October 7, 2025

What Is The Interquartile Range Iqr

Q: What does a large IQR value mean?

A large IQR indicates that the data points in the middle 50% of the set are spread out widely. Conversely, a small IQR signifies that the middle 50% of data points are clustered closely together.

Q: Is the interquartile range the same as the range?

No. The range is the difference between the highest and lowest values in the entire data set. The IQR is the range of only the middle 50% of the data, making it less sensitive to extreme values (outliers).

Q: How is the IQR used to identify outliers?

A common method is the '1.5 x IQR rule.' A data point is often considered an outlier if it falls below Q1 - (1.5 * IQR) or above Q3 + (1.5 * IQR).

Q: Can the IQR ever be a negative number?

No, the IQR cannot be negative. By definition, the third quartile (Q3) must be greater than or equal to the first quartile (Q1). Therefore, the result of Q3 - Q1 will always be zero or a positive value.

Learn what the interquartile range (IQR) is in statistics, how to calculate it, and why it's a useful measure of variability and spread in a data set.

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What Is the Interquartile Range (IQR)?

The interquartile range (IQR) is a measure of statistical dispersion, or spread, in a data set. It is defined as the difference between the third quartile (Q3) and the first quartile (Q1). The IQR represents the range within which the central 50% of the data points lie.

Section 2: How to Calculate the IQR

To calculate the interquartile range, you first order the data from least to greatest. Then, you find the quartiles, which divide the data into four equal parts. The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half. The formula is simply: IQR = Q3 - Q1.

Section 3: A Practical Example

Consider the following data set of test scores: {70, 75, 80, 82, 85, 90, 95}. The median (Q2) is 82. The lower half of the data is {70, 75, 80}, so its median, Q1, is 75. The upper half is {85, 90, 95}, so its median, Q3, is 90. Therefore, the IQR = Q3 - Q1 = 90 - 75 = 15.

Section 4: Importance of the IQR

The primary importance of the IQR is its resistance to outliers. Unlike the total range (maximum value minus minimum value), the IQR is not affected by extremely high or low values in the data set. This makes it a more robust and reliable measure of spread, especially for data that is skewed. It is also commonly used to identify potential outliers in box plots.

Frequently Asked Questions

What does a large IQR value mean?

Is the interquartile range the same as the range?

How is the IQR used to identify outliers?

Can the IQR ever be a negative number?