You have been looking at analyzing different data by the measures of central tendency and the measures of dispersion. In some sets of data, an interesting thing occurs when one or more item is farther away from the rest of the set than most of the others.
See the frequency table of scores on an exam above. Notice that there is one score in the 20 to 29 interval that is far apart from the others. If you put this data in a histogram, here is the result:
When a data member is so far away from the other scores, it affects the mean. It lowers it significantly and gives the appearance that the rest of the students scored more poorly than they did. This data member is called an outlier.
Put the raw data in a stem-and-leaf plot.
The mean of this data in its current form is 75.2. But, if you examine the scores, most students had a better score than 75.
What would happen if you left out the score of 28 and then found the mean? The mean of the scores is 77, which is a good central indicator of how the middle set of the data falls.
Researchers often find results where one piece of data is far removed from the other. If this piece is determined to be an outlier, and not typical of the rest of the results, it can be disregarded. The following explains how to find an outlier.
Instructions for Finding Outliers
Find Q1 and Q3.
Find the inter-quartile range, IQR, by subtracting Q3 – Q1.
Multiply this by 1.5.
Subtract this product from Q1 and add the product to Q3.
If a number is less than or equal to the difference or greater than or equal to the sum, it is an outlier.
If no number fits the standards in number 5, there is no outlier in the set of data.
Example 1: Find the Outlier of the Data
Find the outlier of the scores for the reading exam.
1. First, find the median, Q1, and Q3.
There are 25 numbers. The median is the thirteenth. Q1 is between the sixth and seventh, and Q3 is between the nineteenth and twentieth. Using the stem-and-leaf plot above:
Q1 = 70
Q3 = 86
2. Now, find the inter-quartile range.
IQR = Q3 – Q1 = 86 – 70 = 16
1.5 (IQR) = 1.5 (16) = 24
Q1 – 24 = 70 – 24 = 46
Q3 + 24 = 86 + 24 = 110
So an outlier would any number a ≤ 46 or a ≥ 110.
There is just one number, 28, that fits the criteria, and 28 is an outlier.
Quick Practice
Find the median, Q1, Q3, inter-quartile range, and any outliers for each set of data. You may need to order the data from least to greatest first.
Instructions for Finding Outliers
