The interquartile range (IQR) is a measure of statistical dispersion that is used to describe the spread of data. It is defined as the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset. The IQR is used in many statistical analyses, including box plots, as well as in calculating the upper and lower fences in detecting outliers.
The IQR is a robust measure of dispersion, which means that it is less sensitive to outliers than other measures of dispersion such as the standard deviation. This makes it a useful tool for analyzing datasets that contain outliers or skewed distributions. The IQR is also relatively easy to calculate and interpret, making it a popular choice for descriptive statistics.
To calculate the IQR, you first need to find the values of the first and third quartiles. The first quartile (Q1) is the value that separates the lowest 25% of the data from the rest of the data, while the third quartile (Q3) is the value that separates the highest 25% of the data from the rest of the data. The median (Q2) is the value that separates the lowest 50% of the data from the highest 50%.
Once you have calculated Q1 and Q3, you can find the IQR by subtracting Q1 from Q3. The formula for calculating the IQR is:
IQR = Q3 – Q1
Let’s look at an example to illustrate the calculation of the IQR. Suppose we have the following dataset:
5, 7, 10, 12, 15, 16, 20, 22, 24, 25
To find the first quartile (Q1), we need to find the value that separates the lowest 25% of the data from the rest of the data. There are 10 data points in this dataset, so the first quartile is the 2.5th value when the data is sorted in ascending order. This value can be calculated using the formula:
Q1 = (n + 1) / 4
where n is the number of data points. In this case, n = 10, so Q1 = (10 + 1) / 4 = 2.75. Since Q1 is not a whole number, we need to interpolate between the second and third values in the dataset to find the first quartile. The second value is 7, and the third value is 10, so we can calculate the first quartile as:
Q1 = 7 + 0.75 * (10 – 7) = 8.25
To find the third quartile (Q3), we need to find the value that separates the highest 25% of the data from the rest of the data. Using the same formula as before, we can calculate Q3 as:
Q3 = 3 * (n + 1) / 4 = 7.25
Since Q3 is not a whole number, we need to interpolate between the seventh and eighth values in the dataset to find the third quartile. The seventh value is 20, and the eighth value is 22, so we can calculate the third quartile as:
Q3 = 20 + 0.25 * (22 – 20) = 20.5
Finally, we can calculate the IQR by subtracting Q1 from Q3:
IQR = Q3 – Q1 = 20.5 – 8.25 = 12.25
Therefore, the IQR for this dataset is 12.25.
The IQR is an important measure of dispersion because it is used to identify outliers in a dataset.