Standard Deviation

Date: 07/05/97 at 05:49:21
From: Jeremy Bowell
Subject: Averages involving standard deviation

How do I solve this problem?

The average height of year 10 students is given as 175 cm and the 
standard deviation is 12cm. Find the percentage of students whose
height is:

a) Greater than 175 cm
b) Between 163 and 187 cm
c) Greater than 187 cm
d) Between 151 and 163 cm

What is standard deviation?

Date: 07/05/97 at 07:58:37
From: Doctor Anthony
Subject: Re: Averages involving standard deviation

Standard deviation is a measure of the spread of a distribution about 
the mean.  It is the square root of the VARIANCE, where variance is 
calculated from:
               
     VAR(x) = E(x^2) - Mean^2

E(x^2) is the 'expected value' of x^2. If this term is unfamiliar to 
you, consult a standard textbook or write back.

(a) Because the normal distribution is symmetric about the mean, the 
percentage of students with height greater than the mean, 175 cm, is 
50 percent.  

Using the normal tables, let m = mean and s = standard deviation. Z 
values are those on the horizontal axis giving the number of standard 
deviations from the mean, with z = 0 at the mean. Areas are the areas 
under the normal probability curve between two z values. These areas 
are found using normal tables and entering the appropriate z values.

(b) You calculate the z values using:

      x - m      163 - 175
  z = ------  =  ---------   = -1
       s            12

The area between the mean and -1 s.d. =  .3413

                                187 - 175
Also for the upper limit   z = -----------  = +1
                                    12

Again, the area between the mean and 1 s.d. = .3413

Total probability is then 2 x .3413 = 0.6826, which is 68.26 percent.


(c) Greater than 187 gives area 0.5 - .3413 = .1587, so the number 
greater than 187 is 15.87 percent. 

(d)
           151 - 175
      z =  ---------   = -2     area between mean and -2 is 0.47725
              12 

            163 - 175 
      z =  ----------  = -1     area between mean and -1 is 0.3413 
              12 

The area between -2 and -1 is then .47725 - .3413 =  0.13595

So 13.6 percent of students have heights between 151 and 163 cms.  
                
-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/