When I was a kid in my last class of high school trying to pass the university exams for the Mathematics department I had a mathematics tutor who managed to set us solid basis for what we had to phase in the University.
One thing that we learned at that time, which I would like to share with you is the understanding of what a metric is.
“In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When atopological space has a topology that can be described by a metric, we say that the topological space is metrizable.” Wikipedia (Metric-Mathematics)
The above definition is a beautiful topological definition that most people will not understand. But lets take my tutor’s approach to this definition:
Imaging yourself that you are in a classroom and you want to measure how big the blackboard is, but the only thing you have near by you is a sponge what would you do?
You would take the sponge and start counting how many times it fits inside the black board. The result would be that the blackboard is x times the sponge. So what you did is that you invented a metric called sponge and you applied to measure one space.
From this point you can anticipate that a metric can be anything we want*. In this example our metric unit was a sponge.Of course it could also be your steps, the size of your palms etc’
So a metric is nothing else than a convention on an entity/object that we have created/invented and defined as a unit that we are happy to measure stuff with. From this point onwards you should start feeling creative and free to make your own metrics. Especially that is one of the things I love in web analytics is that we have the opportunity to get creative. However with freedom comes responsibility like anywhere, so after you make up your new metrics you must see if they can be justified and if they are senssible. Even if I plan a seperate post on this the rest of the post will give enough materiall to think about. At the end of the day is not that complicated.
Next we can argue that measuring can be considered as the act of applying a metric. The logical questions that you could have are:
- What do we measure?
- Were do we apply our metric?
The answers are:
- On a topology/space/object
- On the dimensions that this topology/space/object conceals
If we translate this to our blackboard example then we could say:
- On the Blackboard
- The two dimensions of the blackboard, width and height.
We have to highlight on this point that any topology/space/object conceals its own dimensions. This dimension can be considered as properties of the object and can be from 0 to n depending on the nature of the object, for example the blackboard has 2 and a cube has 3. When we say that we can measure a space/object we mean that we can apply a metric on its dimensions.
Again the way we define the dimension is an abstraction that we define, from what is more sensible to us from our senses etc’
At the moment we have to understand that we can create metrics and apply them on the dimensions of an object. The question that we should ask ourselves next is can we apply any metric to any topology/space/object?
Lets stick to our sponge metric, with the sponge we can measure the dimensions of our blackboard even the dimensions of a highway, but we can not be accurate in measuring the surface of a lake and is not possible to measure the sky. We see that the metric is bounded to its context, actually a metric can only leave in a well defined context.
Actually the way we make metrics is not an abstraction that we come up with, rather they come as an answer to a topology/space/object that we want measure and study. So firstly we must have the topology then we define its dimensions and on the last step we define a proper metric that we want to measure with and is justifyible within this context.
However we have to keep in mind that not all topologies can be measured as so there are cases in which we will not be able to generate metrics for some topologies. The last note is more of an issue for the mathematicians rather than the web analysts.
Hopefully now the wikepidia definition will make more sense to you.
Keeping this context in mind in the next posts I will explain the web analytics dimensions and metrics we use in our everyday practice.
