Range and Interquartile Range
Learn how to measure the spread of data using range and IQR.
Definition
Range
Interquartile Range (IQR)
- (First Quartile) = median of the lower half
- (Third Quartile) = median of the upper half
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Worked Examples
Find the range of these test scores:
Identify the maximum value
Looking at all values: → Maximum
Identify the minimum value
The smallest score in the set → Minimum
Calculate the range
→
Answer: The range is points, meaning the scores span 27 points from lowest to highest.
Common Mistakes
Not ordering the data before finding quartiles
Why it's wrong: Quartiles are based on position in ordered data. Unordered data gives wrong Q1 and Q3 values.
Correct: Always sort the data from smallest to largest first, then find the median and quartiles.
Including the median in both halves when finding Q1 and Q3
Why it's wrong: When you have an odd number of data points, the median should be excluded from both halves.
Correct: For odd-sized datasets, split the data excluding the median, then find Q1 and Q3 from those halves.
Confusing range with IQR
Why it's wrong: They measure different things: range uses extreme values, IQR uses quartiles.
Correct: Range Max Min. IQR . Both measure spread, but IQR focuses on the middle 50%.
Interactive Visual
Box Plot Builder
Click on parts of the box plot to learn what they represent.
Click on numbers to select them. Adjust the range to explore different values.
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
15 problemsWhat is the range of this data set: ?
Why It Matters
- Weather: A city with temperatures ranging from to has consistent weather, while to shows extreme variation
- Sports: Consistent athletes have a small IQR in their scores; inconsistent ones have large spreads
- Quality Control: Manufacturers use IQR to ensure products are uniform
- Grades: Teachers analyze test score spread to understand class performance
Real World Applications
Salary Comparisons
Companies use IQR to understand typical salary ranges without being skewed by executive pay.
Example:
If salaries at a company range from 30,000 to 500,000 euros, but the IQR is 35,000 to 55,000 euros, most employees earn in that narrower range.
Employee salaries (in thousands): 32, 38, 42, 45, 48, 52, 58, 250. Find the IQR.
What is the interquartile range of these salaries?
Step 1: Write the mathematical expression
First find and , then calculate :
Athletic Performance
Coaches analyze athlete consistency using IQR of performance metrics.
Example:
A sprinter's 100m times: seconds. A small IQR indicates consistent performance.
A basketball player's points per game: 12, 15, 18, 20, 22, 25, 28, 30. Find the range and IQR.
What is the range and IQR?
Step 1: Write the mathematical expression
Calculate both measures of spread:
Key Takeaways
- 1Range Maximum Minimum (measures total spread)
- 2IQR (measures spread of middle 50%)
- 3 is the median of the lower half; is the median of the upper half
- 4IQR is resistant to outliers; range is not
- 5A smaller range or IQR indicates data points are closer together (less variability)
Frequently Asked Questions
Glossary
- Range
- The difference between the maximum and minimum values in a dataset
- Interquartile Range (IQR)
- The difference between the third quartile () and first quartile (), representing the spread of the middle 50% of data
- Quartile
- Values that divide an ordered dataset into four equal parts
- Q1 (First Quartile)
- The median of the lower half of the data; 25% of values fall below it
- Q3 (Third Quartile)
- The median of the upper half of the data; 75% of values fall below it
- Outlier
- A data point that is significantly different from other values in the dataset