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Teacher Guide: Range and Interquartile Range

Learn how to measure the spread of data using range and IQR.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Spread & Variability. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Calculate the range of a dataset
  • Identify quartiles (Q1, Q2, Q3) in ordered data
  • Calculate the interquartile range (IQR)
  • Compare range and IQR as measures of spread
  • Explain why IQR is resistant to outliers
Prerequisites
  • • Understanding of median and how to find it
  • • Ability to order numbers from least to greatest
  • • Basic subtraction skills
  • • Familiarity with the concept of data sets
Discussion Starters
  • 1. When would you prefer to use range over IQR, and vice versa?
  • 2. A company reports the average salary is 60,000 dollars with an IQR of 10,000 dollars and a range of 500,000 dollars. What does this tell you?
  • 3. How would adding an outlier affect the range? The IQR?
  • 4. If two datasets have the same range but different IQRs, what does that tell you about their distributions?
Common Misconceptions

Thinking a larger range always means more variability in all the data

Remediation: Show examples where one outlier creates a large range while most data is tightly clustered. Compare with IQR to show true spread.

Believing Q1 and Q3 must always be values from the data set

Remediation: Show a half with an even number of values, like 12, 15, 18, 22: Q1 is the average of the two middle values, (15+18)÷2=16.5(15 + 18) \div 2 = 16.5, which is not in the list. Quartiles are cut points, not necessarily data values.

Differentiation Ideas

For Struggling Students:

  • • Use small datasets (5-7 values) with whole numbers
  • • Provide a step-by-step checklist: order data, find median, split, find Q1, find Q3, subtract
  • • Use number lines to visualize quartile positions

For On-Level Students:

  • • Work with datasets of 10-15 values
  • • Include datasets with and without outliers for comparison
  • • Interpret range and IQR in context (test scores, temperatures)

For Advanced Students:

  • • Analyze datasets where mean, median, range, and IQR tell different stories
  • • Explore the 1.5 IQR rule for identifying outliers
  • • Compare distributions using five-number summaries
Standards Alignment
  • 6.SP.B.5c (CCSS.MATH.CONTENT.6.SP.B.5.C)

    Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation)

  • 6.SP.B.4 (CCSS.MATH.CONTENT.6.SP.B.4)

    Display numerical data in plots on a number line, including dot plots, histograms, and box plots

Lesson Resources
  • visualInteractive Box Plot

    Students explore how range and IQR appear on a box plot

  • activityOutlier Investigation

    Compare range and IQR with and without outliers

  • worksheetReal Data Analysis

    Calculate range and IQR for sports statistics, weather data, and test scores

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Range and interquartile range (IQR) are measures of spread: they tell you how spread out the data values are.

Range

The range is the difference between the largest and smallest values:
Range=Maximum−Minimum\text{Range} = \text{Maximum} - \text{Minimum}

Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of the data:
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1
Where:
  • Q1Q_1 (first quartile, also called lower quartile) = median of the lower half
  • Q3Q_3 (third quartile, also called upper quartile) = median of the upper half
How we split the data (in this whole lesson and in the box plot): order the values and find the median. With an odd number of values, leave the median out of both halves; with an even number, split the list exactly in the middle. Some textbooks use a slightly different rule, so their quartiles can differ a little.
The IQR is more reliable than the range when there are extreme values (outliers), because it only looks at the middle half of the data.

Worked Examples

Find the range of these test scores: 72,85,91,68,77,83,9572, 85, 91, 68, 77, 83, 95
1

Identify the maximum value

Looking at all values: 72,85,91,68,77,83,9572, 85, 91, 68, 77, 83, 95 → Maximum =95= 95
2

Identify the minimum value

The smallest score in the set → Minimum =68= 68
3

Calculate the range

Range=95−68\text{Range} = 95 - 68 → Range=27\text{Range} = 27

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 =Q3−Q1= Q_3 - Q_1. Both measure spread, but IQR focuses on the middle 50%.

Why It Matters

Understanding data spread is essential in real life:
  • Weather: A city with temperatures ranging from 10°C10°C to 15°C15°C has consistent weather, while −5°C-5°C to 35°C35°C 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
The range gives a quick overview, but the IQR tells you about typical variation without being skewed by outliers.

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 dollars, but the middle 50% of salaries lie between 35,000 and 55,000 dollars (IQR = 20,000 dollars), most employees earn in that narrower band.

1Try It Yourself

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 Q1Q_1 and Q3Q_3, then calculate Q3−Q1Q_3 - Q_1:

Athletic Performance

Coaches analyze athlete consistency using IQR of performance metrics.

Example:

A sprinter's 100m times: 11.2,11.4,11.3,11.5,11.4,11.6,11.311.2, 11.4, 11.3, 11.5, 11.4, 11.6, 11.3 seconds. A small IQR indicates consistent performance.

2Try It Yourself

A basketball player's points per game: 12, 15, 18, 20, 22, 25, 28, 30. Find the range and IQR.

What is the IQR of the points per game?

Step 1: Write the mathematical expression

Calculate both measures of spread:

Key Takeaways

  • 1Range == Maximum −- Minimum (measures total spread)
  • 2IQR =Q3−Q1= Q_3 - Q_1 (measures spread of middle 50%)
  • 3Q1Q_1 is the median of the lower half; Q3Q_3 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

Why is IQR better than range for some datasets?

The IQR is barely affected by extreme values (outliers) that can inflate the range. It focuses on where the middle 50% of the data lies, giving a better picture of the typical spread.

What does it mean if the IQR is 0?

An IQR of 0 means the middle 50% of your data are all the same value. This indicates very little variability in the central portion of your dataset.

How do I know if I should include the median when finding Q1 and Q3?

With an odd number of data points, exclude the median from both halves. With an even number, split the data exactly in half and include all values.

Glossary

Range
The difference between the maximum and minimum values in a dataset
Interquartile Range (IQR)
The difference between the third quartile (Q3Q_3) and first quartile (Q1Q_1), 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; about 25% of the values lie below it
Q3 (Third Quartile)
The median of the upper half of the data; about 75% of the values lie below it
Outlier
A data point that is significantly different from other values in the dataset

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