Measures of DispersionDate: 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/ |
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