Range & Interquartile Range

What Is the Range?

The range tells us how spread out a set of numbers is. It’s calculated like this:

$$\text{Range} = \text{Largest value} - \text{Smallest value}$$

Example: For the data set {2, 4, 6, 10}, the range is: $10 - 2 = 8$

Watch out for negative numbers: If the set is {-3, 0, 4, 6}, the range is: $6 - (-3) = 9$

What Are Quartiles?

Quartiles break an ordered data set into four equal parts:

Lower Quartile (Q1): One-quarter (25%) of values lie below this point

Median (Q2): The middle value of the data set (also known as the second quartile)

Upper Quartile (Q3): Three-quarters (75%) of values lie below this point

How Do I Find Quartiles?

Step-by-step:

1. Put the data in order
2. Find the median – this is Q2
3. Split the list into lower and upper halves
   • If there is an even number of values: split evenly
   • If there is an odd number of values: exclude the median from both halves
4. Find the median of the lower half (Q1) and upper half (Q3)

What Is the Interquartile Range (IQR)?

The IQR shows how spread out the middle 50% of the data is:

$$\text{IQR} = Q3 - Q1$$

The IQR is not affected by outliers.

It’s more robust than the range when the data has extremes.

Example 1: Finding the Range

Data set:

3.4, 4.2, 2.8, 3.6, 9.2, 3.1, 2.9, 3.4, 3.2, 3.5, 3.7, 3.6, 3.2, 3.1, 2.9, 4.1, 3.6, 3.8, 3.4, 3.2, 4.0, 3.7, 3.6, 2.8, 3.9, 3.1, 3.0

Highest = 9.2

Lowest = 2.8

$$\text{Range} = 9.2 - 2.8 = 6.4$$

Range = 6.4

Example 2: Finding Quartiles & IQR

Data set:

31, 32, 35, 35, 36, 37, 39, 40, 42, 45,
46, 48, 49, 50, 51, 51, 53, 54, 57, 60

• Total values: 20
• Lower half: first 10 values $\to$ Q1 = median of (36, 37) $\to Q1 = 36.5$
• Upper half: last 10 values $\to$ Q3 = median of (51, 51) $\to Q3 = 51$

Now calculate IQR:

$$\text{IQR} = 51 - 36.5 = 14.5$$