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Interquartile, Semi-Interquartile and Mid-quartile Ranges

In a set of data, the quartiles are the values that divide the data into four equal parts. The median of a set of data separates the set in half.

The median of the lower half of a set of data is the lower quartile ( L Q ) or Q 1 .

The median of the upper half of a set of data is the upper quartile ( U Q ) or Q 3 .

The upper and lower quartiles can be used to find another measure of variation call the interquartile range .

The interquartile range or IQR is the range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.

Interquartile range = Q 3 Q 1

In the above example, the lower quartile is 52 and the upper quartile is 58 .

The interquartile range is 58 52 or 6 .

Data that is more than 1.5 times the value of the interquartile range beyond the quartiles are called outliers .

Statisticians sometimes also use the terms semi-interquartile range and mid-quartile range .

The semi-interquartile range is one-half the difference between the first and third quartiles. It is half the distance needed to cover half the scores.  The semi-interquartile range is affected very little by extreme scores.  This makes it a good measure of spread for skewed distributions. It is obtained by evaluating Q 3 Q 1 2 .

The mid-quartile range is the numerical value midway between the first and third quartile.  It is one-half the sum of the first and third quartiles.  It is obtained by evaluating Q 3 + Q 1 2 .

(The median, midrange and mid-quartile are not always the same value, although they may be.)