# understanding variation of data and visualization

1) What is the threshold or measure to say that the variation of data is high? Typically we say high or low variation by comparing with another data set. But there is only one data set.

2) We can use boxplot to visualize standard deviation, mean, etc., of the data. However, I am curious to know whether there is any visualization method or distribution method in R to stress that the variation is high in a graph.

For example, I have a data set of 64 points and it is one dimensional. Please see data below. Assume the values are always positive. Its mean is 32.8, standard deviation(sd) is 37.7, and coefficient of variation(cv) is 1.2.

May be these are silly questions. But I won't get peaceful mind if I don't get a clarification. Please help me. Thank you.

0 19  0 83 88 97  0 50  0 82 89  0  0  0  0  0  0 17 33 34  0  0 50 59 84 0 67 83 89  0 17  0 66  1 59 89  0 14 68  0  0  0 95  0  0 89  4 85 89 25 50 83 34  0  0  0  0  0  0  0 96  0 89 19


The reason why you normally only see comparisons is because there's no absolute standard. What's 'big' in one application might be 'moderate' in a second and 'small' in another.

The fact that the coefficient of variation exceeds 1 might perhaps be of some relevance in some applications; but the large number of zeros might be more interesting than the standard deviation (the data conditional on being positive has nearly as large a sd, but almost twice the mean).

A histogram with narrow binwidth (I suggest 1001 bins or binwidth 0.1, ymmv) perhaps with the mean (or perhaps mean +/- 1 sd) marked on may show almost everything there is to be said.

You're right to think there's no absolute standard for 'high'. What you can compare variability to depends on the type of measurement scale.

The standard deviation doesn't tell you much in absolute terms. It has the units of whatever you're measuring, so it makes sense to compare it only to other measurements in the same units.

The coefficient of variation has no units, but is only meaningful on a ratio scale, not on an interval scale. So it would make sense to say that electrical conductivity varies more than thermal conductivity among a sample of different metals.

For count data the Poisson distribution is a special case (counts from a continuous memoryless process) where the variance is proportional to the mean. Under- or over-dispersion occur when the variance differs significantly from the mean.