## Joint distribution

It is a data table (similar to a relative frequency table) that shows the distribution of one set of data against the distribution of another set of data in percentages.

Weight lost (lbs) | ||||||

0-2 | 2-4 | 4-6 | 6-8 | +8 | ||

Miles walked per day
| 1-3 | 4% | 2% | 21% | 1% | 1% |

3-5 | 12% | 8% | 6% | 2% | 8% | |

5-7 | 1% | 12% | 1% | 0% | 10% | |

+7 | 2% | 3% | 4% | 1% | 1% |

Table 2.1. a data table of a group of 50 individuals, measuring the average number of hours each participant spent walking each day over the course of the study, data about the total number of pounds of weight lost in total by each participant was gathered over that same period of time.

The table 2.1 is an example of joint distribution, it shows that 4 % of the group, which would be 2 out of the 50 people studied, spent between 1 and 3 hours per der exercising, and lost between 0 and 2 pounds.

## Marginal distribution

If we add totals (by totalling up the data in each row and column) to the table 2.1. we get the following data table:

Weight lost (lbs) | |||||||

0-2 | 2-4 | 4-6 | 6-8 | +8 | Total | ||

Miles walked per day
| 1-3 | 4% | 2% | 21% | 1% | 1% | 29% |

3-5 | 12% | 8% | 6% | 2% | 8% | 36% | |

5-7 | 1% | 12% | 1% | 0% | 10% | 24% | |

+7 | 2% | 3% | 4% | 1% | 1% | 11% | |

Total | 19% | 25% | 32% | 4% | 20% | 100% |

Table 2.2. data table with marginal distributions.

## Conditional distribution

Conditional distribution is the distribution of one variable, while the other variable value is already known.

Weight lost (lbs) | |||||||

0-2 | 2-4 | 4-6 | 6-8 | +8 | Total | ||

Miles walked per day | 1-3 | 44% | 22% | 21% | 11% | 2% | 100% |

3-5 | 51% | 19% | 15% | 14% | 1% | 100% | |

5-7 | 61% | 19% | 10% | 0% | 10% | 100% | |

+7 | 22% | 38% | 14% | 5% | 21% | 100% |

Table 2.3. data table with 4 different conditional distributions.

The data table 2.3 shows that people who spent 1 − 3 hours walking per day, 44 % of them lost 0 − 2 pounds, 22 % of them lost 2 − 4 pounds, 21 % of them lost 4 − 6 pounds, 11 % of them lost 6 − 8 pounds and only 2 % of them lost +8 pounds. This distribution is conditional on 1 – 3 walking hours.

If we flip the two distributions, taking the miles walked per day distribution versus each weight loss variable and we calculate the percentages of each conditional variable. We will get the following data table:

Weight lost (lbs) | ||||||

0-2 | 2-4 | 4-6 | 6-8 | +8 | ||

Miles walked per day
| 1-3 | 40% | 52% | 21% | 15% | 11% |

3-5 | 12% | 8% | 16% | 29% | 8% | |

5-7 | 10% | 33% | 3% | 31% | 21% | |

+7 | 38% | 7% | 60% | 25% | 60% | |

Total | 100% | 100% | 100% | 100% | 100% |

Table 2.4. data table with 5 different conditional distributions.