# Interquartile range

In descriptive statistics, the **interquartile range** (**IQR**), also called the **midspread** or **middle 50%**, or technically **H-spread**, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles,[1][2] IQR = *Q*_{3} − *Q*_{1}. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

The IQR is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

## Use

Unlike total range, the interquartile range has a breakdown point of 25%,[3] and is thus often preferred to the total range.

The IQR is used to build box plots, simple graphical representations of a probability distribution.

The IQR is used in businesses as a marker for their income rates.

For a symmetric distribution (where the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).

The median is the corresponding measure of central tendency.

The IQR can be used to identify outliers (see below).

The quartile deviation or semi-interquartile range is defined as half the IQR.[4][5]

## Algorithm

The IQR of a set of values is calculated as the difference between the upper and lower quartiles, Q_{3} and Q_{1}. Each quartile is a median[6] calculated as follows.

Given an even *2n* or odd *2n+1* number of values

*first quartile Q*= median of the_{1}*n*smallest values*third quartile Q*= median of the_{3}*n*largest values[6]

The *second quartile Q _{2}* is the same as the ordinary median.[6]

## Examples

### Data set in a table

The following table has 13 rows, and follows the rules for the odd number of entries.

i | x[i] | Median | Quartile |
---|---|---|---|

1 | 7 | Q_{2}=87(median of whole table) |
Q_{1}=31(median of upper half, from row 1 to 6) |

2 | 7 | ||

3 | 31 | ||

4 | 31 | ||

5 | 47 | ||

6 | 75 | ||

7 | 87 | ||

8 | 115 | ||

Q_{3}=119(median of lower half, from row 8 to 13) | |||

9 | 116 | ||

10 | 119 | ||

11 | 119 | ||

12 | 155 | ||

13 | 177 |

For the data in this table the interquartile range is IQR = Q_{3} − Q_{1} = 119 - 31 = 88.

### Data set in a plain-text box plot

+−−−−−+−+ * |−−−−−−−−−−−| | |−−−−−−−−−−−| +−−−−−+−+ +−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+−−−+ number line 0 1 2 3 4 5 6 7 8 9 10 11 12

For the data set in this box plot:

- lower (first) quartile
*Q*_{1}= 7 - median (second quartile)
*Q*_{2}= 8.5 - upper (third) quartile
*Q*_{3}= 9 - interquartile range, IQR =
*Q*_{3}-*Q*_{1}= 2 - lower 1.5*IQR whisker =
*Q*_{1}- 1.5 * IQR = 7 - 3 = 4. (If there is no data point at 4, then the lowest point greater than 4.) - upper 1.5*IQR whisker =
*Q*_{3}+ 1.5 * IQR = 9 + 3 = 12. (If there is no data point at 12, then the highest point less than 12.)

This means the 1.5*IQR whiskers can be uneven in lengths.

## Distributions

The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function—any other means of calculating the CDF will also work). The lower quartile, *Q*_{1}, is a number such that integral of the PDF from -∞ to *Q*_{1} equals 0.25, while the upper quartile, *Q*_{3}, is such a number that the integral from -∞ to *Q*_{3} equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

where CDF^{−1} is the quantile function.

The interquartile range and median of some common distributions are shown below

Distribution | Median | IQR |
---|---|---|

Normal | μ | 2 Φ^{−1}(0.75)σ ≈ 1.349σ ≈ (27/20)σ |

Laplace | μ | 2b ln(2) ≈ 1.386b |

Cauchy | μ | 2γ |

### Interquartile range test for normality of distribution

The IQR, mean, and standard deviation of a population *P* can be used in a simple test of whether or not *P* is normally distributed, or Gaussian. If *P* is normally distributed, then the standard score of the first quartile, *z*_{1}, is −0.67, and the standard score of the third quartile, *z*_{3}, is +0.67. Given *mean* = *X* and *standard deviation* = σ for *P*, if *P* is normally distributed, the first quartile

and the third quartile

If the actual values of the first or third quartiles differ substantially from the calculated values, *P* is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and −0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such as Q-Q plot would be indicated here.

## Outliers

The interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 − 1.5 IQR or above Q3 + 1.5 IQR. In a boxplot, the highest and lowest occurring value within this limit are indicated by *whiskers* of the box (frequently with an additional bar at the end of the whisker) and any outliers as individual points.

## References

- Upton, Graham; Cook, Ian (1996).
*Understanding Statistics*. Oxford University Press. p. 55. ISBN 0-19-914391-9. - Zwillinger, D., Kokoska, S. (2000)
*CRC Standard Probability and Statistics Tables and Formulae*, CRC Press. ISBN 1-58488-059-7 page 18. - Rousseeuw, Peter J.; Croux, Christophe (1992). Y. Dodge (ed.). "Explicit Scale Estimators with High Breakdown Point" (PDF).
*L1-Statistical Analysis and Related Methods*. Amsterdam: North-Holland. pp. 77–92. - Yule, G. Udny (1911).
*An Introduction to the Theory of Statistics*. Charles Griffin and Company. pp. 147–148. - Weisstein, Eric W. "Quartile Deviation".
*MathWorld*. - Bertil., Westergren, (1988).
*Beta [beta] mathematics handbook : concepts, theorems, methods, algorithms, formulas, graphs, tables*. Studentlitteratur. p. 348. ISBN 9144250517. OCLC 18454776.CS1 maint: extra punctuation (link)

## External links

Media related to Interquartile range at Wikimedia Commons