# 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?

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Only with some additional assumptions... –  Glen_b Feb 1 '14 at 3:42
Gung, to clarify, there is nothing i can say about the PERCENTAGE OF INCREASE between the quartiles being about the same between the 2 data sets. (the percentage of increase is about 125 % in both data sets between the 2nd and 3rd quartile) thanks –  angela grant Feb 3 '14 at 23:41
angela, if you want to be sure @gung will see your comment, you need to comment under his answer rather than your question, or put "@" in front of his name. Either will notify him. This comment should do it. –  Glen_b Feb 3 '14 at 23:55
Without additional information / assumptions, you cannot say the variances are equal given the same multiplicative change from the 2nd to the 3rd quartile, @angelagrant. You can certainly note the fact of the similar multiplicative change, if that is interesting. It may be that you can say something beyond that, but I wouldn't know what it is & it wouldn't be that the variances are equal. –  gung Feb 3 '14 at 23:58

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 $21.3$.