IQR (Interquartile Range) - Algebra-1

1. Fundamental Concepts

Interquartile Range (IQR): The full name is "interquartile range", which is a statistical measure describing the dispersion of data. It is calculated as IQR = Q3 - Q1 (where Q3 is the upper quartile and Q1 is the lower quartile).
Meaning: It represents the range covered by the middle 50% of the data in a dataset, also known as "midspread". Specifically, it refers to the interval of data distribution from the 25th percentile to the 75th percentile.
Prerequisite for calculation: It is necessary to first determine Q1 and Q3 of the data, and the calculation of Q1 and Q3 relies on sorting the data and dividing it into quartiles.

Insensitivity to extreme values: Unlike the range (maximum value - minimum value), IQR only focuses on the middle 50% of the data. Therefore, extreme values (outliers) do not interfere with it, and it can more stably reflect the central tendency and dispersion of the data.
Relationship with Box and Whisker Plot: In a box and whisker plot, the two ends of the box correspond to Q1 and Q3 respectively, and the length of the box is the IQR, which intuitively shows the distribution range of the middle 50% of the data.

1.
Data: 3, 5, 7, 9, 11 (sorted)
- Calculate Q1 and Q3: Q1 = 4 (median of the first half: 3, 5), Q3 = 10 (median of the second half: 9, 11)
- Calculate IQR: IQR = Q3 - Q1 = 10 - 4 = 6
2.
Data: 2, 4, 4, 6, 7, 8, 9, 10 (sorted)
- Calculate Q1 and Q3: Q1 = 4 (median of the first half: 2, 4, 4, 6), Q3 = 8.5 (median of the second half: 7, 8, 9, 10)
- Calculate IQR: IQR = 8.5 - 4 = 4.5
3.
Data: 0, 2, 3, 4, 4, 5, 6, 7, 8, 9, 17
- Calculate Q1 and Q3: Q1 = 3 (median of the first half: 0, 2, 3, 4, 4), Q3 = 8 (median of the second half: 6, 7, 8, 9, 17)
- Calculate IQR: IQR = 8 - 3 = 5

Strictly sort the data: The first step in calculating IQR is to sort the data in ascending (or descending) order to ensure the accurate positions of Q1 and Q3.
Calculate Q1 and Q3 step by step:
- First, determine the median to divide the data into two parts (the first half is the data below the median, and the second half is the data above the median);
- Calculate the median of the first half (Q1) and the median of the second half (Q3) respectively.
Verify the rationality of IQR: The result of IQR should be non-negative (since Q3 ≥ Q1). If a negative value appears, it indicates that the calculation of Q1 and Q3 is incorrect, and it is necessary to recheck the sorting and quartile division.
Analyze in combination with actual scenarios: A smaller IQR indicates that the middle 50% of the data is more concentrated; a larger IQR indicates a higher dispersion of the middle data. The characteristics of the data can be further interpreted in combination with the box and whisker plot or the outlier rule.