**Calculating Quartile Deviation**

Quartile deviation is a measure of dispersion that indicates the spread or variability within a dataset. It is calculated by finding the difference between the upper quartile (Q3) and the lower quartile (Q1). Quartiles divide a dataset into four equal parts, with each quartile representing 25% of the data.

To calculate quartile deviation, we need to follow these steps:

**Step 1: Arrange the Data in Increasing Order**
The first step is to arrange the given data in ascending order. This will make it easier to find the quartiles later on.

**Step 2: Identify the Lower Quartile (Q1) and Upper Quartile (Q3)**
The lower quartile (Q1) is the median of the lower half of the dataset, while the upper quartile (Q3) is the median of the upper half of the dataset.

**Step 3: Calculate the Quartile Deviation**
Quartile deviation is the difference between the upper quartile (Q3) and the lower quartile (Q1). It gives us an idea of how spread out the data is within the middle 50% range.

Here is an example to illustrate the calculation of quartile deviation:

Consider the following dataset: 12, 15, 18, 22, 25, 28, 32, 35, 39, 42

**Step 1: Arrange the Data in Increasing Order**
12, 15, 18, 22, 25, 28, 32, 35, 39, 42

**Step 2: Identify the Lower Quartile (Q1) and Upper Quartile (Q3)**
Since we have 10 data points, the median (Q2) will be the average of the 5th and 6th values, which are 25 and 28 respectively. Therefore, Q2 = (25 + 28) / 2 = 26.5.

To find Q1, we consider the lower half of the dataset: 12, 15, 18, 22, 25. Since we have an odd number of values, the median of this lower half will be the middle value, which is 18. Therefore, Q1 = 18.

To find Q3, we consider the upper half of the dataset: 28, 32, 35, 39, 42. Again, since we have an odd number of values, the median of this upper half will be the middle value, which is 35. Therefore, Q3 = 35.

**Step 3: Calculate the Quartile Deviation**
Quartile deviation (QD) = Q3 - Q1 = 35 - 18 = 17

Therefore, the quartile deviation for the given dataset is 17.

Quartile deviation provides a measure of the spread within the middle 50% of the data. A smaller quartile deviation indicates less variability and a more concentrated dataset, while a larger quartile deviation suggests greater variability and a more dispersed dataset.