How to Identify an Outlier

An outlier is a data value that is significantly higher or lower than the rest of the data set. Outliers can result from measurement errors, unusual events, or genuinely extreme observations. On the GED Math test you may need to identify outliers, explain their effect on the mean, or determine which values fall outside the normal range using the IQR (Interquartile Range) fence method.

What Is an Outlier?

Informally, an outlier “doesn’t fit” with the rest of the data. Formally, the most common rule is:

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• A value is an outlier if it is more than \(\color{blue}{1.5 \times \text{ IQR }}\) above Q3 or more than \(\color{blue}{1.5 \times \text{ IQR }}\) below Q1.

How to Identify an Outlier Using the IQR Method

Step 1: Find Q1, Q3, and IQR

• Sort the data from least to greatest.
• Q1 = median of the lower half of the data
• Q3 = median of the upper half of the data
• IQR = \(\color{blue}{Q3 – Q1}\)

Step 2: Calculate the fences

• Lower fence = \(\color{blue}{Q1 – 1.5 \times \text{ IQR }}\)
• Upper fence = \(\color{blue}{Q3 + 1.5 \times \text{ IQR }}\)

Step 3: Identify outliers

Any data value below the lower fence or above the upper fence is an outlier.

Step-by-Step Summary

1. Sort the data set.
2. Find Q1 (median of lower half) and Q3 (median of upper half).
3. Calculate \(\color{blue}{\text{ IQR } = Q3 – Q1}\).
4. Lower \(\color{blue}{\text{ fence } = Q1 – 1.5 \times \text{ IQR }}\); Upper \(\color{blue}{\text{ fence } = Q3 + 1.5 \times \text{ IQR }}\).
5. Any value outside these fences is an outlier.

Watch: How to Determine if a Set of Numbers Has Outliers (Video Lesson)

This video demonstrates the full IQR fence method with step-by-step calculations:

Worked Examples

Example 1: Identify any outliers in: 3, 7, 8, 9, 10, 11, 12, 50.

Sorted: 3, 7, 8, 9, 10, 11, 12, 50
Lower half: 3, 7, 8, 9 → \(\color{blue}{Q1 = \frac{(7+8)}{2} = 7.5}\)
Upper half: 10, 11, 12, 50 → \(\color{blue}{Q3 = \frac{(11+12)}{2} = 11.5}\)
\(\color{blue}{\text{ IQR } = 11.5 – 7.5 = 4}\)
Lower \(\color{blue}{\text{ fence } = 7.5 – 1.5(4) = 7.5 – 6 = 1.5}\)
Upper \(\color{blue}{\text{ fence } = 11.5 + 1.5(4) = 11.5 + 6 = 17.5}\)
50 > 17.5 → 50 is an outlier; 3 > 1.5, so 3 is not an outlier.

Example 2: Does the data set 10, 12, 13, 14, 15, 16, 17, 100 contain an outlier?

\(\color{blue}{Q1 = \frac{(12+13)}{2} = 12.5}\); \(\color{blue}{Q3 = \frac{(15+16)}{2} = 15.5}\); \(\color{blue}{\text{ IQR } = 3}\)
Upper \(\color{blue}{\text{ fence } = 15.5 + 4.5 = 20}\)
100 > 20 → 100 is an outlier

Example 3: How does an outlier affect the mean?

Data: 5, 6, 7, 8, 9 → \(\color{blue}{\text{ Mean } = \frac{35}{5} = 7}\)
Replace 9 with 90: 5, 6, 7, 8, 90 → \(\color{blue}{\text{ Mean } = \frac{116}{5} = 23.2}\)
The outlier (90) raised the mean from 7 to 23.2, far from the typical value.

Example 4: Find the outlier: 2, 4, 5, 6, 7, 8, 9, 1.

Sorted: 1, 2, 4, 5, 6, 7, 8, 9
\(\color{blue}{Q1 = \frac{(2+4)}{2} = 3}\); \(\color{blue}{Q3 = \frac{(7+8)}{2} = 7.5}\); \(\color{blue}{\text{ IQR } = 4.5}\)
Lower \(\color{blue}{\text{ fence } = 3 – 6.75}\) = −3.75; Upper \(\color{blue}{\text{ fence } = 7.5 + 6.75 = 14.25}\)
All values are \(\color{blue}{\text{ between } -3.75}\) and 14.25 → No outliers

More Practice: How to Find Outliers (Video)

My Secret Math Tutor explains the outlier identification process with clear examples:

Exercises

1. Find Q1, Q3, IQR, and both fences for: 5, 10, 11, 12, 13, 14, 15, 60.
2. Is 2 an outlier in the set: 2, 20, 22, 24, 25, 28, 30?
3. A student scored 45, 88, 92, 90, 87, 91, 89. Is 45 an outlier?
4. Data: 3, 5, 6, 7, 8, 9, 10. Are there any outliers?
5. Which measure — mean or median — is more affected by an outlier? Explain.

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Answers

1. \(\color{blue}{Q1=\frac{(10+11)}{2}=10.5}\); \(\color{blue}{Q3=\frac{(13+14)}{2}=13.5}\); \(\color{blue}{\text{ IQR }=3}\); Lower \(\color{blue}{\text{ fence }=10.5-4.5=6}\); Upper \(\color{blue}{\text{ fence }=13.5+4.5=18}\). Values 5 and 60 are outside the fences: 5 and 60 are outliers
2. Sorted: 2,20,22,24,25,28,30; \(\color{blue}{Q1=20}\); \(\color{blue}{Q3=28}\); \(\color{blue}{\text{ IQR }=8}\); Lower \(\color{blue}{\text{ fence }=20-12=8}\). Since 2 < 8: yes, 2 is an outlier
3. Sorted: 45,87,88,89,90,91,92; \(\color{blue}{Q1=87.5}\); \(\color{blue}{Q3=90.5}\); \(\color{blue}{\text{ IQR }=3}\); Lower \(\color{blue}{\text{ fence }=87.5-4.5=83}\). Since 45 < 83: yes, 45 is an outlier
4. \(\color{blue}{Q1=5}\); \(\color{blue}{Q3=9}\); \(\color{blue}{\text{ IQR }=4}\); Lower fence=−1; Upper \(\color{blue}{\text{ fence }=15}\). All values in range: no outliers
5. The mean is more affected because it includes every value in its calculation. The median only depends on the middle value(s).

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Frequently Asked Questions

Is an outlier always an error in the data?

No. An outlier might be a genuine extreme observation (e.g., an exceptionally talented athlete in a group of beginners). Before removing an outlier, investigate whether it represents a data-entry error or a truly unusual but valid case.

Why use \(\color{blue}{1.5 \times \text{ IQR }}\) as the rule?

The \(\color{blue}{1.5 \times \text{ IQR }}\) rule (sometimes called Tukey’s fences) is a widely accepted convention. It identifies about 0.7% of data as outliers in a normal distribution, which is a reasonable balance between sensitivity and specificity.

What happens to the median when an outlier is removed?

The median is resistant to outliers, so removing an extreme value usually changes the median very little. In contrast, the mean can shift substantially.

Related Topics

• Finding Range, Quartile, and Interquartile Range
• Mean, Median, Mode, and Range
• Distributions in Line Plot
• Line Plot
• How to Solve the Frequency Distribution Table