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Finite Set, Infinite Set and Statistics

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Contents:

  1. Finite Set and Infinite Set
  2. Finite and Infinite Statistics

Finite Set and Infinite Sets

A finite set has a certain, countable number of objects. For example, you might have a fruit bowl with ten pieces of fruit. That’s a finite set.

If you can count the number of objects in your set, that’s a finite set.

Now try counting the number of stars in the universe. You won’t be able to, because there are an infinite number of items in the set of all stars.

If you can’t count the number of objects, it’s the opposite: an infinite set.

Notation

In notation, a finite set is:
{1, 2, 3, 4, 5}
Where you can replace 1 through 5 with any amount of any number. For example:
{101, 222, 433, 97894, 5213457}
or
{.21, .22, .43, .7654, .975}

On the other hand, if you see three dots “…” at the end of a set, that means it contains an infinite number of items.

Usually, but not always, the items in the infinite set will give you a clue to the missing contents. For example:

  • {1, 2, 3, 4, 5 …} indicates it goes on and on to 6, 7, 8, 9, 10 … and beyond (basically, keep counting and never stop).
  • {100, 200, 300 …} indicates you keep counting by one hundred until infinity.

Finite and Infinite Statistics

Finite statistics are statistics calculated from finite sets. Basically, you have a sample that you’re using to make a calculation (like the sample variance). If you have a countable number of data points in your sample, what you end up with is a finite statistic.

On the other hand, infinite statistics are those calculated from infinite sets. For example, a probability density function has, for practical purposes, an infinite number of data points under its curve.

The normal distribution is another example of an area that uses infinite statistics: the z-table on this site lists just a few hundred points, but technically the table has an uncountable number of points on it (e.g. z=2.1 is listed, but z = 2.1249865 is not). This is for a couple of reasons:

  1. Space: there simply isn’t room on any page in existence for a table of infinite values!
  2. Practical Purposes: Even if you could list every possible z-value, there comes a point where the values are so similar, a finite set is “good enough”. Take a look at this snapshot from the table:
    infinite set


    Any value between 3.7 and 3.8 would also be an area of 0.4999, so there’s really no point in listing them all.
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Finite Set, Infinite Set and Statistics was last modified: June 1st, 2018 by Stephanie