Q:

Can you determine if variances are equal when quartiles are similar?

I am comparing 2 different data sets.  I have broken down each set into quartiles.  I am looking at the percentage of increase between the quartiles (example there was a 125 percent increase between the 2nd and 3rd quartile for one data set and a 123 percent increase between the 2nd and 3rd increase of the other data set) can I say that the variability is the same for both groups if the percentage of increase between all the quartiles are about the same between the two datasets?

A:

Based on nothing beyond this information, you cannot say that the variances are the same. It is possible that the variance is the same, though.  
The fact that the the 3rd quartile is a (e.g.) $125\%$ increase over the 2nd quartile is a multiplicative change.  The arithmetic increase will depend on the raw value of the 2nd quartile. Consider that one group has a median (i.e., 2nd quartile) of $100$, then the 3rd quartile will be $225$ (i.e., $100 \times 2.25 = 225$), but if the median of another group is $1000$, then the 3rd quartile will be $2250$.  Obviously, these won't have the same variance.  So the first additional piece of information that you would need to know is if the 2nd quartiles are approximately equal.    
The next thing you need to know is whether the two distributions have the same shape and form.  It is possible to construct two different types of distributions, say a Poisson distribution and a uniform distribution, that have equal 2nd and 3rd quartile values, but that would not necessarily have the same variance.  For example, a Poisson distribution with $\lambda = 30$, will have a 2nd quartile $\approx 30$ and 3rd quartile $\approx 34$, as will a uniform distribution on the interval $(22,\ 38)$, but the variance of the Poisson will be $30$, whereas the variance of the uniform will be $21.3$.  
In addition, the above discussion is only guaranteed to hold for populations.  In any two given samples, it is possible for the distributions to be of the same kind, with (roughly) equal medians and 2nd to 3rd quartile multiplicative changes, but still not have the similar variances.  Consider samples taken from two normal distributions with the same means and variances.  It is possible to draw sets of values such that all three quartiles are the same, but that one has an extreme value many standard deviations away from the median.  This value would be an 'outlier' in the sense that it would lie outside of the range of the bulk of the values, but not in the sense of having actually come from a different distribution.  Of course, it is incredibly unlikely for such a thing to occur, but as Charlie Chaplin once said, "strange events permit themselves the luxury of occurring".  In this situation, the sample variances would not be similar, even though the population variances were identical (by stipulation).  
Lastly, I should note that saying two groups have the same variance could only possibly be meaningful if they are measures of the same thing in the same units.  For example, it doesn't mean anything if the variance of the weights of some people (in some units) happens to be the same number as the variance of the heights of another group of people (in some units).  Nor would it be meaningful to say that the variance of the weights of some wheels, measured in grams, is the same number as the weights of some other wheels, measured in kilograms.  

