# Ogives (Cumulative Frequency Graphs)

**Cumulative Frequency** is the progressive total of the frequencies. It assists us to find the median, quartiles and percentiles from large quantities of data organized into tables and graphs. To find the cumulative frequency, you **add** up the frequencies row by row.

An **Ogive** is a graph of the cumulative frequency. Interpreting quartiles, median and percentiles from a graph is more accurate than from a table.

## Example - Teenagers' CD Collections

The following table shows the number of music CDs owned by students in a Math class. Complete the table with two more columns for the cumulative frequency and cumulative percentage. Draw an ogive (a cumulative frequency graph). From the ogive, find the 1st quartile, median, 3rd quartile and 80th percentile.

NUMBER OF CDs | NUMBER OF TEENAGERS |

0 | 2 |

1 | 3 |

2 | 5 |

3 | 10 |

4 | 5 |

**Answer:**

NUMBER OF CDs | NUMBER OF STUDENTS | CUMULATIVE FREQUENCY | CUMULATIVE PERCENTAGE |

0 | 2 | 2 | ^{2}⁄_{23} × 100 = 8.7% |

1 | 3 | 5 | ^{5}⁄_{23} × 100 = 21.7% |

2 | 5 | 10 | ^{10}⁄_{23} × 100 = 43.5% |

3 | 10 | 20 | ^{20}⁄_{23} × 100 = 87% |

4 | 3 | 23 | ^{23}⁄_{23} × 100 = 100% |

Σ f = 23 |

**From Table:**

n = 25

1st Quartile | = (n + 1) ÷ 4 | = (23 + 1) ÷ 4 | = 6th student | = 2 CDs |

Median | = (n + 1) ÷ 2 | = (23 + 1) ÷ 2 | = 12th student | = 3 CDs |

3rd Quartile | = (n + 1) ÷ 4 × 3 | = (23 + 1) ÷ 4 × 3 | = 18th student | = 3 CDs |

80th percentile | = 80% of 25 students | = 20th student | = 3 CDs |

**From Graph (more accurate):**

1st Quartile | = (n + 1) ÷ 4 | = (23 + 1) ÷ 4 | = 6th student | = 1.2 CDs |

Median | = (n + 1) ÷ 2 | = (23 + 1) ÷ 2 | = 12th student | = 2.2 CDs |

3rd Quartile | = (n + 1) ÷ 4 × 3 | = (23 + 1) ÷ 4 × 3 | = 18th student | = 2.8 CDs |

80th percentile | = 80% of 25 students | = 20th student | = 3.0 CDs |

## Did You Know That...?

Michael Jackson's "Thriller" album with sales of 110 million is the best-selling album of all time.