Baseline, then drift

Weight each value by its share, or total deviations from a baseline: for 30 units at 98 and 70 at 102 with baseline 100, the signed deviations are 30 x -2 = -60 and 70 x 2 = 140, totaling 80 over 100 units, so the average is 100 + 0.8 = 100.8. Deviations keep every intermediate small; the direct total 2940 + 7140 = 10080 checks it.

Why it works

The mean equals any baseline plus the weight-averaged deviation from that baseline, which equals the direct weight total divided by total weight.

mean = B + sum(w x (x - B)) / sum(w) = sum(w x x) / sum(w)

Spot it: Values cluster near a round baseline such as 100, so deviations are one or two digits.

Technique reference and sources

Picture 30 units pulling 2 below 100 and 70 units pulling 2 above. The upward pull wins by 140 - 60 = 80 across 100 units, so the center drifts 0.8 above 100 to 100.8. The baseline carries the scale (100) and the deviations carry the small correction (0.8). Only one running total lives in your head: -60, then 80, then 0.8.

When this method fits

Values cluster near a round baseline such as 100, so deviations are one or two digits.
Weights are counts, notionals, or percents that sum to a round total such as 100.
One value or weight is missing and the average pins it down.
You can verify by the direct total: weights times values divided by total weight.

Pick the baseline. For 98 and 102 use 100. For 90, 100, 110 use 100. The baseline should sit inside the values so deviations stay small.
Total the signed deviations. 30 x (98 - 100) = -60 and 70 x (102 - 100) = 140, so -60 + 140 = 80. For 40 at 97 and 60 at 103: -120 + 180 = 60.
Divide by total weight. 80/100 = 0.8 and 60/100 = 0.6. This drift is the only division in the method.
Add back and verify direct. 100 + 0.8 = 100.8. Direct: 30 x 98 = 2940 and 70 x 102 = 7140, total 10080/100 = 100.8.

Two-point drift: Drift = (w1 x d1 + w2 x d2) / (w1 + w2) with di = xi - B; mean = B + drift.

30 at 98, 70 at 102 from 100: (-60 + 140)/100 = 0.8, mean 100.8
40 at 97, 60 at 103 from 100: (-120 + 180)/100 = 0.6, mean 100.6

Multi-point and missing values: Same drift with more terms; for a missing value solve total = mean x total weight.

20 at 90, 30 at 100, 50 at 110: total 10300/100 = 103
40 at 80 plus 60 at x averaging 92 gives x = 100

Deviations beat raw totals when clustered: Deviations hold one- and two-digit products where raw totals hold four-digit ones. Raw totals are the alternative and the check; they win when weights are equal and values are already small, because then no baseline shortens anything.

30 x -2 + 70 x 2 = 80 vs 2940 + 7140 = 10080
50 at 80, 50 at 120: deviations -1000 + 1000 = 0, mean 100

Common traps

Averaging 98 and 102 as (98 + 102)/2 = 100 when weights are 30 and 70. Equal weights give 100, but 70 units pull harder: drift is 80/100 = 0.8, so 100.8. Weights, not values, set the pull.

Dividing the deviation total by a single weight (80/70) instead of total weight (80/100). Drift spreads across every unit: 80/100 = 0.8. Dividing by 70 answers a different question.

Forgetting to add the baseline back and answering 0.8. 0.8 is the drift, not the mean. Mean = baseline + drift = 100.8. The direct total 10080/100 = 100.8 catches the dropped baseline.

Another example: 40 at 80 plus 60 at unknown value averages 92

Total needed: 100 x 92 = 9200
Known part: 40 x 80 = 3200
Unknown: (9200 - 3200)/60 = 6000/60 = 100

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

Gerard W. Kelly, Short-Cut Math (Dover 1984), Chapter VI
Chapter VI on fractions, mixed numbers, and percentages (contents page seen). Lesson examples are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.