Show that the two formula for the standard deviation of ungrouped data

are equivalent, where .

The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases :

(i) If wrong item is omitted

(ii) if it is replaced by 12.

The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on, it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations were omitted.

Find the standard deviation for the following distribution :

Table below shows the frequency f with which ‘x’ alpha particles were radiated from a diskette

Calculate the mean and variance.