Inter-quartile range

Inter-quartile Range (IQR) is a measure of statistical dispersion or variability that describes the spread of the middle 50% of a dataset. It is calculated as the difference between the 75th percentile (the upper quartile) and the 25th percentile (the lower quartile). The IQR is used to build box plots, a type of graphical display that shows the central tendency, dispersion, and skewness of a dataset.

Definition

The Inter-quartile Range is defined as:

IQR = Q3 − Q1

where Q3 is the third quartile (the 75th percentile) and Q1 is the first quartile (the 25th percentile). These quartiles can be found by sorting the dataset in ascending order and then dividing the dataset into four equal parts. The values that separate these parts are the first, second (the median), and third quartiles.

Calculation

To calculate the IQR, follow these steps:

1. Arrange the data in ascending order.
2. Find the median (the middle value) of the dataset. If the dataset has an even number of observations, the median is the average of the two middle numbers.
3. Divide the dataset into two halves at the median. If the dataset has an odd number of observations, include the median in both halves.
4. The first quartile (Q1) is the median of the lower half of the dataset, and the third quartile (Q3) is the median of the upper half of the dataset.
5. Subtract Q1 from Q3: IQR = Q3 - Q1.

Uses

The IQR is particularly useful for identifying outliers and understanding the variability of a dataset. It is less sensitive to extreme values than other measures of spread, such as the range. Therefore, the IQR provides a more stable measure of variability when the dataset contains outliers or is skewed.

Box Plots

A box plot is a graphical representation of the five-number summary of a dataset, which includes the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. The IQR is represented by the length of the box in the box plot. Outliers, if any, are shown as individual points outside the "whiskers" of the box plot.

Comparison with Other Measures

The IQR is often compared with the range and standard deviation to understand the spread of a dataset. While the range gives the difference between the maximum and minimum values, it can be heavily influenced by outliers. The standard deviation measures the average distance of each data point from the mean but also can be affected by outliers. The IQR, by focusing on the middle 50% of the data, provides a measure of variability that is resistant to extreme values.

Limitations

The IQR does not provide information about the shape of the distribution outside the middle 50% of the data. It also does not take into account how the data within the quartiles is distributed.