Skill: Identify outliers in a data set

An outlier is a data value that is unusually far from the other values. You can often spot one as a value separated from the main group in a table or dot plot.

When quartiles are available, use these steps:

1. Find the interquartile range:

$$\text{IQR}=Q_3-Q_1$$

2. Find the lower and upper fences:

$$\text{Lower fence}=Q_1-1.5(\text{IQR})$$ $$\text{Upper fence}=Q_3+1.5(\text{IQR})$$

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

Example

The numbers of minutes 8 students spent commuting to school are:

$$8,\ 9,\ 10,\ 11,\ 12,\ 13,\ 14,\ 30$$

The data is already in order.

Step 1: Find $Q_1$ and $Q_3$.

The lower half is $8,9,10,11$, so

$$Q_1=\frac{9+10}{2}=9.5$$

The upper half is $12,13,14,30$, so

$$Q_3=\frac{13+14}{2}=13.5$$

Step 2: Find the IQR.

$$\text{IQR}=13.5-9.5=4$$

Step 3: Find the fences.

$$\text{Lower fence}=9.5-1.5(4)=3.5$$ $$\text{Upper fence}=13.5+1.5(4)=19.5$$

Step 4: Compare the data values with the fences.

No value is below $3.5$. However, $30$ is above $19.5$.

Therefore, $30$ minutes is an outlier.

The value $30$ is not automatically an error. It could represent a student who lives much farther away. An outlier can affect the mean and range greatly, while usually having less effect on the median.