 + 20×90)/(30+20) = (2100 + 1800)/50 = 3900/50 = 78. It leans toward the larger (30-student) group.
2. A student scores 82, 88 and 91 on three exams weighted 20%, 30% and 50%. Find the weighted average.
0.20×82 + 0.30×88 + 0.50×91 = 16.4 + 26.4 + 45.5 = 88.3.
3. Five numbers average 24. Four of them are 18, 22, 26 and 30. Find the fifth.
Required sum = 24 × 5 = 120. Known four sum to 96. Fifth = 120 − 96 = 24.
4. A shopper buys 3 kg of rice at $2/kg and 2 kg at $4.50/kg. What is the average price per kg?
Weight by quantity: (3×2 + 2×4.50)/(3+2) = (6 + 9)/5 = 15/5 = $3.00 per kg — not the simple average $3.25.

✏️ Practice — try these, take hints as needed

1. A team of 40 averages 60 and a team of 10 averages 85. Combined average?
Weight by team size.
(40×60 + 10×85)/50.
(2400 + 850)/50.
65
2. Four numbers average 50. Three are 40, 55 and 65. Find the fourth.
Required total = 50 × 4 = 200.
Known three sum to 160.
200 − 160.
40
3. Scores of 90 and 70 carry weights 3 and 1. Weighted mean?
(3×90 + 1×70)/(3+1).
(270 + 70)/4.
340/4.
85
4. A student has averaged 78 over 4 exams. What is needed on a 5th to raise the average to 80?
Old total = 78 × 4 = 312.
New target total = 80 × 5 = 400.
400 − 312.
88
5. A 10-litre mix is 20% acid; a 5-litre mix is 50% acid. Overall acid percent when combined?
Acid volume, not percent, is what adds.
10×0.20 + 5×0.50 = 2 + 2.5 = 4.5 L.
4.5 / 15.
30%

📝 Topic test — 8 questions

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