Average After Adding or Removing a Number: Formulas and Examples

To find the average after adding a number, add it to the old total and increase the count by one. To remove a number, subtract it from the old total and decrease the count by one. If n numbers have average A, adding x gives (nA + x)/(n + 1). Removing x gives (nA − x)/(n − 1), provided n is greater than 1.

The key is to recover the total first: total = average × number of observations. An average is not a total, so adding a new score directly to the old average will usually give the wrong answer.

Which average formula should you use?

Change	New total	New count	New average
Add one value x	nA + x	n + 1	(nA + x)/(n + 1)
Remove one value x	nA − x	n − 1	(nA − x)/(n − 1)
Replace x with y	nA − x + y	n	(nA − x + y)/n

Here A is the old arithmetic mean, n is the old count, and all observations must use the same units. Removing the only observation leaves an empty set, whose arithmetic mean is undefined.

Example 1: Find the average after adding a score

Five quiz scores have an average of 18 marks. A sixth score of 24 marks is added. Find the new average.

1. Old total = 5 × 18 = 90 marks.
2. New total = 90 + 24 = 114 marks.
3. New count = 6.
4. New average = 114 ÷ 6 = 19 marks.

Check: 24 is above the old average of 18, so the average should rise. The answer 19 is between 18 and 24, as expected. The increase is only 1 mark because the extra 6 marks are spread across all six scores.

Example 2: Find the average after removing a value

Eight observations have an average of 15. One observation, 29, is removed. What is the average of the remaining seven?

1. Old total = 8 × 15 = 120.
2. Remaining total = 120 − 29 = 91.
3. Remaining count = 7.
4. New average = 91 ÷ 7 = 13.

Removing a value above the mean lowers the mean. Removing a value below it raises the mean. Removing a value equal to the mean leaves it unchanged, as long as observations remain.

Example 3: Find the number that was added

The average of six numbers is 12. After one number is added, the average becomes 14. Find that number.

Old total = 6 × 12 = 72. New total = 7 × 14 = 98. The added number is therefore 98 − 72 = 26.

General rule: added number = (n + 1) × new average − n × old average. The increase in average, 2, is not the added number.

Example 4: Find the number that was removed

Ten numbers have an average of 20. After one is removed, the average becomes 18.

Old total = 10 × 20 = 200. Remaining total = 9 × 18 = 162. Removed number = 200 − 162 = 38. Check: (200 − 38)/9 = 18.

A quick way to check your result

For one addition, new average = A + (x − A)/(n + 1). For one removal, new average = A + (A − x)/(n − 1). These are the same formulas rearranged; the total-and-count method is often easier to remember.

Yaad rakho: pehle total nikalo, phir total aur count dono badlo. In English: find the total first, then update both the total and the count.

Practice questions with answers

1. Four numbers average 10. Add 20. Answer: (40 + 20)/5 = 12.
2. Six numbers average 9. Remove 4. Answer: (54 − 4)/5 = 10.
3. Nine numbers average 16. Adding one number makes the average 17. Answer: added number = 170 − 144 = 26.
4. Seven numbers average 11. Removing one makes the average 10. Answer: removed number = 77 − 60 = 17.

Common questions

Does replacing a number change the count?

No. One value goes out and another comes in, so the count stays n. Use the corrected mean formula for replacement or recording-error questions.

What if several numbers are added?

Add their total and increase n by their count. If a second group has k observations with mean B, the new mean is (nA + kB)/(n + k). See the combined mean worked examples.

Can these formulas be used for a median?

No. These formulas concern the arithmetic mean. A median depends on the ordered middle position, so you need information about the individual values.

Still stuck on the count or total? Try writing those two quantities separately before solving. You can also ask a Maths question on DoubtPay AI; check the current account and credit requirements before requesting an answer.