A distribution decribes a variable by communicating two important pieces of information.

1. It specifies all the possible values that a variable may take (these can be called the possible outcomes).
2. It quantifies the relative frequency of each possible value (this can be called the probability of an outcome).

A distribution is very good at describing uncertainty because it can describe events that have multiple possible outcomes.

Below are four examples

Rolling a dice is a classic example of qualitative data (each face of the dice is labelled).

There are six possible outcomes from rolling a dice.

For an unbiased dice each outcome is equally likely, meaning the distribution of outcomes is:

The probability of each outcome is a sixth.

Later, we'll see that this is an example of a categorical distribution.

For one week I recorded the number of times I saw my neighbours' cats in my garden and their coat colour.

Here are my data as a frequency distribution.

Their are four outcomes (the coat colours). The frequency distribution gives the number of times each outcome occurred (e.g. I saw the Tabby cat eight times).

Human height data is quantitative continuous: Every individual has a different height.

To display this continuous distribution we divide the x-axis (height) into bins (1 cm bins are used above) and count the number of data points within each bin (called FREQUENCY on the y-axis).

Below is the distribution of mean daily temperatures for January across Ireland. The data are from 1995-2016

This temperature data is quantitative continuous.

To display the distribution a bin width of 0.5 deg C has been used.

Below is the empirical distribution of cortisol concentrations measured from a sample of 103 wolves

This module uses these data on cortisol concentrations measured from hair samples of wolves in Canada (described here).

The Normal distribution is a bell shaped curved that has a well defined mathematical description.

Above is a Normal distribution (blue curve) being used to mimic human height data (described here)

The categorical distribution describes the probabilities of a finite number of outcomes. The mathematical description of the categorical distribution is the probabilities for each outcome.

Above is a catagorical distribution for the outcomes of rolling an unbiased dice. Each outcome has a probability of 1/6.

The Gamma distribution is a skewed theoretical distribution. It has a well defined mathematical description.

Above are five Gamma distributions with differing amounts of (right) skew.

The log-normal distribution is another skewed theoretical distribution. It has a well defined mathematical description.

Above are four log-normal distributions with differing amounts of (right) skew.

The binomial distribution describes the number of successes and failures from repeatedly performing a task with a constant probability of success. It has a well defined mathematical description.

Above is the binomial distribution for tossing a coin 10 times. The outcomes are the number of heads (ranging from zero to ten) and the distribution gives the probability of each outcome.

The Chi-squared distribution describes the distribution of outcomes from squaring values from a normal distribution and then adding them up. It has a well defined mathematical description.

Above is the distribution of outcomes by taking three values drawn from a normal distribuiton (with mean of zero and standard deviation of one), squaring each and adding up the results. This is a Chi-squared with three degrees of freedom.

The Poisson distribution describes a quantitative discrete variable and is related to the binomial distribution. It has a well defined mathematical description.

Above is the Poisson distribution for describing the number of times a person will be hit by lightning in their lifetime, assuming the probability that a person is struck by lightning in their lifetime is about 15,000 to one.

The Gamma or log-normal distributions could both mimic the empirical distribution of the wolf cortisol data

Above is the empirical distribution of cortisol (grey bars), a Gamma distribution (red) and a log-normal distribution (blue).

A distribution can be broadly described by its shape.

Below are some words used to describe a distribution's shape

A symmetrical distribution looks identical when it is reflected around its centre

Below is a Normal distribution with mean=12 and standard deviation=5

The Normal distribution is symmetrical about its mean

A skewed distribution is asymmetrical.

Below right is a Gamma distribution with mean=12 and standard deviation=5. The symmetrical Normal distribution is shown in grey.

Right-skew (positive skew) has a distribution with an extended tail on the right

Left-skew (negative skew) is the opposite of right-skew

Right-skew (left-skew) commonly causes the mean to be larger (smaller) than the median, because the mean is influenced by extreme values.

An over-dispersed distribution has an excess of extreme values (i.e. it has fat tails).

Below left is a t-distribution distribution, shifted to have mean=12 and standard deviation=5. The equivalent Normal distribution (no over-dispersion) is shown in grey. On the right is a zoom into the tail of the distribution showing the 'fat-tail'

Over-dispersed is also known as platykurtic (platy- means broad)

An under-dispersed distribution has a deficit of extreme values (i.e. it has thin tails).

Below is a Uniform distribution with mean=12 and standard deviation=5. The equivalent Normal distribution (no under-dispersion) is shown in grey.

Under-dispersed is also known as leptokurtic (lepto- means slender)