Corrected Mean Formula: Wrong Observation Solved Examples

Corrected mean = incorrect mean + (correct value − wrong value) ÷ number of observations. Use this formula when a wrong observation is replaced and the number of observations stays the same. Equivalently, corrected total = old total − wrong value + correct value; divide that total by the unchanged count.

Why the shortcut works

The old mean represents a total divided by n observations. If one entry was typed incorrectly, only that entry changes. Removing the wrong value and adding the correct value updates the total exactly.

Worked example

The mean of 8 scores was recorded as 18. One score written as 12 should have been 21. Old total = 8×18 = 144. Corrected total = 144 − 12 + 21 = 153. Corrected mean = 153/8 = 19.125.

Two corrections

If 15 was entered instead of 5 and 9 instead of 19 for 10 observations with old mean 20, corrected total = 200 − 15 − 9 + 5 + 19 = 200. The corrected mean remains 20.

Check the direction

If the correct value is larger than the wrong value, the mean must increase. If your calculation moves in the opposite direction, revisit the subtraction and addition signs.

Practice

The mean of 6 numbers is 14. A value 8 should be 20. Answer: corrected mean = (84 − 8 + 20)/6 = 16.

Need a quick statistics check? Use DoubtPay AI, include n and both values, and ask for the total before the mean.

Corrected mean formula and when to use it

Corrected mean = old mean + (correct value − wrong value) / n. Here n is the number of observations, and n must stay unchanged. This shortcut assumes the old mean is exact; if it was rounded, use the original total when available.

For several corrected entries, add all the changes before dividing by n: corrected mean = old mean + Σ(correct − wrong)/n. Each observation has equal weight in this arithmetic-mean formula.

Why the formula works

Let the old mean be m. The old total is nm. Replacing one wrong observation w with the correct observation c gives a new total nm − w + c. Dividing by the same count n gives m + (c − w)/n.

A complete check using the actual observations

The recorded values are 10, 12, 14 and 20. Their total is 56 and their mean is 14. Suppose 20 should be 16. The shortcut gives 14 + (16 − 20)/4 = 13. Checking the corrected list independently: (10 + 12 + 14 + 16)/4 = 52/4 = 13.

What if an observation was omitted or counted twice?

Then the count changes, so the replacement shortcut alone is not enough.

• One omitted value c: new mean = (nm + c)/(n + 1).
• One extra entry w removed: new mean = (nm − w)/(n − 1), provided n is greater than 1.

Example: four recorded values have mean 14, and an omitted fifth value is 24. The corrected mean is (4 × 14 + 24)/5 = 16. Dividing by 4 would be wrong because the corrected list has five observations.

Three practice questions with worked answers

1. Mean decreases: Ten values have mean 25. An entry of 42 should be 12. Corrected mean = 25 + (12 − 42)/10 = 22.
2. Two errors: Five values have mean 18. Entries 7 and 23 should be 17 and 18. Net correction = 10 − 5 = 5. Corrected mean = 18 + 5/5 = 19.
3. An extra entry: Six recorded values have mean 15. One entry of 30 was included by mistake. New mean = (6 × 15 − 30)/5 = 12.

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