Math160.Section3.2.doc - Math A160 - Cooley Introduction to...

Math A160 - Cooley Introduction to Statistics OCC
SECTION 3.2 – Measures of Dispersion
Dispersion – The degree to which the data are spread out.
Range – The range , R , of a variable is the difference between the largest and the smallest data value. That is,
Range = R = largest data value – smallest data value.
Standard Deviation
The standard deviation measures variation by indicating how far, on average, the observations are from the
mean. For a data set with a large amount of variation, the observations will, on average, be far from the mean; so
the standard deviation will be large. For a data set with a small amount of variation, the observations will, on
average, be close to the mean; so the standard deviation will be small.
Population Standard Deviation,

The population standard deviation ,

(lowercase Greek sigma) of a variable is the square root of the sum of
squared deviations about the population mean divided by the number of observations in the population, N . That
is, it is the square root of the mean of the squared deviations about the population mean.
       
2 2 2 2
1 2 N i
x x x x
N N
   

      
 
where
1 2
, ,...,
N
x x x
are the N observations in the population and  is the population mean.
Conceptual Formula & Computational Formula
There are two ways to calculate the population standard deviation,

.
Conceptual Formula Computational Formula
 
2
i
x
N


 

2
2 i
i
x
x
N
N

  
 
 
 

Sample Standard Deviation, s
The sample standard deviation , s , of a variable is the square root of the sum of squared deviations about the
population mean divided by n – 1, where n is the sample size.
       
2 2 2 2
1 2 N i
x x x x
N N
   

      
 
where
1 2
, ,...,
N
x x x
are the n observations in the sample and x is the sample mean.