Determine mean and standard deviation of first n terms of an A.P. whose first term is a and common difference is d.

Given: first n terms of an A.P. whose first term is a and common difference is d

To find: mean and standard deviation

The given AP in tabular form is as shown below,

Here we have assumed a as mean.

Given the AP have n terms. And we know the sum of all the terms of AP can be written as,

Now we will calculate the actual mean,

Substituting the corresponding values, we get

Now we other two columns sums, i.e.,

Now we know standard deviation is given by,

Substituting the corresponding values, we get

Cancelling the like terms, we get

Taking out common terms we get

By taking the LCM, we get

Hence the mean and standard deviation of the given AP is and respectively.