Question
The mean of 10 observations is 25. After computing, it was discovered that one observation was misread as 35 instead of the correct value 53. Find the corrected mean. Also, if the original variance was 16, find the corrected variance.
Solution — Step by Step
Original sum . Wrong observation contributed instead of — a deficit of .
Corrected sum .
Original , so .
So (with the wrong value).
Replace with .
Corrected mean = 26.8, corrected variance = 81.16.
Why This Works
Mean and variance are summary statistics — they only depend on and . So when one observation changes, we don’t need to recompute from scratch. Just adjust the two sums and recompute.
The variance jumps from 16 to 81 because the corrected observation 53 is much further from the new mean than the original 35 was from the original mean — pulling out the spread.
Alternative Method
Recompute from scratch by listing all 10 observations and using . Possible if the original data is given, but slow for an exam — the sum-update approach is much faster.
Common Mistake
Students update the mean correctly but forget to update in the variance formula — they compute , which is wrong. Always use the new mean (26.8) when recomputing variance after a correction.