Cumulative Frequency - tandrageemaths



Cumulative Frequency

Example

Here are the marks for 80 students for a recent mathematics exam.

|Marks |Frequency |

|0 ≤ m < 10 |0 |

|10 ≤ m < 20 |3 |

|20 ≤ m < 30 |7 |

|30 ≤ m < 40 |35 |

|40 ≤ m < 50 |21 |

|50 ≤ m < 60 |10 |

|60 ≤ m < 70 |4 |

To look clearly at the totals we create a cumulative frequency table. Instead of the frequency we look at the running totals.

|Marks |Frequency |

|< 10 |0 |

|< 20 |3 |

|< 30 |10 |

|< 40 |45 |

|< 50 |66 |

|< 60 |76 |

|< 70 |80 |

We can plot these cumulative frequencies on a graph.

[pic]

Here are the heights of 48 students.

|Height (cm) |Frequency |

|140 ≤ h < 145 |2 |

|145 ≤ h < 150 |4 |

|150 ≤ h < 155 |8 |

|155 ≤ h < 160 |9 |

|160 ≤ h < 165 |12 |

|165 ≤ h < 170 |7 |

|170 ≤ h < 175 |4 |

|175 ≤ h < 180 |2 |

Complete the cumulative frequency table below.

|Height (cm) |Cumulative Frequency |

|< 145 | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

Draw the cumulative frequency curve for these results on the graph paper provided.

Using a Cumulative Frequency Curve

From our curve we can find useful values

MEDIAN - Halfway mark, at this point 50% of the results are above and 50% below

LOWER QUARTILE – quarter mark, 75% of the results are above and 25% below

UPPER QUARTILE – three quarter mark, 25% of the results are above and 75% below

INTERQUARTILE RANGE – the spread of the middle 50%, we work this out by

upper quartile – lower quartile

We can also use our curve to work out other totals.

1) The number of students who are taller than 160cm

2) The number of students who are between 145cm and 170cm.

-----------------------

How many students got less than 30? 0 + 3 + 7 = 10

How many got more than 50? 10 + 4 = 14

How many got less than 60? 80 – 4 = 76

Sometimes we are interested in the totals

The final cumulative number is the total frequency

Notice that the Marks column now only shows the upper class boundary.

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