# 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 ?

##### 1 Answer

#### Explanation:

Recall the definition of standard deviation:

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

Where

Let's first find the mean. By definition:

#mu=(sumx_i)/N#

Recall that

#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 *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

#absa=2#