Quartiles

Numbers separating an ordered distribution into four groups, each containing (as closely as possible) equal numbers of values. The most common names for these three numbers, in order from lowest to highest, are lower quartile, median, and upper quartile.

The lower quartile is a number that is a quarter of the way through the ordered distribution, from the lower end. The upper quartile is a number that is a quarter of the way through the ordered distribution, from the upper end.

There are several different methods for calculating quartiles. For reasonably small data sets, it is recommended that the values are sorted into order (or displayed on a suitable graph) and then the median is calculated. This allows the distribution to be split into a ‘lower half’ and an "upper half". The lower quartile is the median of the "lower half", and the upper quartile is the median of the ‘upper half’. Use software for large data sets.

Note that different software may use different methods for calculating quartiles that may give different values for the quartiles. This is of no concern because, in most cases, any differences will be slight.

Example 1 (odd number of values)

The maximum temperatures, in degrees Celsius (°C), in Rolleston for the first 9 days in November 2008 were: 18.6, 19.9, 20.6, 19.4, 17.8, 18.1, 17.8, 18.7, and 19.6.

Ordered values: 17.8, 17.8, 18.1, 18.6, 18.7, 19.4, 19.6, 19.9, and 20.6.

The median is 18.7°C.

The values in the "lower half" are 17.8, 17.8, 18.1, and 18.6. Their median is the mean of 17.8 and 18.1, which is 17.95. The lower quartile is 17.95°C.

The values in the "upper half" are 19.4, 19.6, 19.9, and 20.6. Their median is the mean of 19.6 and 19.9, which is 19.75. The upper quartile is 19.75°C.

The data and the quartiles are displayed on the dot plot below.

Notice that there are 2 values below the lower quartile, 2 values between the lower quartile and the median, 2 values between the median and the upper quartile, and 2 values above the upper quartile.

Example 2 (even number of values)

The maximum temperatures, in degrees Celsius (°C), in Rolleston for the first 10 days in November 2008 were: 18.6, 19.9, 20.6, 19.4, 17.8, 18.1, 17.8, 18.7, 19.6, and 18.8.

Ordered values: 17.8, 17.8, 18.1, 18.6, 18.7, 18.8, 19.4, 19.6, 19.9, 20.6

The median is 18.75°C.

The values in the "lower half" are 17.8, 17.8, 18.1, 18.6, 18.7. Their median is 18.1. The lower quartile is 18.1°C.

The values in the "upper half" are 18.8, 19.4, 19.6, 19.9, and 20.6. Their median is 19.6. The upper quartile is 19.6°C.

The data and the quartiles are displayed on the dot plot below.

Notice that there are 2 values below the lower quartile, 2 values between the lower quartile and the median, 2 values between the median and the upper quartile, and 2 values above the upper quartile.

See: Median

Curriculum achievement objectives references

Statistical investigation: Levels 5, 6, 7, 8

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