Quartile Calculator

How to Calculate Quartiles and Interquartile Range

Quartiles and the Interquartile Range (IQR) are important measures in descriptive statistics that help us understand the spread and central tendency of a dataset. They are particularly useful for identifying outliers and comparing distributions.

Formulas and Their Meanings

For an ordered dataset of n values:

1. Q1 (First Quartile): The median of the lower half of the data
2. Q2 (Median): \[Q2 = \begin{cases} \frac{x_{(n+1)/2}}{2}, & \text{if n is odd} \\ \frac{x_{n/2} + x_{(n/2)+1}}{2}, & \text{if n is even} \end{cases}\]
3. Q3 (Third Quartile): The median of the upper half of the data
4. IQR (Interquartile Range): \(IQR = Q3 - Q1\)

Where x_i represents the i-th value in the ordered dataset.

Calculation Steps

1. Sort the dataset in ascending order.
2. Find the median (Q2) of the entire dataset.
3. Find Q1 by calculating the median of the lower half of the data (values below the median).
4. Find Q3 by calculating the median of the upper half of the data (values above the median).
5. Calculate the IQR by subtracting Q1 from Q3.

Example

Let's calculate the quartiles and IQR for the dataset: 2, 4, 7, 9, 12, 15, 18

1. The data is already sorted.
2. Q2 (Median) = 9 (middle value)
3. Q1 = median of (2, 4, 7) = 4
4. Q3 = median of (12, 15, 18) = 15
5. IQR = Q3 - Q1 = 15 - 4 = 11

Visual Representation

This box plot illustrates the quartiles and range of our example dataset. The box represents the IQR, with Q1, median, and Q3 marked. The whiskers extend to the minimum and maximum values.