2.11 The Zero-Sum Property of Deviations from the Actual Mean

A property that frequently resolves “find the missing value” questions in seconds: the sum of deviations of all observations from their own (true) average is always exactly zero. This follows directly from the definition — if Average = Sum/n, then Sum − n×Average = 0, which is exactly the sum of (each value − average) added up.

(x_i - x) = 0

This property is different from the deviation method in Section 2.10, where you choose an assumed mean (which need not equal the true average, so the deviations need not sum to zero). Here, the deviations are taken from the actual, correct average, and they are guaranteed to sum to zero. This distinction is important: it lets you verify whether a value you assumed to be the average is in fact correct — if the deviations you compute do NOT sum to zero, your assumed value is not the true average, and the required correction is exactly (sum of deviations)/n, which is the deviation method formula itself.

Solved Example 29: Five numbers have deviations of −8, −3, +2, +5, and x from their average. Find the value of x.

Since deviations from the true average must sum to zero: −8 −3 +2 +5 + x = 0 −4 + x = 0 x = 4.

Solved Example 30: The average marks of 6 students is 72. Five of the students scored 65, 70, 75, 80, and 68. Find the marks of the sixth student, and verify using the zero-sum deviation property.

Direct method: Total = 72×6 = 432. Sum of known 5 = 65+70+75+80+68 = 358. Sixth student’s marks = 432−358 = 74. Verification via deviations from average 72: (65−72)+(70−72)+(75−72)+(80−72)+(68−72)+(74−72) = (−7)+(−2)+(3)+(8)+(−4)+(2) = 0 ✓. This confirms 74 is correct.

This zero-sum check is an excellent self-verification tool for exam use: after solving any average-based “find the missing value” question, quickly sum the deviations from your computed average — if it isn’t zero, you have made an arithmetic error somewhere and should recheck before marking your answer.

Solved Example 30(a): Six numbers have deviations of −12, −5, +3, +8, +10, and x from their average. Find the value of x.

Sum of the known deviations = −12−5+3+8+10 = 4. Since all deviations from the true average must sum to zero: 4 + x = 0 → x = −4. (A negative deviation value is perfectly valid — it simply means this particular number lies below the average.)

Solved Example 30(b): The average of 7 numbers is 50. Six of the numbers are 42, 48, 55, 60, 38, and 65. Find the seventh number, and verify your answer using the zero-sum deviation property.

Direct method: Total = 50×7 = 350. Sum of the known 6 numbers = 42+48+55+60+38+65 = 308. Seventh number = 350 − 308 = 42. Verification via deviations from average 50: (42−50)+(48−50)+(55−50)+(60−50)+(38−50)+(65−50)+(42−50) = (−8)+(−2)+(5)+(10)+(−12)+(15)+(−8) = 0 ✓. This confirms 42 is correct.