Measures of Spread: Quartiles and the IQR
Quick Reference
Textbook: Moore, McCabe, and Craig, Introduction to the Practice of Statistics (IPS), 9th edition. Location: Chapter 1 (Looking at Data: Distributions), Section 1.3 (Describing Distributions with Numbers), subsection “Measuring spread: the quartiles and the five-number summary.” Pages: Page numbers pending faculty verification. Role in the course: A center alone hides how tightly or loosely the data cluster. The quartiles and the interquartile range supply a resistant measure of spread, complete the five-number summary, and set up the boxplot and the variance/standard deviation that follow.
Try This First
📋 A two-minute warmup before any definition (click to open)
Two study groups each report their five quiz scores out of 10.
- Group A: 5, 5, 6, 6, 6
- Group B: 1, 3, 6, 9, 10
Both groups have the same median, 6. Without computing anything formal, answer two questions.
- Which group’s scores are bunched close together, and which are spread far apart?
- If a single number had to capture how far apart the middle of each group is, what would you measure?
Compare your reasoning
Group A is bunched: every score sits within one point of the median. Group B is spread: scores reach from 1 all the way to 10.
A natural single number is the distance between the values that sit a quarter of the way in from each end. In Group A that distance is small. In Group B that distance is large. That distance is the interquartile range, and the values a quarter of the way in are the quartiles. The section below makes both ideas precise.
The median answers “what is a typical value.” It says nothing about whether the other values huddle near that typical value or scatter far from it. Two data sets can share a median and still describe very different situations, as the warmup shows.
Spread is the missing half of the description. One way to report spread is the range, the largest value minus the smallest. The range uses only the two most extreme observations, so a single unusual value can stretch it without warning. A measure that looks at the middle of the data instead is steadier. The quartiles mark off the middle, and the interquartile range reports its width.
Prerequisite Hub
Builds on
| Skill | Why it comes first |
|---|---|
| Measures of Center: Mean and Median | A quartile is the median of one half of the data, so the median locate-and-average procedure must already be automatic. |
| Describing a Distribution: Shape, Center, Spread | Spread is one of the three features every distribution is described by. Quartiles and the IQR are a specific way to report spread. |
Unlocks
| Skill | What this enables |
|---|---|
| Measures of Spread: Variance and Standard Deviation | The second major spread measure, used when the mean is the chosen center. |
| Density Curves | A smooth model of a distribution, where quartiles reappear as areas under the curve. |
Cross-course prerequisites: none.
Prerequisite Map
Quick Reference
| Property | Value |
|---|---|
| Chapter.Section | IPS 1.3 |
| Course | MATH380 |
| Difficulty | Intermediate |
| Time | ~30 minutes |
The Quartiles at a Glance
| Symbol | Name | What it marks |
|---|---|---|
| $Q_1$ | First quartile | The median of the lower half of the ordered data; about one quarter of the values fall below it |
| $M$ | Median | The center of the ordered data; about half the values fall below it |
| $Q_3$ | Third quartile | The median of the upper half of the ordered data; about three quarters of the values fall below it |
| $\text{IQR}$ | Interquartile range | The width of the middle half, $Q_3 - Q_1$ |
Procedure in one line: order the data, find the median to split the list, then take the median of each half.
Official Definitions
Quartiles ($Q_1$ and $Q_3$).
To calculate the quartiles, arrange the observations in increasing order and locate the median $M$. The first quartile $Q_1$ is the median of the observations whose position in the ordered list is to the left of the location of the overall median. The third quartile $Q_3$ is the median of the observations to the right of the location of the overall median.
Source: Moore, McCabe, Craig, IPS 9th edition, Section 1.3 (Describing Distributions with Numbers); corroborated by OpenStax Introductory Statistics 2e, quartiles separate the data into quarters (openstax.org).
Interquartile range (IQR).
The interquartile range IQR is the distance between the first and third quartiles, $\text{IQR} = Q_3 - Q_1$. It is the spread of the middle half of the data.
Source: Moore, McCabe, Craig, IPS 9th edition, Section 1.3 (Describing Distributions with Numbers); corroborated by OpenStax Introductory Statistics 2e, the IQR is the range of the middle 50 percent found by subtracting $Q_1$ from $Q_3$ (openstax.org).
Read the quartile definition slowly, because it contains the one rule beginners most often miss. After the overall median splits the ordered list, the lower half is “the observations to the left of the location of the overall median” and the upper half is “the observations to the right.” When the data set has an odd count, the median is an actual data value, and that value belongs to neither half. It is left out when each half’s median is taken.
There is no theorem in this section. The two definitions are the entire toolkit. The skill is following the locate-then-split-then-locate-again procedure carefully and reading $\text{IQR} = Q_3 - Q_1$ as a width, the size of the box that holds the middle half of the data.
Worked Examples
Example 1: An odd count, so the median is a data value
A clinic records the wait time in minutes for seven patients: 12, 4, 18, 7, 9, 15, 6.
Predict first. The values run from 4 to 18. Predict whether the interquartile range will be closer to 5 or closer to 14, then check after the calculation.
Step 1: Order the data.
$$4, \quad 6, \quad 7, \quad 9, \quad 12, \quad 15, \quad 18.$$
Step 2: Locate the overall median. There are seven values, an odd count. The median is the fourth value, $M = 9$.
Step 3: Split into halves, leaving out the median. The lower half is the three values to the left of the median’s position: $4, 6, 7$. The upper half is the three values to the right: $12, 15, 18$. The median value 9 is in neither half.
Step 4: Take the median of each half.
$$Q_1 = \text{median of } (4, 6, 7) = 6, \qquad Q_3 = \text{median of } (12, 15, 18) = 15.$$
Step 5: Compute the IQR.
$$\text{IQR} = Q_3 - Q_1 = 15 - 6 = 9 \text{ minutes}.$$
Check against your prediction. The IQR is 9 minutes, between the two guesses and closer to 14 than to 5. The middle half of the patients waited within a 9-minute window. The full range, $18 - 4 = 14$ minutes, is larger because it includes the two most extreme waits.
Example 2: An even count, so the median splits cleanly between two values
A coffee shop records the number of customers in eight 10-minute windows: 22, 30, 18, 25, 27, 19, 24, 31.
Predict first. Predict whether $Q_1$ will be below or above 20, then check.
Step 1: Order the data.
$$18, \quad 19, \quad 22, \quad 24, \quad 25, \quad 27, \quad 30, \quad 31.$$
Step 2: Locate the overall median. There are eight values, an even count. The median is the average of the fourth and fifth values:
$$M = \frac{24 + 25}{2} = 24.5.$$
Step 3: Split into halves. Because the count is even, the median falls between two values, and every observation lands in one half. The lower half is the first four values, $18, 19, 22, 24$. The upper half is the last four, $25, 27, 30, 31$.
Step 4: Take the median of each half. Each half has four values, an even count, so each quartile is an average of the two middle values of its half.
$$Q_1 = \frac{19 + 22}{2} = 20.5, \qquad Q_3 = \frac{27 + 30}{2} = 28.5.$$
Step 5: Compute the IQR.
$$\text{IQR} = Q_3 - Q_1 = 28.5 - 20.5 = 8 \text{ customers}.$$
Check against your prediction. $Q_1 = 20.5$, just above 20. The middle half of the busy-window counts spans 8 customers.
Example 3: The IQR resists a single extreme value
Start with the wait times from Example 1, where the IQR was 9 minutes and the range was 14 minutes. Now suppose the last patient’s record is corrected to a 90-minute wait by mistake, so the data become 4, 6, 7, 9, 12, 15, 90.
Predict first. The range will clearly grow. Predict whether the IQR changes at all, then check.
Work it out. The data are already ordered. With seven values, the median is still the fourth, $M = 9$. The lower half is still $4, 6, 7$ and the upper half is still $12, 15, 90$.
$$Q_1 = \text{median of } (4, 6, 7) = 6, \qquad Q_3 = \text{median of } (12, 15, 90) = 15.$$
$$\text{IQR} = 15 - 6 = 9 \text{ minutes}.$$
Compare the two spread measures.
| Measure | Original data | After the 90-minute value |
|---|---|---|
| Range | $18 - 4 = 14$ | $90 - 4 = 86$ |
| IQR | $15 - 6 = 9$ | $15 - 6 = 9$ |
Conclusion. Changing the largest value from 18 to 90 multiplied the range by more than six, yet left the IQR unchanged. The quartiles depend on the position of values in the ordered list, not on the size of the most extreme ones. This is what it means to call the IQR a resistant measure of spread. The same property makes the IQR the spread reported alongside the median, while the standard deviation is reported alongside the mean.
Common Misconceptions
the first quartile is the value one quarter of the way along the number line between the smallest and largest values. The quartiles are positions in the ordered list of observations, not points cut from the numerical range. In Example 1 the smallest value is 4 and the largest is 18, so a quarter of the way along the number line would be $4 + 0.25(18 - 4) = 7.5$. The actual first quartile is 6, the median of the lower half. The quartiles count observations, they do not slice the axis. Predict-then-check: compute the “quarter of the way along the axis” value, then compute the real $Q_1$, and confirm the two disagree.
when the count is odd, the median value should be placed in both halves before finding the quartiles. The definition states that the lower half is the observations to the left of the median’s position and the upper half is the observations to the right. The median value itself sits at neither side, so it is excluded from both halves when an odd count makes it an actual data value. In Example 1, including 9 in each half would change $Q_1$ to the median of $(4, 6, 7, 9)$, which is 6.5, and would change $Q_3$ to the median of $(9, 12, 15, 18)$, which is 13.5, giving a different IQR. Different software packages handle this boundary case in different ways, so always follow the textbook rule for this course: exclude the overall median from both halves.
a larger IQR always means a larger range, and a small range guarantees a small IQR. The IQR measures the width of the middle half, while the range measures the full extent. Example 3 shows them coming apart: the range jumps from 14 to 86 while the IQR holds at 9. The IQR ignores the extreme values that the range is built from. A distribution can have a wide range from one outlier and still have a narrow, tightly clustered middle. The two numbers report different things on purpose.
Practice Problems
Fill in each blank using the official definitions.
(a) The interquartile range is computed as $\text{IQR} = \rule{2cm}{0.4pt}$.
(b) The first quartile $Q_1$ is the median of the observations whose position is to the $\rule{2cm}{0.4pt}$ of the overall median.
(c) The IQR describes the spread of the middle $\rule{2cm}{0.4pt}$ of the data.
The ages of nine members of a hiking club are 21, 34, 27, 45, 30, 52, 38, 29, 41. Find $Q_1$, $M$, $Q_3$, and the IQR.
The prices in dollars of six used textbooks are 35, 12, 48, 20, 27, 41. Find $Q_1$, $M$, $Q_3$, and the IQR.
A neighborhood’s home prices are strongly right-skewed, with a few very expensive houses pulling the high end far out. A report needs one center and one matching spread. Which pairing is the most appropriate?
(A) Report the mean for the center and the IQR for the spread.
(B) Report the median for the center and the IQR for the spread.
(C) Report the median for the center and the range for the spread.
(D) Report the mean for the center and the range for the spread.
A data set has five values: 2, 5, 8, 11, 20. A classmate computes $Q_1$ by including the overall median in the lower half, while the textbook rule excludes it. Find $Q_1$ both ways, and state which one this course uses and why the two disagree.
Construct a set of eight whole-number test scores between 0 and 100 that has a median of 70 and an interquartile range of exactly 20. Show the quartiles for your set, and explain the two free choices you made.
Mental Model
The box that holds the middle half.
Picture the ordered data as a row of dots on a number line. The median is one cut that splits the row in two. Two more cuts, one in the center of each half, mark $Q_1$ and $Q_3$. Those two outer cuts fence off the middle half of the data. The interquartile range is the width of that fence, $Q_3 - Q_1$.
This is exactly the box drawn in a boxplot. A narrow box means the middle of the data is tightly packed. A wide box means it is loose. Because the fence ignores the values beyond $Q_1$ and $Q_3$, one or two extreme observations can sit far outside the box without changing its width at all. That is the resistance shown in Example 3, seen as a picture.
Mastery Checklist
Novice (Level 1-2):
Competent (Level 3-4):
Proficient (Level 5):
Connections
Looking back:
- Measures of Center: Mean and Median supplies the median procedure that each quartile reuses on a half of the data.
- Describing a Distribution: Shape, Center, Spread frames spread as one of the three features that describe any distribution.
Looking ahead:
- Measures of Spread: Variance and Standard Deviation gives the spread measure paired with the mean for symmetric data.
- Density Curves reintroduces the quartiles as areas under a smooth model of a distribution.
Real-world connections:
- Standardized test reports often publish the 25th and 75th percentiles, which are $Q_1$ and $Q_3$, to show the middle half of scorers.
- Salary surveys report the interquartile range so a few very high earners do not distort the sense of a typical spread.
Resources
| Resource | Where it points |
|---|---|
| IPS 9th edition, Section 1.3 (Describing Distributions with Numbers) | Moore, McCabe, Craig, IPS 9e, Chapter 1, Section 1.3. The primary source for the quartile and IQR definitions in this skill. |
| OpenStax Introductory Statistics 2e, 2.3 Measures of the Location of the Data | openstax.org. Free open companion treatment of quartiles and percentiles. |
| OpenStax Introductory Statistics 2e, 2.4 Box Plots | openstax.org. The boxplot that displays the five-number summary built from the quartiles. |
| Previous | Up | Next |
|---|---|---|
| Measures of Center: Mean and Median | Skills Index | Measures of Spread: Variance and Standard Deviation |
Last updated: 2026-06-16