How To Find The Median In A Cumulative Frequency Graph
Unit of measurement 16 Section iii : Cumulative Frequency
Cumulative frequencies are like shooting fish in a barrel to summate from a frequency table. Cumulative frequency graphs tin can so be used to estimate the median of a set of data. In this section we likewise expect at the idea of quartiles, the interquartile range and the semiinterquartile range.
When y'all have a fix of n values, in order,
| Lower quartile | = thursday value |
| Median | = th value |
| Upper quartile | = th value |
| Interquartile range | = upper quartile – lower quartile |
| Semi-interquartile range | = |
If the information is bundled in an ordered list, and the number of data values, northward, is odd then the th value volition be a single detail from the list, and this will be the median. For instance, if n = 95 the median will exist the
For large sets of data, we estimate the lower quartile, median and upper quartile using the thursday, th and thursday values. For example, if due north = 2000 , then we would guess the lower quartile, median and upper quartile using the 500th, 1000th and 1500th values.
Example 1
For the following set of data,
| iv | vii | xviii | 3 | 9 | 5 | 10 |
(a)
determine the median,
First list the values in gild:
| 3 | 4 | v | 7 | 9 | 10 | xviii |
As there are 7 values, the median volition exist the = 4th value.
Median = 7.
(b)
calculate the interquartile range,
The lower quartile volition be the = 2nd value.
Lower quartile = 4.
The upper quartile will exist the = 6th value.
Upper quartile = x.
The interquartile range = upper quartile – lower quartile = 10 – four = 6
The semi-interquartile range = = = 3
Example 2
(a)
Draw a cumulative frequency graph for the post-obit data:
| Acme (cm) | 150 ≤ h < 155 | 155 ≤ h < 160 | 160 ≤ h < 165 | 165 ≤ h < 170 | 170 ≤ h < 175 |
|---|---|---|---|---|---|
| Frequency | 4 | 22 | 56 | 32 | five |
From the data table we can run across that there are no heights under 150 cm.
Under 155 cm there are the first iv heights.
Nether 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 meridian 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 = iv | 4 + 22 = 26 | 26 + 56 = 82 | 82 + 32 = 114 | 114 + 5 = 119 |
The cumulative frequency graph tin now be plotted using the points in the table, (150, 0), (155, four), (160, 26), (165, 82), (170, 114) and (175, 119).
To obtain the cumulative frequency polygon, we draw straight line sections to bring together these points in sequence.
(b)
Guess the median from the graph.
There are 119 values, so the median volition be the = 60th value.
This can be read from the graph equally 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 = vi cm
Example iii
Estimate the semi-interquartile range of the data illustrated in the following cumulative frequency graph:
The sample contains fifteen values, so the lower quartile will exist the = 4th value.
Similarly, the upper quartile volition exist the twelfth value.
These can be obtained from the graph, as follows:
Lower quartile = i.four kg
Upper quartile = three kg
Interquartile range = 3 – 1.4 = 1.6 kg
Semi-interquartile range = 0.8 kg
Exercises
Source: https://www.cimt.org.uk/projects/mepres/book9/bk9i16/bk9_16i3.html
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