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# Identifying outliers with the 1.5xIQR rule

An outlier is a data point that lies outside the overall pattern in a distribution.
The distribution below shows the scores on a driver's test for 19 applicants. How many outliers do you see?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.
Some people may say there are 5 outliers, but someone else might disagree and say there are 3 or 4 outliers. Statisticians have developed many ways to identify what should and shouldn't be called an outlier.
A commonly used rule says that a data point is an outlier if it is more than 1, point, 5, dot, start text, I, Q, R, end text above the third quartile or below the first quartile. Said differently, low outliers are below start text, Q, end text, start subscript, 1, end subscript, minus, 1, point, 5, dot, start text, I, Q, R, end text and high outliers are above start text, Q, end text, start subscript, 3, end subscript, plus, 1, point, 5, dot, start text, I, Q, R, end text.
Let's try it out on the distribution from above.

## Step 1) Find the median, quartiles, and interquartile range

Here are the 19 scores listed out.
5, 7, 10, 15, 19, 21, 21, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25
What is the median?
start text, m, e, d, i, a, n, end text, equals

What is the first quartile?
start text, Q, end text, start subscript, 1, end subscript, equals

What is the third quartile?
start text, Q, end text, start subscript, 3, end subscript, equals

What is the interquartile range?
start text, I, Q, R, end text, equals

## Step 2) Calculate $1.5\cdot\text{IQR}$1, point, 5, dot, start text, I, Q, R, end text below the first quartile and check for low outliers.

problem a
Calculate start text, Q, end text, start subscript, 1, end subscript, minus, 1, point, 5, dot, start text, I, Q, R, end text
start text, Q, end text, start subscript, 1, end subscript, minus, 1, point, 5, dot, start text, I, Q, R, end text, equals

problem b
How many data points can we say are low outliers?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.

## Step 3) Calculate $1.5\cdot\text{IQR}$1, point, 5, dot, start text, I, Q, R, end text above the third quartile and check for high outliers.

problem a
Calculate start text, Q, end text, start subscript, 3, end subscript, plus, 1, point, 5, dot, start text, I, Q, R, end text
start text, Q, end text, start subscript, 3, end subscript, plus, 1, point, 5, dot, start text, I, Q, R, end text, equals

problem b
How many data points can we say are high outliers?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.