Cumulative frequencies are easy to calculate from a frequency table. Cumulative frequency graphs can then be used to estimate the median of a set of data. In this section we also look at the idea of *quartiles*, the *interquartile range* and the *semiinterquartile range*.

When you have a set of *n* values, in order,

Lower quartile | = th value |

Median | = th value |

Upper quartile | = th value |

Interquartile range | = upper quartile – lower quartile |

Semi-interquartile range | = |

If the data is arranged in an ordered list, and the number of data values, *n*, is odd then the th value will be a single item from the list, and this will be the median. For example, if *n* = 95 the median will be the *n* is even then will determine the two central values that must be averaged to obtain the median. For example, if *n* = 156 then = 78.5, which tells us that we must average the 78th and 79th values to get the median.

For large sets of data, we estimate the lower quartile, median and upper quartile using the th, th and th values. For example, if *n* = 2000 , then we would estimate the lower quartile, median and upper quartile using the 500th, 1000th and 1500th values.

For the following set of data,

4 | 7 | 18 | 3 | 9 | 5 | 10 |

(a)

determine the *median*,

First list the values in order:

3 | 4 | 5 | 7 | 9 | 10 | 18 |

As there are 7 values, the median will be the = 4th value.

Median = 7.

(b)

calculate the *interquartile range*,

The lower quartile will be the = 2nd value.

Lower quartile = 4.

The upper quartile will be the = 6th value.

Upper quartile = 10.

The interquartile range = upper quartile – lower quartile = 10 – 4 = 6

The semi-interquartile range = = = 3

(a)

Draw a cumulative frequency graph for the following data:

Height (cm) | 150 ≤ h < 155 | 155 ≤ h < 160 | 160 ≤ h < 165 | 165 ≤ h < 170 | 170 ≤ h < 175 |
---|---|---|---|---|---|

Frequency | 4 | 22 | 56 | 32 | 5 |

From the data table we can see that there are no heights under 150 cm.

Under 155 cm there are the first 4 heights.

Under 160 cm there are the first 4 heights plus a further 22 heights that are between 155 cm and 160 cm, giving 26 altogether.

Under 165 cm we have the 26 heights plus the 56 that are between 160 cm and 165 cm, giving 82 altogether.

Continuing this process until every height has been counted gives the following *cumulative frequency table*.

Height (cm) | Under 150 | Under 155 | Under 160 | Under 165 | Under 170 | Under 175 |
---|---|---|---|---|---|---|

Cumulative Frequency | 0 | 0 + 4 = 4 | 4 + 22 = 26 | 26 + 56 = 82 | 82 + 32 = 114 | 114 + 5 = 119 |

The cumulative frequency graph can now be plotted using the points in the table, (150, 0), (155, 4), (160, 26), (165, 82), (170, 114) and (175, 119).

To obtain the *cumulative frequency polygon*, we draw straight line sections to join these points in sequence.

(b)

Estimate the *median* from the graph.

There are 119 values, so the median will be the = 60th value.

This can be read from the graph as shown above.

Median ≈ 163 cm.

(c)

Estimate the *interquartile range* from the graph.

The lower quartile will be given by the th value.

Lower quartile ≈ 160.5 cm.

The upper quartile will be given by the th value.

Upper quartile ≈ 166.5 cm.

Using these values gives:

Interquartile range = 166.5 – 160.5 = 6 cm

Estimate the semi-interquartile range of the data illustrated in the following cumulative frequency graph:

The sample contains 15 values, so the lower quartile will be the = 4th value.

Similarly, the upper quartile will be the 12th value.

These can be obtained from the graph, as follows:

Lower quartile = 1.4 kg

Upper quartile = 3 kg

Interquartile range = 3 – 1.4 = 1.6 kg

Semi-interquartile range = 0.8 kg