Although the 5 number summary is very useful
for describing a data set, it is not the most
widely used. The most common measures are
the mean for the center and the standard
deviation to measure spread. Standard deviation
measures how far the observations are from their
mean.
*
* The variance of a set of observations is the
average of the squares of the deviations of
each observation from the mean.
* The standard deviation (s) is simply the square
root of the variance. Your TI-83 calls this Sx.
*
*
2
2
2
( x1 - x ) + ( x2 - x ) + ...+ ( xn - x )
Variance = s =
n-1
2
1
2
( xn - x )
s =
n-1
2
* Seven men took part in a study of metabolic
rates. Here are the calories burned in 24 hours
by the men:
1792
1666
1362
1460
1867
1439
* Find s.
* s=189.24
*
1614
* The sum of the deviations from the mean will
always be 0. This is why we square and square
root when finding s.
*
* Now the question comes up as to why we divide by n-1
instead of n. Because the sum of the deviations is always
0, the last deviation can be found once we know the first
n-1 deviations. Since only n-1 of the squared deviations
can vary freely, we average by dividing the total by n-1.
The number n-1 is called the degrees of freedom of the
variance or standard deviation.
*
1. s measures spread about the mean - so, use s only when
using xbar.
2. s=0 if there is no spread (ie, all observations are the
same). Else, s>0. A big s value implies the data are spread
out.
3. s is nonresistant.
*
* The five number summary is usually better
than the mean and standard deviation for
describing a skewed distribution.
* Use x-bar and s for reasonably symmetric
distributions.
* Remember, always graph the data!
*