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Continuous in Probability and Statistics

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Information in probability and statistics can be classified in different ways. The data could be considered in light of what level of measurement it falls in. But determining the level of measurement is not the only way to categorize data. A different way to classify a data set is to determine whether it is continuous or discrete.

Discrete objects in mathematics are those things that can be separated out and counted. It helps to think of a discrete set as a series of dots starting at one point and going to another. There are certain places where there is blank space. Now, if we were to “connect the dots” we would have a series of line segments with no gaps. This connecting of the dots has provided us with an example of something that is continuous.

Counting and the Discrete

One of the first things that we learn in mathematics is how to count. The set of numbers that we use is called the set of natural numbers: {0, 1, 2, 3, 4, . . . }. We could represent these numbers as points along a number line. Here in this situation we note that there are numerous gaps between the dots. This is because the counting numbers are what is known as a discrete set.

When we expand the set of numbers we’re working with to include fractions, we are now using the set of rational numbers. There are more numbers and more points along our number line. The number 1/2 is midway between 0 and 1. The number 1/4 is midway between 0 and 1/2. There is nothing special about these examples. Between any two of our fractions, there is another fraction. The technical mathematical term for this is the density property of the rational numbers.

Real Numbers

If we were to plot every rational number on a number line, we would see the line filling in. In fact, for all appearances it would seem that we had filled in all the holes and connected all of the dots. However, there are still gaps in the number line if we just use fractions as our numbers. To see this, we can look at the decimal expansion of a number. Just as 1/2 = 0.5 and 1/8 = 0.125, every point along the number line corresponds to a possibly infinite string of digits called a decimal expansion.

The decimal expansions of fractions either terminate (like 1/5 = 0.2) or they repeat forever (like 1/3 = 0.33333333333. . .). But some decimal expansions do not do either of these things. A number such as 0.12345678910111213 . . . does not terminate and does not repeat, so it is not a fraction. It is instead a different kind of number called a real number. When we work with this set of numbers we have a smooth number line with no holes in it.

Connection to Statistics

This discussion about the real numbers might have seemed like a bit of a digression, but there is a clear connection to statistics. When we have data that is discrete, it is usually from us counting something. For example you could toss a coin three times an end up with zero, one, two or three heads. There is no possibility of getting anything other than one of these three outcomes.

If we instead contrast this with a measurement, then we see that we have something different. If we want to measure the time that a lightbulb lasts, then we need to have a very precise measurement of time. It may not be good enough to count the number of days that a lightbulb lasts – we may want to know its lifetime down to the minute. And rounding to the nearest minute might not be good enough either, we might measure to the nearest second.

Our unit of measurement could be subdivided even further into milliseconds, microseconds or even femtoseconds, supposing we can measure to this level of precision. We see that the lifetime of a lightbulb is a number that is more like a real number on a number line than a counting number used with discrete data.

A helpful rule of thumb for distinguishing between discrete and continuous data follows:

  • Discrete data typically involves counting.
  • Continuous data typically involves measurement.

Continuous Probability Distributions

Discrete ideas are used in probability and statistics when experiments have outcomes that can be separated from one another. Other times our outcomes can instead fall anywhere along a connected spectrum of values. In this case a continuous probability distribution is appropriate. Rather than a distribution that is a histogram, the function that defines a continuous probability distribution has a smooth graph with no jumps or graphs. Indeed, this is a continuous function.

Continuous probability distributions abound in the study of statistics. A prime example of a continuous probability distribution is the normal distribution or bell curve. Others include Student’s t distribution and the F distribution.

Mathematical Techniques With the Continuous

Discrete data uses an area of mathematics known as combinatorics. However, any time that we deal with continuous data, we should be prepared to use some calculus. Calculus gives us powerful mathematical tools to deal with real numbers and continuous data. For example we may need to find the area of a region that has curved sides. This region is not something with a straightforward formula such as a triangle or rectangle.

Even if we don’t know how to do any calculus, many times it has been done for us. For example, the list of probabilities in a standard normal table of z-scores involves an area of calculus called integration. But if we have the table, then there is no need for us to do the calculus to calculate the areas. This has already been accomplished.

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