For 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. Find the correct mean and standard deviation.
Given ∑xi18=7(∵ mean =7 and n=18)
⇒∑xi=18×7=126
Since an observation 12 was miscopied as 21, therefore,
correct ∑xi=126−21+12=117
Hence true mean =correct ∑xi18=11718=6.5
Also variance is given to be 42=16,
therefore, ∑x2i18−(mean)2=16
⇒∑x2i18=42+(mean)2=16+72
⇒∑x2i=18(16+49)=1170
But, in the summation on R.H.S., one observation (=12) was miscopied as 21, therefore,
correct ∑x2i=1170−212+122=1170−441+144=873.
Hence true variance =correct∑x2i18−(true mean)2
=87318−(6.5)2=48.5−42.25=6.25
∴ True S.D=√true variance=√6.25=2.5
∴ Correct mean =6.5 and
correct standard Deviation =2.5