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Distributions

Introduction

A 'distribution' is as the name implies how the data is distributed. For example, if 4 people said yes, 8 said maybe, and 3 said no, then the 'distribution' of 'yes', 'maybe' and 'no' is 4, 8 and 3

In statistics when the term 'distribution' is used, it will refer not to how the actual counts were distributed, but the probability of those. In the example 4/15 for yes, 8/15 for maybe, and 3/15 for no.

Many names exists for different distributions. One very simplistic one is the uniform distribution. This is a distribution where each outcome is equally likely. If we had 5 yes, 5 maybe and 5 no, it would result in a probability of each of 5/15, and we could then say we have a uniform distribution

A fair die is another example of something that has a uniform distribution, i.e. the probability for each outcome is the same 1/6.

The two examples above for the uniform distribution, were more specifically examples of a discrete uniform distribution. A large categorisation of the various distributions in statistics is in 'discrete' and 'continuous' distributions. The difference is in what the possible outcomes could be. Discrete would indicate no 'in-between values'. An example of continuous would be the daily temperature. Continuous distributions are a bit more complicated to understand.

Since in a continuous distribution we can have an infinite many 'in-between' values the probability of any specific value will be 0 (or strickly 1/infinity). Let's use an example. We ask someone to pick any number between 0 and 1. This could be 0.3, 0.002, 0.784521, etc. Each number is equally likely to occur. So this is an example of a continuous uniform distribution. To visualise this we draw a rectangle with the horizontal going from 0 to 1, and a height of 1. This doesn't mean each value has a probability of one, but instead we can use the area as the probability. For example the chance of someone picking a number below 0.25 would then be the area highlighted in yellow in Figure 1.

Figure 1
Continuous Uniform Distribution Example

Note that it doesn't really matter if we say below 0.25 or below or equal to 0.25. The chance of exactly 0.25 is next to 0 so the difference between those two is also next to 0.

Perhaps the most important thing to remember with continuous distributions is that it is not the hight of the curve, but the area under it that determines the probability

Since significance is the probability of a result as in the sample, or more extreme if the assumption about the population is true, we are often interested in cumulative probabilities. This is the so-called cumulative distribution function (cdf). The height of the curve is known as the probability density function (pdf), and for discrete distribution this is the probabability of that specific category and called the probability mass function (pmf)

So what's the point of all these distributions? The advantage of the distributions that we have, is that we know how to calculate the probabilities from them. So if we know something in the population has a specific distribution given an assumption about the population, we can calculate the probability of having a result as in the sample, or more extreme. This is exactly the significance.

For example. We know that the sample means from a population will follow a so-called normal distribution, with the population mean at its center. We can therefor calculate the probability of obtaining a value as we had in our sample, or less if indeed that would be the population mean. This is exactly the definition of significance.

With many statistical tests we therefor have to calculate some specific value, and some very smart person has proven that that value will be following a specific type of distribution. Which then means we can use that distribution to calculate the significance.

Common values to calculate are z, t, chi2 and F. Each follows the distribution with the same name (although z follows a normal distribution)

For more details on each specific distribution you can click on it in the menu on the left.

Distributions 

Introduction

Binomial

Bivariate Normal

Chi-Square

Normal

Student t

Wilcoxon Signed Rank

 

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