Why do we need both range and IQR if they both measure spread?

Range is the quickest way to see how far apart the extremes are, but it ignores everything in between. IQR tells you how spread out the middle 50 percent of your data is and is not fooled by outliers. If your data has extreme values, the range will be large even if most of your data is bunched together. Using both gives you a complete picture: the range shows the full span, and the IQR shows where most of the data actually sits.

How do I find quartiles if my data set has an even number of values?

Arrange the data in order and find the median as usual (the average of the two middle values). The lower half includes all values up to but not including the median, and the upper half includes all values after the median. Then find the median of each half. For example, in the data set 2, 5, 7, 9, 11, 14, the median is (7 + 9) / 2 = 8. The lower half is 2, 5, 7 (median Q1 = 5), and the upper half is 9, 11, 14 (median Q3 = 11). So IQR = 11 − 5 = 6.

What does it mean if a box plot is skewed?

A skewed box plot has one whisker or tail that is longer than the other, or the median line inside the box is off to one side rather than centered. This means the distribution is not symmetric: more data is bunched on one side, and the tail stretches further out on the other. For example, if the right whisker is much longer than the left, the data is right-skewed (positively skewed), with a few unusually high values pulling the tail to the right. This often happens in real life—for example, house prices are often right-skewed because most homes are in a typical price range but a few luxury homes pull the distribution to the right.

When is an outlier an outlier?

A common rule is that a value is an outlier if it lies more than 1.5 times the IQR below Q1 or above Q3. Specifically, a value is a low outlier if it is less than Q1 - 1.5 × IQR, or a high outlier if it is greater than Q3 + 1.5 × IQR. For example, if Q1 = 50, Q3 = 70, then IQR = 20, and 1.5 × IQR = 30. A value below 50 - 30 = 20 or above 70 + 30 = 100 is flagged as an outlier. On a box plot, outliers are often shown as separate dots, and the whiskers extend only to the farthest non-outlier values.

M6MATH-10.4

Measures of Spread & Summarizing Data

Learn how to find range, interquartile range, and create box plots to describe how spread out data is and what it tells us.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Measures of Spread & Summarizing Data, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Measures of Spread & Summarizing Data free

What this lesson covers

You already know how to find the mean and median—numbers that describe the center of a data set. But the center tells only part of the story. Two classes might have the same average test score, yet one class has scores clustered tightly around that average while the other has scores all over the place. To really understand data, you need to describe the spread: how scattered or bunched together the values are. This lesson teaches you three tools to measure spread—range, interquartile range, and box plots—and how to use them to summarize what a data set actually shows.

What Is Spread and Why It Matters

Spread refers to how far apart data values are from each other. Imagine two runners training for a race over ten days. Runner A completes 5 miles, 5.1 miles, 4.9 miles, 5 miles, and 5.2 miles. Runner B completes 2 miles, 8 miles, 3 miles, 7 miles, and 5 miles. Both have a mean of about 5 miles, but Runner A is consistent (low spread) while Runner B is all over the place (high spread). In the real world, spread matters: a restaurant with consistent wait times is more reliable than one where waits vary wildly, even if the average is the same. When you summarize a data set, you describe both its center and its spread to give a complete picture.

Range: The Simplest Measure of Spread

The range is the difference between the largest and smallest values in a data set.

\text{Range} = \text{Maximum} - \text{Minimum}

Range is easy to compute and gives you a quick sense of how far apart the extreme values are. For Runner A's distances (5, 5.1, 4.9, 5, 5.2), the range is

5.2 - 4.9 = 0.3

miles. For Runner B's distances (2, 8, 3, 7, 5), the range is

8 - 2 = 6

miles. The much larger range for Runner B confirms that this runner's performance is inconsistent.

However, range has a limitation: it depends only on the two extreme values and ignores everything in between. A single very high or very low outlier can make the range huge even if most of the data is tightly bunched. That's why we also use the interquartile range.

Interquartile Range (IQR) and Quartiles

The interquartile range describes the spread of the middle half of your data, so it is not fooled by extreme outliers.

First, you need to find quartiles. Arrange your data in order and find the median (the middle value). The median divides the data into two halves. The first quartile (Q1) is the median of the lower half, and the third quartile (Q3) is the median of the upper half. The second quartile (Q2) is just another name for the median itself.

Then:

\text{IQR} = Q3 - Q1

For example, suppose test scores are: 72, 75, 78, 82, 85, 88, 90, 92, 95. First, find the median: it's 85 (the middle value). The lower half is 72, 75, 78, 82; Q1 is the median of this half, which is

(75 + 78) / 2 = 76.5

. The upper half is 88, 90, 92, 95; Q3 is the median of this half, which is

(90 + 92) / 2 = 91

. So IQR =

91 - 76.5 = 14.5

. The IQR tells you that the middle 50 percent of scores are spread across a 14.5-point range.

Box Plots: Visualizing Spread and Outliers

A box plot is a diagram that shows the five-number summary of a data set: the minimum, Q1, median (Q2), Q3, and maximum. It gives you a visual snapshot of the spread and shape of your distribution.

To create a box plot: First, find all five numbers for your data set. On a number line, mark the minimum and maximum with short lines (whiskers). Draw a rectangle (box) from Q1 to Q3 with a line inside at the median. Connect the whiskers to the box.

A box plot instantly shows whether data is symmetric (the median line is near the center of the box) or skewed (the median is off to one side). If one whisker is much longer than the other, that tail of the distribution stretches further out. A common rule flags a value as an outlier if it lies more than 1.5 times the IQR below Q1 or above Q3. Many box plots mark outliers separately with dots and draw whiskers only to the farthest non-outlier values.

Box plots are especially powerful for comparing distributions: place two or more side by side on the same number line and you can instantly see which distribution is more spread out, where they overlap, and how they are shaped.

Summarizing Data in Context

When you describe a data set, always tie your numbers back to the situation. Don't just say 'the range is 20'; explain what that means: 'the test scores ranged from 70 to 90 points, showing a spread of 20 points across the class.' Use measures of center and spread together. For instance: 'The median time to solve the puzzle was 3.2 minutes (center), and the interquartile range was 1.5 minutes (spread), indicating that most students solved it in 1.85 to 3.35 minutes.' This kind of summary is far more informative than any single number. When you present a box plot, note whether the distribution is roughly symmetric or skewed, and explain what that tells you about the real-world variable—for example, whether student performance was fairly consistent or whether a few students struggled much more than the rest.

Key terms

Spread.: A measure of how far apart or how bunched together data values are; also called variability or dispersion.
Range.: The difference between the maximum and minimum values in a data set. It measures spread but is sensitive to outliers.
Quartile.: A value that divides ordered data into four equal parts. Q1 is the 25th percentile, Q2 (the median) is the 50th, and Q3 is the 75th percentile.
Interquartile Range (IQR).: The difference between the third quartile and the first quartile: $\text{IQR} = Q3 - Q1$ . It measures the spread of the middle 50 percent of the data.
Five-Number Summary.: The set of five values that describe a data set: minimum, Q1, median, Q3, and maximum.
Box Plot.: A visual diagram that displays the five-number summary, showing the median, quartiles, and extreme values, useful for comparing distributions.
Outlier.: A data value that is unusually far from the other values in the set, often defined as a value more than 1.5 times the IQR below Q1 or above Q3.
Skewed.: A description of a distribution that is not symmetric; one tail is longer than the other, pulling the median away from the center of the box.

Worked example

A teacher recorded how long it took 11 students to complete a quiz (in minutes): 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 18. Find the five-number summary, calculate the IQR, and describe what the data tells you.

Start by arranging the data in order. It's already in order: 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 18.

Find the minimum and maximum: Minimum = 8 minutes, Maximum = 18 minutes.

Find the median (Q2): With 11 values, the median is the 6th value: 12 minutes.

Find Q1: This is the median of the lower half (values below the median). The lower half is 8, 9, 10, 10, 11. The median of these 5 values is the 3rd value: 10 minutes.

Find Q3: This is the median of the upper half (values above the median). The upper half is 13, 14, 15, 18. The median of these 4 values is

(14 + 15) / 2 = 14.5

minutes.

Calculate the IQR:

\text{IQR} = Q3 - Q1 = 14.5 - 10 = 4.5

minutes.

Summarize: The five-number summary is 8, 10, 12, 14.5, 18. The range is

18 - 8 = 10

minutes, meaning quiz times spanned a 10-minute window. The IQR is 4.5 minutes, so the middle 50 percent of students finished between 10 and 14.5 minutes—a fairly tight window. The median time was 12 minutes. The data is slightly skewed toward longer times (the upper whisker would be longer than the lower), suggesting a few students took noticeably longer than the rest.

Practice questions

A sports team recorded the points scored in each game over a season. The five-number summary is: minimum = 45, Q1 = 62, median = 71, Q3 = 78, maximum = 92. What is the interquartile range?

Answer: 16

The interquartile range is Q3 − Q1 = 78 − 62 = 16 points. This tells you that the middle 50 percent of games had scores spread across a 16-point range. The range of all games (92 − 45 = 47) is much larger because of the high and low outliers, but the IQR focuses on the tightly bunched middle of the distribution.

The box plot below shows quiz scores for two classes.

[Imagine a box plot with Class A: box from 70 to 85, median line at 78, whiskers to 65 and 92; Class B: box from 75 to 88, median line at 80, whiskers to 72 and 89.]

Based on the box plots, which class had more consistent quiz scores and why?

Answer: Class B had more consistent scores because the IQR is smaller and the whiskers are shorter.

Class B's box is narrower (smaller IQR) and the whiskers are shorter, meaning values are more tightly clustered. Class A has a wider spread from Q1 to Q3 and longer whiskers reaching to 65 and 92, showing more variation. When a box plot is taller and more spread out, the distribution is less consistent.

A data set has a range of 50 and an IQR of 8. Explain why the range alone would give you an incomplete picture of the spread.

Answer: The range tells you the difference between the highest and lowest values (50), but the IQR tells you how spread out the middle 50 percent of data is (8). A large range might be caused by just one or two extreme outliers; the IQR is not fooled by extreme values and shows you the spread where most of the data actually is. Together, they tell you that the extremes are far apart, but most of the data is relatively tightly bunched.

This question tests understanding of when and why we use different measures. The range depends only on the two extreme values, so a single very high or very low outlier can make it large even if the rest of the distribution is tightly bunched. The IQR focuses on the middle 50 percent and is resistant to outliers. In this case, the large range but small IQR suggests the data has a few extreme values with most values clustered in a narrow band—a very different picture than if both the range and IQR were large.

FAQ

Why do we need both range and IQR if they both measure spread?: Range is the quickest way to see how far apart the extremes are, but it ignores everything in between. IQR tells you how spread out the middle 50 percent of your data is and is not fooled by outliers. If your data has extreme values, the range will be large even if most of your data is bunched together. Using both gives you a complete picture: the range shows the full span, and the IQR shows where most of the data actually sits.
How do I find quartiles if my data set has an even number of values?: Arrange the data in order and find the median as usual (the average of the two middle values). The lower half includes all values up to but not including the median, and the upper half includes all values after the median. Then find the median of each half. For example, in the data set 2, 5, 7, 9, 11, 14, the median is (7 + 9) / 2 = 8. The lower half is 2, 5, 7 (median Q1 = 5), and the upper half is 9, 11, 14 (median Q3 = 11). So IQR = 11 − 5 = 6.
What does it mean if a box plot is skewed?: A skewed box plot has one whisker or tail that is longer than the other, or the median line inside the box is off to one side rather than centered. This means the distribution is not symmetric: more data is bunched on one side, and the tail stretches further out on the other. For example, if the right whisker is much longer than the left, the data is right-skewed (positively skewed), with a few unusually high values pulling the tail to the right. This often happens in real life—for example, house prices are often right-skewed because most homes are in a typical price range but a few luxury homes pull the distribution to the right.
When is an outlier an outlier?: A common rule is that a value is an outlier if it lies more than 1.5 times the IQR below Q1 or above Q3. Specifically, a value is a low outlier if it is less than $Q1 - 1.5 \times \text{IQR}$ , or a high outlier if it is greater than $Q3 + 1.5 \times \text{IQR}$ . For example, if Q1 = 50, Q3 = 70, then IQR = 20, and $1.5 \times \text{IQR} = 30$ . A value below $50 - 30 = 20$ or above $70 + 30 = 100$ is flagged as an outlier. On a box plot, outliers are often shown as separate dots, and the whiskers extend only to the farthest non-outlier values.

Learn this with a teacher, not a page

The Crimsora tutor teaches Measures of Spread & Summarizing Data live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

Try a live session free