When we arrange the results we have received in groups, and calculate the relative weight of each group, the resulting picture is called the results distribution.

The picture of the results distribution is usually displayed in a table, such as in the following table which contains the results of the arithmetic grades of first graders.

Actually, row 1 in the table is unnecessary, since the mark of each group (row 2) is also the group name.

In general, the groups are usually named according to the result they “store”.

For example, the group with a score of 10 points is called “the 10-point group” for short.

The group with a score of 7 points is called “the 7-point group”. The group with 4 visits to the garage is called “the 4-visit group”.

Table 3.19- The result distribution (of grades of first-graders in arithmetic)

Group name | Group 6 | Group 5 | Group 4 | Group 3 | Group 2 | Group 1 |

Marks in points | 10 pts. | 9 pts. | 8 pts. | 7 pts. | 6 pts. | 5 pts. |

Relative weight of the group | 5% | 10% | 40% | 40% | 10% | 5% |

The results distribution table is sufficient to calculate the expectation (or average), by multiplying the two figures for each group, and adding up the results. Whenever you are asked to calculate the expectation, fill in a results distribution table.

## Example

A white rat gives birth to between 2 and 8 offspring per litter (Note: None of the data in this question are zoologically accurate; they are designed solely to demonstrate how to calculate the expectation). At an experimental rat farm, 100,000 litters of white rats are recorded in a given year. Observation of the number of offspring in each litter gives the results displayed in the table in columns 2 and 3.

Table 3.20

No. of the Group | Characteristics of the Group | No. of Litters Included in the Group | Weight of the Group | Contribution of the Group to the Expectation |

(1) | (2) | (3) | (4) | (5) = (2) X (4) |

Group 1 | 2 offspring in the litter | 5,000 litters | 5% | 0.1 |

Group 2 | 3 offspring in the litter | 15,000 litters | 15% | 0.45 |

Group 3 | 4 offspring in the litter | 20,000 litters | 20% | 0.8 |

Group 4 | 5 offspring in the litter | 25,000 litters | 25% | 1.25 |

Group 5 | 6 offspring in the litter | 20,000 litters | 20% | 1.2 |

Group 6 | 7 offspring in the litter | 10,000 litters | 10% | 0.7 |

Group 7 | 8 offspring in the litter | 5,000 litters | 5% | 0.4 |

Total | 100,000 litters | 100% | 4.9 expectation |

**Explanation of rat distribution table: **

In the table, the litters are sorted into groups, according to the number of offspring:

- All the litters with 2 offspring are included in Group 1 (a total of 5,000 litters).
- All the litters with 3 offspring are included in Group 2 (a total of 15,000 litters). And so forth.

The expected number of offspring in a litter is 4.9.