Suppose, we have a number of sets of data and we want to compare the variations of the sets. We know that coefficient of variation (CV) is the measure which is used and/or can be used in this case. The formula for computing CV is given by
CV = (SD/AM) x 1 00
where SD and AM are standard deviation and arithmetic mean respectively.
Now if the AM of any of the sets of data is 0, the corresponding CV does not exist.
In fact I have found several sets of data such that the AM of each of the data sets is 0.
In such situation, is there any technique/method of comparing variations of the sets of data ?
Thus the question here is
"Is there any technique/method of comparing of variations of several sets of data if AMs are 0 for the sets ?"
You could use standard deviation or variance. Or RMSE, using the mean, even if it's zero, as the predicted value.
What's nice about the coefficient of variation is that it makes the standard deviation relative to the mean, so that it's somewhat comparable across data sets that have different scales. (Like, if you measured something in kilometers or in meters, the CV would come out the same.)
However, if you have both positive and negative numbers in the data, statistics that normalize for the mean (CV, NRME) may not make much sense.
As a followup to my answer, here's an example.
The following is R code that can be run at: https://rdrr.io/snippets/ .
Note that A, B, and C all have the same standard deviation. They are essentially the same data, except shifted to the negative.
Standard deviation is the same for all three.
CV(%) varies, and will approach infinity as the mean goes to zero.
RMSE stays the same for all three.
But normalized RMSE (NRMSE) would have the same problem as CV.
I didn't mention median absolute error (MedAE), but that would work as well.
A = c(0,1,2,3,4,5,6,7,8,9)
B = c(-1,0,1,2,3,4,5,6,7,8)
C = c(-2,-1,0,1,2,3,4,5,6,7)
sd(A)
sd(B)
sd(C)
sd(A) / mean(A) * 100
sd(B) / mean(B) * 100
sd(C) / mean(C) * 100
sqrt(mean((A - mean(A))^2))
sqrt(mean((B - mean(B))^2))
sqrt(mean((C - mean(C))^2))
@ Salvatore S. Mangiafico,
The situation is
"Units of observed values in different data sets are different and the arithmetic mean of the observed values in the data set is 0 in the case of some or all the data sets."
In this situation, how to compare the variations of the data sets is the vital problem.
In summary, you can use standard deviation, variance, RMSE, median absolute error. SD, RMSE, and MedAE have the same units as the original. If the units different and there isn't a way to make them commensurate, it probably makes no sense to compare their dispersion.
@ Sal Mangiafico ,
The problem is that the units of observations in different sets are not same but different.
Automatically, the values of dispersion of different sets of data will be different.
Thus, we need a measure of dispersion which is free from unit or we need some other suitable technique.
Dhritikesh Chakrabarty , I think...
I may be wrong.
@ Sal Mangiafico ,
I have understood the content in your answer.
Sincere thanks to you for providing fruitful explanation of the problem mentioned in the question.
However, I have not found sufficient logic to accept the following statement as mentioned by you:
" If we have two variables which assume positive & negative values and which aren't on the same scale then it makes no sense to compare the variances of the two variables."
Dhritikesh Chakrabarty ,
- It it may be a failure of imagination on my part, but I can't imagine a case where having variables that can't be made commensurate --- or where coefficient of variation wouldn't make sense --- where there would be any reasonable way to compare their variances. .... That was toad and temperature example in last post. Can you give an example where you think this would make sense ? Or, maybe, let us know what actual variables you are thinking of ? Maybe we're just misunderstanding each other.
- I'll also note, though, that even comparing CV's for different kinds of variables may not be very informative. For example in my toad and temperature example, you could convert temperature from C to K, but the coefficient of variation for temperature would be quite small. (Imagine the temperature varies from say 274 to 278 K, and the average is 275. The CV would be 0.5 % . This is fine. But trying to compare it to some variable like weight, where the CV might be 10 %. Is this informative in any way ?
- Thinking about your original question, I think your best bet is just use a measure of dispersion that is in the same units as the measured variable. Like sd, RMSE, MAE, or MedAE. This eliminates any weird influence of dividing by the mean. And if the units are interpretable by the reader, these measures will be also.
