17th sum...

17.  From a frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4 respectively. But on comparison with the original data, it was found that a figure 12 was miscopied as 21 in calculations. Calculate the correct mean and standard deviation. 

Dear Student,

Mean x¯=xinincorrect mean=77=xi18xi=18×7=126But 12 was read as 21, hence there was additional 21-12=9 addedso the currect sum is 126-9=117New Mean=11718=6.5Now Standard deviation =4variance=42=16Also variance σ2=1n∑x2-x¯216=118∑x2-7216×18=∑x2-49×18∑x2=16×18+49×18∑x2=65×18=1170Bow we need to subtract the additional digit addedHence new ∑x2=1170-212+122=1170-441+144=873Corrected variance σ2=1n∑corrected x2-corrected x¯2=118×873-6.52=48.5-42.25=6.25Standard deviation =σ2=6.252.5Hence correct answer are 6.5 and 2.5

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