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title: "Parameters of a probability distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574965/Parameters%20of%20a%20probability%20distribution"
format: markdown
---
Descriptive parameters can be calculated for a probability distribution providing one knows the [probability density function](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574888/) (or [probability mass function](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574887/)) f(x) and the inverse F-1(x) of the distribution function F(x). [The one exception is the [Cauchy distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575220/) for which the mean, variance and higher moments are not defined.] We will use the height of a person (i.e. a [continuous quantity](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575199/)) randomly selected from a population as our variable in the examples below:

  


| Continuous variable |
| --- |
| Parameter | Description | Definition |
| Percentiles x<sub>P</sub> | P% probability the person will be shorter than x<sub>P</sub> | F<sup>-1</sup>(P) |
| Location |  |  |
| Mode | The most probable height | Value x s.t. f(x) = max |
| Median x<sub>50</sub> | 50% probability the person will be shorter than x<sub>50</sub> | F<sup>-1</sup>(0.5) |
| Mean *m* | The expected height of the person | ![image](media://c99fdd24-2985-4670-8582-57299cae7366)<br>> Macro (mathinline) |
| Spread |  |  |
| Standard deviation *s* | The square root of the distribution's variance | ![image](media://0dfe1194-a204-4204-98d4-b1a60bc4c1c1)<br>> Macro (mathinline) |
| Variance V | The probability-weighted squared deviation of height from the expected height | ![image](media://e94f59d2-116c-4986-9a77-cbdf2b5c86be)<br>> Macro (mathinline) |
| Mean absolute deviation | The probability-weighted average difference in height from the expected height | ![image](media://e83d3463-6ec9-42c4-9b47-d9cdfcbe591e)<br>> Macro (mathinline) |

  


In the next example, we use a discrete quantity, the number of earthquakes next year, to illustrate the difference in equations and definitions:

  


| Discrete variable |
| --- |
| Parameter | Description | Definition |
| Percentiles x<sub>P</sub> | P% probability there will be less than x<sub>P</sub> earthquakes | F<sup>-1</sup>(P) |
| Location |  |  |
| Mode | The most probable number of earthquakes | Value x s.t. f(x) = max |
| Median x<sub>50</sub> | 50% probability there will be less than x<sub>50</sub> earthquakes | F<sup>-1</sup>(0.5) |
| Mean *m* | The expected number of earthquakes | ![image](media://6b74692a-8d0d-4e9a-af64-8e6bbc7dd662)<br>> Macro (mathinline) |
| Spread |  |  |
| Standard deviation *s* | The square root of the distribution's variance | > Macro (mathinline) |
| Variance V | The probability-weighted squared deviation of number of possible earthquakes from the expected value | ![image](media://07210b6d-8421-432d-9ab5-30e87217fce0)<br>> Macro (mathinline) |
| Mean absolute deviation | The probability-weighted average difference between the number of earthquakes and the expected height | ![image](media://f460471f-2811-4942-ac59-f5bb363d0401)<br>> Macro (mathinline) |

  


  


  


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