The Five-Number Summary, Range, and IQR
The five-number summary describes data spread using five values: minimum, Q1 (median of lower half), Q2 (median), Q3 (median of upper half), and maximum. For the ordered set 1, 3, 5, 8, 9, 10: Min=1, Q1=3, Median=6.5, Q3=9, Max=10. The IQR = Q3 - Q1 = 9 - 3 = 6, measuring the spread of the middle 50%. A small IQR means consistent data; a large IQR means widely scattered. This framework from Reveal Math, Course 1, Module 10 is essential for 6th grade box plots and data comparison.
Key Concepts
Property To describe the spread (variability) of data, we calculate specific distances. Range: Maximum value minus the Minimum value. 5 Number Summary: A breakdown of the data into four equal quarters using five values: Minimum, Q1 (median of the lower half), Median (Q2), Q3 (median of the upper half), and Maximum. Interquartile Range (IQR): The difference between the third and first quartiles ($Q 3 Q 1$). It measures the spread of the middle 50% of the data.
Examples Calculating the 5 Number Summary: Given an ordered set {1, 3, 5, 8, 9, 10}. Min: 1. Max: 10. Median: The average of 5 and 8 is 6.5. Q1 (median of lower half {1, 3, 5}): 3. Q3 (median of upper half {8, 9, 10}): 9. The summary is (1, 3, 6.5, 9, 10). Interpreting IQR: Player A has an IQR of 3 points. Player B has an IQR of 12 points. Player A is the more consistent scorer because their typical, middle 50% of scores vary within a much tighter window.
Explanation Why do we need the IQR if we already have the total Range? The total Range is easily distorted by just one extreme outlier. The IQR acts as a statistical filter: it chops off the lowest 25% and the highest 25% of the data, focusing strictly on the core middle 50%. A smaller IQR means your data is consistent and tightly clustered; a larger IQR means your data is widely scattered.
Common Questions
What is the five-number summary?
The five-number summary consists of minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), and maximum. These five values capture the full spread of a data set.
How do I find Q1 and Q3?
After finding the median, split the data into a lower half and an upper half. Q1 is the median of the lower half, and Q3 is the median of the upper half.
What is the IQR and how do I calculate it?
The Interquartile Range (IQR) = Q3 minus Q1. It measures the spread of the middle 50% of data, ignoring the extreme top and bottom quarters.
What does a large vs. small IQR tell you about data?
A small IQR means the middle 50% of values are tightly clustered and consistent. A large IQR means they are widely spread and variable.
Why is the IQR more useful than the total range?
The total range is distorted by a single outlier. The IQR cuts off the extremes and focuses only on the core middle 50% of data, making it a more reliable measure of typical spread.
When do 6th graders learn the five-number summary?
Module 10 of Reveal Math, Course 1 covers the five-number summary and IQR in the Statistical Measures and Displays unit.