# Outlier Calculator

Find the outliers in a data set by entering the numbers in the calculator below.

Separate numbers using a comma (,)

## Outliers:

### Steps to Solve

#### Step One: Find the Quartiles

The first step is to find the quartiles for the data set.

For this data set, the quartiles are:

Q1 = ?
Q3 = ?

#### Step Two: Find the Interquartile Range

The interquartile range is the difference between the first and third quartiles.

IQR = Q3 - Q1

#### Step Three: Find the Fences

The next step is to set up a fence beyond the first and third quartiles using the interquartile range.

Lower Fence = Q1 - (1.5 × IQR)

Upper Fence = Q3 + (1.5 × IQR)

#### Step Four: Find the Outliers

Any numbers in the data that are above or below the fences are outliers.

Learn how we calculated this below

## How to Find Outliers

In statistics, outliers are observations that lie an abnormal distance from other values in a set of data. These data points are considered unusual and are often problematic in statistical analyses because they tend to distort the results.

There are a few methods that you can use to find outliers in the data.

### Method One: Visually Identify Outliers

The first method is to visually identify outliers. Looking at the scatter plot graph below, it’s easy to spot the outliers shown in red.

Looking at the graph, two values look unusually compared to the other data, which follow the line closely and are outliers in the data.

### Method Two: Using the IQR Method

The second method to find outliers in the data is to use the interquartile range method. To use this method, find the quartiles and interquartile range for the data.

The first quartile (lower quartile) is the median value of the first half of the data. Split the data in half, then the first quartile is the median for the half.

The third quartile (upper quartile) is the median value of the second half of the data.

The interquartile range (IQR) is the difference between the first and third quartiles. You can find the interquartile range using the formula:

IQR=Q_{3}\ –\ Q_{1}

Then, using the quartiles and interquartile range, set fences beyond the quartiles. Any values outside of these fences are considered outliers.

You can find the fences using the following formula:[1]

\text{lower fence }=Q_{1}-\left (1.5 \times IQR \right )
\text{upper fence }=Q_{3}+\left (1.5 \times IQR \right )

Any values in the data that are smaller than the lower fence or larger than the upper fence are outliers.

### Method Three: Using Z-Scores

The third method to find outliers in the data is to use z-scores. To use this method, calculate z-scores for each observation.

The further the value’s z-score is from zero, the more unusual the value is. Or rather, the more standard deviations away from the mean the value is, the more unusual.

Values with a z-score above 3.0 or below -3.0 are considered outliers using this method.[2]

This is similar to the Grubbs’ Test below.

### Method Four: Using Grubbs’ Test

Given a normally distributed data set with a minimum of seven values, the Grubbs’ Test can also be used to identify outliers. To use Grubb’s Test, you need to find the G test statistic, find the critical value for it, compare the test statistic to the critical value, and then reject the value as an outlier if the test statistic is greater than the critical value for G.

To find the Grubbs’ test statistic, which is the largest deviation from the sample mean, use the following formulas:[3]

#### Left-Tailed Test

G = \frac{\bar{x}-x_{min}}{s}

#### Right-Tailed Test

G = \frac{x_{max}-\bar{x}}{s}

Where:
= sample mean
xmin = minimum value in the sample
s = sample standard deviation

#### Important Value

Now, find the critical value to compare to. When the test statistic is greater than the critical value, the value can be rejected as an outlier.

G\gt\frac{n-1}{\sqrt{n}}\sqrt{\frac{t^{2}}{n-2+t^{2}}}

Where:
n = sample size
t = tn-2,p
p = 1 – α/(2n)

Grubbs’ Test is also known as the extreme studentized deviate test, or ESD test.

## References

1. PennState Eberly College of Science, Elementary Statistics - 3.2 - Identifying Outliers: IQR Method, https://online.stat.psu.edu/stat200/lesson/3/3.2
2. Mahmood, M. S., Outlier Uncoverion (Part 1), Towards Data Science, https://towardsdatascience.com/outlier-detection-part1-821d714524c
3. National Institute of Standards and Technology, Engineering Statistics Handbook - 1.3.5.17.1. Grubbs' Test for Outliers, https://www.itl.nist.gov/div898/handbook/eda/section3/eda35h1.htm