How to Finding Range, Quartile and Interquartile Range

How to find range, quartiles, and interquartile range, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Once you know the mean, median, and mode, the next step in statistical analysis is measuring spread — how tightly or loosely the data is clustered. The range, quartiles, and interquartile range (IQR) are the key spread measures you need for the GED Math test. They also form the backbone of box-and-whisker plots.

What Are Range, Quartiles, and IQR?

• Range = \(\color{blue}{\text{ Maximum } – \text{ Minimum }}\)   (total spread)
• Q1 \(\color{blue}{(\text{ First Quartile }) = \text{ median }}\) of the lower half of the data (25th percentile)
• Q2 \(\color{blue}{(\text{ Second Quartile }) = \text{ median }}\) of the entire data set (50th percentile)
• Q3 \(\color{blue}{(\text{ Third Quartile }) = \text{ median }}\) of the upper half of the data (75th percentile)
• IQR = \(\color{blue}{Q3 – Q1}\)   (spread of the middle 50% of the data)

How to Find Quartiles and IQR

Step-by-Step Method

1. Sort the data from least to greatest.
2. Find the median (Q2): this is the middle value (or average of two middle values for even count).
3. Find Q1: find the median of all values below Q2.
4. Find Q3: find the median of all values above Q2.
5. Compute \(\color{blue}{\text{ IQR } = Q3 – Q1}\).

Key Notes

• If the data set has an odd number of values, exclude the median when finding Q1 and Q3.
• IQR represents the range of the “middle half” of the data and is more resistant to outliers than the full range.

Step-by-Step Summary

1. Sort the data.
2. \(\color{blue}{\text{ Range } = \text{ max } – \text{ min }}\).
3. \(\color{blue}{Q2 = \text{ median }}\) of the full data set.
4. \(\color{blue}{Q1 = \text{ median }}\) of the lower half; \(\color{blue}{Q3 = \text{ median }}\) of the upper half.
5. \(\color{blue}{\text{ IQR } = Q3 – Q1}\).

Watch: Interquartile Range (IQR) (Video Lesson)

Math with Mr. J walks through finding the IQR with step-by-step examples:

Worked Examples

Example 1: Find the range, Q1, Q3, and IQR of: 4, 7, 10, 12, 15, 18, 20.

Sorted: 4, 7, 10, 12, 15, 18, 20
\(\color{blue}{\text{ Range } = 20 – 4}\) = 16
Q2 \(\color{blue}{(\text{ median }) = 12}\) (4th of 7)
Lower half: 4, 7, 10 → \(\color{blue}{Q1 = \frac{(7+10)}{2}}\) = 8.5
Upper half: 15, 18, 20 → \(\color{blue}{Q3 = \frac{(15+18)}{2}}\) = 16.5
\(\color{blue}{\text{ IQR } = 16.5 – 8.5}\) = 8

Example 2: Find the IQR of: 2, 5, 6, 8, 10, 11, 14, 16.

\(\color{blue}{Q1 = \frac{(5+6)}{2} = 5.5}\); \(\color{blue}{Q3 = \frac{(11+14)}{2} = 12.5}\)
\(\color{blue}{\text{ IQR } = 12.5 – 5.5}\) = 7

Example 3: A data set has \(\color{blue}{Q1 = 20}\) and \(\color{blue}{Q3 = 35}\). What is the IQR?

\(\color{blue}{\text{ IQR } = 35 – 20}\) = 15

Example 4: Scores: 60, 65, 70, 75, 80, 85, 90, 95. Find Q1, Q3, and IQR.

\(\color{blue}{Q1 = \frac{(65+70)}{2} = 67.5}\); \(\color{blue}{Q3 = \frac{(85+90)}{2} = 87.5}\)
\(\color{blue}{\text{ IQR } = 87.5 – 67.5}\) = 20

More Practice: Calculating IQR (Video)

Khan Academy’s 6th-grade data and statistics lesson explains the IQR with visual dot-plot examples:

Exercises

1. Find the range of: 13, 27, 5, 41, 9, 33, 18.
2. Find Q1, Q3, and IQR of: 3, 6, 7, 9, 12, 15, 18, 21.
3. A data set has \(\color{blue}{\text{ range } = 30}\) and \(\color{blue}{\text{ min } = 12}\). What is the max?
4. Find the IQR of: 10, 20, 30, 40, 50.
5. Data: 1, 4, 6, 8, 10, 12, 14. Find Q1, Q3, IQR, and range.
6. If \(\color{blue}{Q1 = 15}\) and \(\color{blue}{\text{ IQR } = 12}\), what is Q3?

Answers

1. \(\color{blue}{41 – 5}\) = 36
2. \(\color{blue}{Q1=\frac{(6+7)}{2}=6.5}\); \(\color{blue}{Q3=\frac{(15+18)}{2}=16.5}\); IQR=10
3. \(\color{blue}{12 + 30}\) = 42
4. Sorted: 10,20,30,40,50; \(\color{blue}{Q1=20}\); \(\color{blue}{Q3=40}\); IQR=20
5. \(\color{blue}{Q1=\frac{(4+6)}{2}=5}\); \(\color{blue}{Q3=\frac{(10+12)}{2}=11}\); IQR=6; \(\color{blue}{\text{ Range }=14-1}\)=13
6. \(\color{blue}{Q3 = 15 + 12}\) = 27

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Common Mistakes When Finding the IQR

Most wrong IQR answers come from one of a few slips. Watch for these:

• Not sorting first. Quartiles only make sense once the values are in order from least to greatest.
• Putting the median in both halves. With an odd number of values, leave the middle value out before you split. Mixing the two ways of splitting is the most common reason two students get different answers.
• Subtracting the wrong numbers. The IQR is Q3 – Q1, not maximum – minimum. That second one is the range.
• Taking the middle of a half incorrectly. If a half has an odd count, its median is the middle value itself. Average only when a half has an even count.

Odd example: 3, 5, 7, 8, 12, 13, 14, 18, 21 (9 values). The median is 12. Leave it out. Lower half: 3, 5, 7, 8, so Q1 = (5 + 7) ÷ 2 = 6. Upper half: 13, 14, 18, 21, so Q3 = (14 + 18) ÷ 2 = 16. IQR = 16 – 6 = 10.

If you had kept 12 in both halves by mistake, you would get Q1 = 7 and Q3 = 14, and an IQR of 7. Some calculators and textbooks do use that method on purpose, so always check which rule your class follows. On this page, we leave the median out.

Frequently Asked Questions

What does a small IQR mean about a data set?

A small IQR means the middle half of the data is clustered closely together — the data is relatively consistent. A large IQR means the middle half is spread out — the data has more variability.

Do I include the median when finding Q1 and Q3?

For an odd number of values, exclude the median (Q2) before splitting the data into lower and upper halves. For an even number of values, simply split the data in half without excluding any value.

How is IQR used to find outliers?

The standard rule: any value below \(\color{blue}{Q1 – 1.5 \times \text{ IQR }}\) or above \(\color{blue}{Q3 + 1.5 \times \text{ IQR }}\) is considered an outlier. This is known as Tukey’s fence method.

What is the difference between range and interquartile range?

The range is the maximum minus the minimum, so it uses only the two most extreme values. The interquartile range is Q3 minus Q1, which measures the spread of the middle half of the data. One very large or very small value can change the range a lot, but it barely moves the IQR.

How do you find the IQR when there is an even number of values?

Sort the data and split it into two equal halves; no value is left out. Q1 is the median of the lower half and Q3 is the median of the upper half. For 2, 5, 6, 8, 10, 11, 14, 16, the halves are 2, 5, 6, 8 and 10, 11, 14, 16, so Q1 = 5.5, Q3 = 12.5, and IQR = 7.

Why do different calculators give different IQR answers?

There is more than one accepted way to find quartiles. Some methods leave the median out of both halves when the count is odd, and others keep it in. Spreadsheets and calculators may use yet another formula. The answers are usually close but not identical, so use the method your teacher or test expects.

Can the interquartile range be zero?

Yes. The IQR is zero when Q1 and Q3 are equal, which happens when the middle half of the data is all the same value. For example, in 4, 5, 5, 5, 5, 5, 9, Q1 and Q3 are both 5, so the IQR is 0. It can never be negative, because Q3 is never less than Q1.

Related Topics

• How to Identify an Outlier
• Mean, Median, Mode, and Range
• Distributions in Line Plot
• Line Plot
• How to Solve the Frequency Distribution Table