Outlier Detection Statistics Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

A data set has mean

\bar{x} = 100

and standard deviation

s = 15

. Using the z-score method, determine whether the values 60, 145, and 155 are outliers (using the threshold

|z| > 2

Solution

1
Step 1: Calculate z-scores: $z_{60} = \frac{60-100}{15} = -2.67$ , $z_{145} = \frac{145-100}{15} = 3.0$ , $z_{155} = \frac{155-100}{15} = 3.67$ .
2
Step 2: All three have $|z| > 2$ : 60 ( $|z|=2.67$ ), 145 ( $|z|=3.0$ ), 155 ( $|z|=3.67$ ). All three are outliers by the z-score criterion, with 155 being the most extreme.

Answer

All three values are outliers:

z_{60} = -2.67

z_{145} = 3.0

z_{155} = 3.67

. All have

|z| > 2

The z-score method for outlier detection flags values more than a specified number of standard deviations from the mean. A threshold of

|z| > 2

catches values in the outer 5% of a normal distribution, while

|z| > 3

is more conservative (outer 0.3%). This method assumes approximately normal data.

About Outlier Detection

Outlier detection is the process of identifying data points that are unusually far from the rest of the dataset, using techniques like the IQR rule, z-scores, or visual inspection of box plots and scatter plots. These anomalous values may indicate measurement errors, data entry mistakes, or genuinely extreme observations.

Learn more about Outlier Detection →

More Outlier Detection Examples

Example 1 easy

The data set is: 10, 12, 11, 13, 12, 14, 11, 50. Identify the outlier and explain how you know.

Example 2 medium

Test scores: 72, 75, 78, 80, 82, 85, 88, 90, 92, 95. A new student's score of 25 is added. How does

Example 3 medium

A scientist records reaction times (ms): 245, 260, 255, 270, 250, 980, 265, 258. Use the [formula] r

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Related Concepts

Interquartile Range (IQR)Z-Score (Standard Score)Mean vs Median