What Is Interquartile in Math?

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Interquartile is a term used in statistics. In particular, the interquartile range is one measure of the spread of a distribution. A distribution is a record of the values of some variable. For example, if we found the incomes of 100 people, that would be the distribution of income in our sample. Another common measure of spread is the standard deviation.

Interquartile Range

  • The quartiles of a distribution are the three points that divide it into four equally numerous parts. The first quartile is the point where 1/4 of the values are lower and 3/4 are higher; the second quartile, better known as the median, divides the distribution into equal parts; the third quartile is just the opposite of the first.

    The interquartile range is the range between the first and third quartiles. It is sometimes written as two numbers with a hyphen between them, and sometimes as the difference between those numbers.

Example

  • If you collect income data on 12 people, and the results are $10,000, $12,000, $13,000, $14,000, $15,000, $21,000, $22,000, $25,000, $30,000, $35,000, $40,000 and $120,000 then the quartiles should divide the results into four groups of three. The first quartile is midway between $13,000 and $14,000 (that is, $13,500) and the third quartile is midway between $30,000 and $35,000 (that is, $32,500) so the interquartile range is $13,500 - $32,500.

Use

  • The interquartile range is a good measure of the spread of a distribution that is skewed; that is, one that has a long tail to the right or left. Income distributions often have a long tail to the right, because there are a few people who make a great deal of money. If the median (rather than the mean) is used for a measure of central tendency, the interquartile range (rather than the standard deviation) should probably be used as the measure of spread.

Alternatives

  • Alternatives to the interquartile range include the median absolute deviation and the full range. You find the former by taking the difference between each value and the mean, taking the absolute values of those differences and then finding the median of that. The latter is simply the range from the lowest to the highest value.

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