The mean and standard deviation of 20 observations are found to be 10 and 2 , respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is
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Official solution
To find the correct standard deviation, we first need to adjust the mean and sum of squares of the observations based on the correction of the erroneous entry. The original erroneous observation was 8, and the correct observation is 12. The mean and standard deviation of the 20 observations before correction were 10 and 2, respectively.
The sum of all 20 original observations (Soriginal) can be calculated from the mean formula:
Mean=Number of observationsSum of all observations10=20SoriginalSoriginal=10×20=200
The corrected sum of observations (Scorrect) will replace the incorrect observation (8) with the correct one (12):
Scorrect=Soriginal−8+12=200−8+12=204
The corrected mean (μcorrect) is:
μcorrect=20Scorrect=20204=10.2
To find the corrected standard deviation, we need the sum of squares of the deviations from the mean for both the original and corrected data. The original sum of squares (SSoriginal) is calculated from the original standard deviation formula, where σ=2:
σ2=nSS4=20SSoriginalSSoriginal=4×20=80
To calculate the corrected sum of squares (SScorrect), we need to adjust SSoriginal by removing the square of the deviation of the incorrect observation and adding the square of the deviation of the correct observation:
SScorrect=SSoriginal−(8−10)2+(12−10.2)2SScorrect=80−(−2)2+(1.8)2SScorrect=80−4+3.24=79.24
Finally, the corrected standard deviation (σcorrect) is:
σcorrect=20SScorrectσcorrect=2079.24σcorrect=3.962
The corrected standard deviation is closest to the value given in Option B, which is
3.96