Measures of Dispersion

Date: 6/30/96 at 1:24:37
From: Anonymous
Subject: Measures of Dispersion

Good day!

  I'm from Manila, Philippines. My question is:
  
What are the different characteristics of these different measures of 
dispersion:

    a.) Range
    b.) Mean Absolute Deviation
    c.) Standard Deviation
    d.) Variance
    e.) Quantiles (percentile, decile, quartiles)
    f.) CV

I`m not quite sure about that last one's real description...
I've gone through different books but still I can't find any answers.  
     
Thanks..:)

Date: 7/1/96 at 8:55:45
From: Doctor Anthony
Subject: Re: Measures of Dispersion

Let us go through the list you have given.

(a) Range - Very simple.  This is simply the difference between the 
largest and smallest members of the population.  So if age is what you 
are looking at, and the oldest is 90, the youngest 35, then the range 
is 90 - 35 = 55

(b) Mean deviation.  First calculate the mean (= total of all the 
measurements divided by number of measurements)  If the population is 
made up of x1, x2, x3, ....xn, then mean = (x1+x2+x3+...+xn)/n
If mean = m then to get mean deviation we calculate the numerical 
value i.e. ignore negative signs of |x1-m|, |x2-m|, |x3-m|, ... |xn-m|  
now add all the numbers together (they are all positive) and divide 
by n.

(c) Standard deviation - This is the square root of the variance, so I 
will describe that in section (d), and then you get the s.d. by taking 
the square root of the variance.

(d) Variance.  To avoid the problem of negative numbers we encountered 
with mean deviation, we could square the deviations and then average 
those. This is the variance.  So variance = 
{(x1-m)^2 + (x2-m)^2 + (xn-m)^2}/n.  As mentioned earlier, the 
standard deviation is then found by taking the square root of the 
variance.

(e) Quantiles.  You need to plot a cumulative frequency diagram to 
make use of these.  The cumulative frequency shows the total number of 
the population less than any given value of x.  If you plot x 
horizontally and cumulative frequency on the vertical axis, then to 
find the median you go half way up the vertical axis, across to the 
curve and down to the x axis to read off the median value of x.  This 
is the value of x which divides the population exactly in half. i.e. 
half the population have values below x and half have values above x.  
The interquartile range is found by going 1/4 and then 3/4 of the way 
up the vertical axis, across to the curve and down to the x axis. 1/4 
of the population have values of x below the lower quartile and 1/4 of 
the population have values of x above the upper quartile.  This means 
that the middle half of the population have values within the 
interquartile range.  A decile is found by going 10% up the vertical 
axis, and corresponds to 10% of the population.  A percentile 
represents 1% of the population, so the 30th percentile is the value 
of x below which 30% of the population lies.

(f) I think you mean covariance by the letters CV.  This applies to 
situations where you have two variables x and y which affect each 
other. (If x and y are independent the covariance is 0).  When finding 
the variance of (x+y) we have:

   var(x+y) = var(x) + var(y) + 2*cov(xy)

Cov(xy) is calculated from {x1y1 + x2y2+ ..+ xnyn}/n - mean(x)*mean(y)

As mentioned above, cov(xy) = 0 if x and y are independent.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: Thu, 23 Oct 1997 16:49:16 -0400 (EDT)
From: Amal Jasentuliyana
Subject: Clarification of a Dr. Math answer

Hi. I think this is a wonderful site.

I noticed a point of possible confusion in one of the archived 
Dr. Math answers. The question asks about a number of different 
measures of the spread of a distribution, and the last one is "CV". 
In the answer, CV was assumed to be covariance. I _think_ that 
coefficient of variation might have been what was intended in the 
question (since everything else was univariate), and that this may 
also be a more common usage of the abbreviation. 

Coeff. of variation is defined as (sd * 100)/mean (see Sokal 
and Rohlf [1981] _BIOMETRY_, 2nd ed., ch. 4.10). 

Keep up the excellent work, - Amal.

    

For more on the meanings of "quartile" and mathematicians' 
disagreements about them, see

  Defining Quartiles
  http://mathforum.org/library/drmath/view/60969.html

- Doctor Melissa, The Math Forum
  http://mathforum.org/dr.math/