# Box plots with outliers. What is an outlier and how to find these extreme points.

Outliers are extreme values that lie within a set of data. If you had to draw a box and whisker diagram then the outliers should be marked on with a cross.

In order to identify an outlier first find the Lower Quartile, Upper Quartile and Inter Quartile Range.

Now an outlier will occur if it’s;

*More than* **Upper Quartile + 1.5(Inter Quartile Range)**

**OR**

*Less than* **Lower Quartile – 1.5(Inter Quartile Range)**

Other scale factors can be used instead of 1.5 but 1.5 is the most commonly used. The exam question will tell you if it’s any different.

Let’s take a look at an example:

Example 1

A class of 15 students took a Science exam. Here are the results of the class:

8%, 9%, 40%, 52%, 53%, 60%, 62%, 62%, 64%, 68%, 70%, 70%, 71%,71%, 98%

Work out the lower quartile, upper quartile and inter quartile range. Also identify any outliers that occur within the test scores:

First calculate the lower quartile:

¼ of 15 = 3.75 = 4^{th} person (always round upwards if it comes out as a decimal)

So the lower quartile is 52% as this is the 4^{th} persons score.

Next work out the upper quartile:

¾ of 15 = 11.25 = 12^{th} person (again round upwards as the answer came out as a decimal)

So the upper quartile is 70% as the 12^{th} highest score in the data was 70%

Since you have the upper quartile and lower quartile, find the difference of these to give you the inter quartile range, 70 – 52 = 18%

So our three values that you need to work out the outliers are:

Lower Quartile = 52%

Upper Quartile = 70%

Inter Quartile Range = 18%

First let’s work out the outliers that occur at the upper end of the data set:

**Upper Quartile + 1.5(Inter Quartile Range)**

= 70 + 1.5x18 = 97%

Any values which are more than 97% will be outliers. So the person who scored 98% is an outlier.

Next work out the outliers that occur at the lower end of the data set:

**Lower Quartile – 1.5(Inter Quartile Range)**

= 52 – 1.5 x 18 = 25%

So any values which are less than 25% are also outliers. So looking back at the data there are two outliers that are less than 25% which are 8% and 9%.

So altogether there are 3 outliers in this data set; 8%,9% and 98%.

**Summary**

So remember that outlier are extreme values. They are either more than **Upper Quartile + 1.5(Inter Quartile Range) **or less than **Lower Quartile – 1.5(Inter Quartile Range).**

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