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Visualising lower and upper quartiles

You have come across the median before. The idea of the median is to partition a data set into two halves, but of course this isn’t strictly possible if the data set contains an odd number of values. Nevertheless, whether the data set has an odd or even number of values, if listed in order, there will be an equal number of values to the left of the median and to the right. Note that some values to the left and/or right might be equal to the median; but the number of values to the left and right will be equal. You can see this in the applet below. Drag the slider to change the number of values in the data set.

The idea of quartiles is to partition a data set into four quarters. Again, this isn’t strictly possible unless the number of values in the set is a multiple of 4. We can however do a pretty good job as follows:

1. Make sure the values are in order, and find the median.
2. Look at the values below the median position, and find the median of these. This gives you the lower quartile.
3. Look at the values above the original median position, and find the median of these. This gives you the upper quartile.

By doing this, the following will all be equal:

• the number of values to the left of the lower quartile,
• the number of values between the lower quartile and the median,
• the number of values between the median and the upper quartile, and
• the number of values to the right of the upper quartile.

Here are some caterpillars, arranged in order of length. The lengths of the caterpillars, in cm, are shown. Use the applet to explore the median, range, quartiles, interquartile range, and boxplots:

Interquartile range and box plots

Visual representation of boxplots

Suppose the points on the number line below represent the scores (out of 100) achieved by 23 students on a test. Move the points around to see how the boxplot is affected: