Interquartile data analysis (also known as Interquartile Range or IQR analysis) is a method used in statistics to measure the spread or variability in a data set by focusing on the middle 50% of the data. The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1). Here's a step-by-step guide to perform interquartile data analysis:

1. Arrange the Data in Ascending Order

• First, arrange the dataset in increasing order. This ensures the values are sorted from smallest to largest, which is essential for calculating quartiles.

2. Find the Median (Q2)

• The median divides the data set into two equal halves.
• If the data set has an odd number of data points, the median is the middle value.
• If the data set has an even number of data points, the median is the average of the two middle numbers.
• The median is also referred to as the second quartile (Q2).

3. Determine the First Quartile (Q1)

• The first quartile (Q1) represents the 25th percentile. It is the median of the lower half of the data (not including Q2 if there’s an odd number of data points).
• Q1 is the value at which 25% of the data falls below it.

4. Determine the Third Quartile (Q3)

• The third quartile (Q3) represents the 75th percentile. It is the median of the upper half of the data (again, excluding Q2 if there’s an odd number of data points).
• Q3 is the value at which 75% of the data falls below it.

5. Calculate the Interquartile Range (IQR)

• The IQR is the difference between Q3 and Q1: $IQR = Q3 - Q1$
• This gives you the spread of the middle 50% of the data.

6. Identify Potential Outliers

• To find potential outliers, calculate the lower and upper bounds using the following formulas: $\text{Lower bound} = Q1 - 1.5 \times IQR$ $\text{Upper bound} = Q3 + 1.5 \times IQR$
• Any data points below the lower bound or above the upper bound may be considered outliers.

---

Example:

Consider the data set: 2, 4, 7, 8, 9, 10, 12, 15, 18

1. Ordered data: 2, 4, 7, 8, 9, 10, 12, 15, 18
2. Median (Q2): 9
3. First Quartile (Q1): Median of [2, 4, 7, 8] = 5.5
4. Third Quartile (Q3): Median of [10, 12, 15, 18] = 13.5
5. IQR: Q3 - Q1 = 13.5 - 5.5 = 8
6. Outlier bounds: $\text{Lower bound} = 5.5 - 1.5 \times 8 = -6.5$ $\text{Upper bound} = 13.5 + 1.5 \times 8 = 25.5$ No outliers since all data points fall within the range [-6.5, 25.5].

---

Tip:

The IQR is resistant to outliers because it only focuses on the middle 50% of the data, making it a robust measure of spread compared to the range.

Related Questions:

1. How can you use IQR analysis to detect outliers in larger datasets?
2. What are the differences between the interquartile range and standard deviation?
3. In what situations is IQR a better measure of spread than the range?
4. How would you apply IQR in box plot visualization?
5. Can IQR analysis be used to compare the variability of two different datasets?

Would you like further clarification or details on any step?