Average Changes After Addition, Removal or Replacement

Learning outcome

Update averages after addition, removal or replacement, using the correct new total and a positive remaining count.

Meaning and method

Start with total = count × mean. For original count n > 0 and mean a, write T = n × a.

Addition: Adding k values with mean b gives new mean = (T + k × b) ÷ (n + k).

Removal: If k existing values have mean b, subtract their total. New mean = (T - k × b) ÷ (n - k). Require 0 < k < n; removing every value leaves an undefined mean.

For single-value addition or removal, k = 1 and b is that value. All stated counts are positive integers; removal must leave a positive count.

Individual replacement: Replacing x with y keeps count n and changes total to T - x + y. New mean = a + (y - x) ÷ n. The total change is n times the mean change.

These formulas work because each observation contributes its value to the total and one place to the count. Adding an above-average group raises the mean; removing an above-average group lowers it. Below-average groups reverse these directions. A group matching the old mean leaves it unchanged.

Worked examples

Example 1 — Add a group. Nine values have mean 26. Add three values with mean 34. Old total = 9 × 26 = 234; added total = 3 × 34 = 102. New mean = (234 + 102) ÷ (9 + 3) = 336 ÷ 12 = 28, higher as expected.

Example 2 — Remove a group. Fifteen values average 32. Remove five values averaging 26. Old total = 15 × 32 = 480; removed total = 5 × 26 = 130. Remaining mean = (480 - 130) ÷ (15 - 5) = 350 ÷ 10 = 35. Removing the lower-average group raises the mean.

Example 3 — Find a replacement. Twelve readings average 27. Replacing the reading 19 raises the mean to 30. Totals change from 12 × 27 = 324 to 12 × 30 = 360. The replacement must exceed 19 by 360 - 324 = 36, so its value is 19 + 36 = 55.

Common mistakes

Do not subtract a removed group's mean instead of its total. Replacement changes no count. Its direction depends on new value minus old value, not on comparing the new value with the old mean.

Practice questions

1. Seven values average 18. Adding one value makes the mean 20. Find that value.
2. Ten values average 24. Add five values averaging 36. Find the new mean.
3. Sixteen values average 29. Remove six values averaging 24. Find the remaining mean.
4. Nine values average 41. Replace 53 with 26. Find the new mean.

Worked answers

1. Old total = 7 × 18 = 126. New count = 8 and new total = 8 × 20 = 160. Added value = 160 - 126 = 34.
2. Totals are 10 × 24 = 240 and 5 × 36 = 180. New mean = (240 + 180) ÷ 15 = 420 ÷ 15 = 28.
3. Old total = 16 × 29 = 464. Removed total = 6 × 24 = 144. Remaining mean = (464 - 144) ÷ 10 = 320 ÷ 10 = 32.
4. Old total = 9 × 41 = 369. New total = 369 - 53 + 26 = 342. Count remains 9, so new mean = 342 ÷ 9 = 38.