Question

In a series of 2n observations, half of them equal a and remaining half -a.If the standard deviation of the observations is 2,then |a| equals ?

Answer 1

`absa=2`

Explanation:

Recall the definition of standard deviation:

`sigma = sqrt((sum(x_i-mu)^2)/N)`

Where `mu` is the mean, `x_i` are the observations, `sum` is the sum, and `N` is the number of observations.

Let's first find the mean. By definition:

`mu=(sumx_i)/N`

Recall that `sumx_i` represents the sum of all the observations. Since we have `n` observations that are `a` and `n` observations that are `-a`, we write:

`mu=(overbrace(a+a+...+a)^(n" times")+overbrace(-a -a-...-a)^(n " times"))/(2n)`

Which is equivalent to:

`mu=(na+n(-a))/(2n)`

And, simplifying:

`mu=0`

This makes our calculation for the standard deviation much simpler as well:

`sigma=sqrt((sum(x_i-0)^2)/N)=sqrt((sumx_i^2)/(2n))`

Recall that `sumx_i^2` means to take the sum of the square of every observation we have. This translates into:

`sigma=sqrt((overbrace(a^2+a^2+...+a^2)^(n" times")+overbrace((-a)^2+(-a)^2+...+(-a)^2)^(n" times"))/(2n))`

Which we can rewrite with more mathematical precision as:

`sigma=sqrt((n(a^2)+n(a^2))/(2n))`

Then, we simplify:

`sigma = sqrt((2na^2)/(2n))=sqrt(a^2)=absa`

We are told that `sigma=2`, so

`absa=2`