Expert: Ahmed Salami - 7/30/2004

Question
Hi Dr. Salami,

My question for you is about standard deviation.
This is a question from the GRE(Graduate Record Exam) Quantitative section:

1. Which is greater?:
(A) The standard deviation of the data 5, 5, 8, 14 and 18.

(B) The standard deviation of the data 6, 6, 8, 14 and 16.

The definition of Standard deviation is "the average distance from the arithmetic mean for the N measurments". So my study guide says. So if I apply this concept to the question above, first I must find the arithmetic mean of both data which is 10 for both(the sum of 5,5,8,14,18 devided by 6 is 10 for A; the same for B..). After this, what do I need to find out the standard deviation of (A) and (B)?

Thanks for your help!

Aes  

Answer
Hi Aes,
Sorry about the delay.
I'll take you through the steps of finding the standard deviation of a set of measurements.
After finding the mean, you find individual deviations(distances from the mean)and square each of them. You add all squared numbers and divide by N. The square root of this is the standard deviation.
For the first data, the deviations are -5,-5,-2,4,8. The squares of these are 25,25,4,16,64. The sum is 25+25+4+16+64 = 134
Dividing by 5 gives 26.8 and hence, the standard deviation which is the square root is about 5.18
For the second data, the deviations are -4,-4,-2,4,6. The squares of these are 16,16,4,16,36. The sum is 16+16+4+16+36 = 88
Dividing by 5 gives 17.6 and hence, the standard deviation which is the square root is about 4.2
So you can now see that it is greater for set A.
But mentally, you can determine which one is greater by considering the different deviations from the mean.The 5's and 18 fron set A are more deviated from the mean(10)than the 6's and 16 from set B.
I hope all these helps.
Regards.