As an example of the cumulative distribution, we will build and analyze the cumulative frequency distribution for rolling two standard number cubes.

**Analysis:**

If we sum all possible combinations from two standard cubes, we will get the following values:

N/N | 1 |
2 |
3 |
4 |
5 |
6 |
---|---|---|---|---|---|---|

1 |
2 | 3 | 4 | 5 | 6 | 7 |

2 |
3 | 4 | 5 | 6 | 7 | 8 |

3 |
4 | 5 | 6 | 7 | 8 | 9 |

4 |
5 | 6 | 7 | 8 | 9 | 10 |

5 |
6 | 7 | 8 | 9 | 10 | 11 |

6 |
7 | 8 | 9 | 10 | 11 | 12 |

Each value has its frequency and probability, see table below:

Value | Frequency | Cumulative frequency | Probability |
---|---|---|---|

⩽ 2 | 1 | 1 | 1/36 |

⩽ 3 | 2 | 2+1=3 | 2/36 |

⩽ 4 | 3 | 3+3=6 | 3/36 |

⩽ 5 | 4 | 6+4=10 | 4/36 |

⩽ 6 | 5 | 10+5=15 | 5/36 |

⩽ 7 | 6 | 15+6=21 | 6/36 |

⩽ 8 | 5 | 21+5=26 | 5/36 |

⩽ 9 | 4 | 26+4=30 | 4/36 |

⩽ 10 | 3 | 30+3=33 | 3/36 |

⩽ 11 | 2 | 33+2=35 | 2/36 |

⩽ 12 | 1 | 35+1=36 | 1/36 |

The cumulative frequency graph can be built according to the values in the table:

Position of the median:

The Median:

– from the graph, round to the nearest integer as for discrete structure

Position of the lower quartile:

Lower quartile:

– from the graph, round to the nearest integer as for discrete structure

Position of the upper quartile:

Upper quartile:

– from the graph, round to the nearest integer as for discrete structure

Interquartile range:

This tells us that the difference between maximal and minimal value in the middle 50% is equal 4.

**References**

Liu, Stanley T. Experimental And Analytical Investigation Of Solar Radiant Flux Distribution On Interior Surfaces Of A Sunspace. Gaithersburg, MD: U.S. Dept. of Commerce, National Bureau of Standards, 1986. Print.

Salkind, Neil J. Encyclopedia Of Research Design. Thousand Oaks, Calif.: Sage, 2010. Print.

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