Find all possible outliers for the following set of data: 108, 31, 75, 87, 79, 88, 89, 118, 51, 89, 174, 95, 51, 70, and 73.
An outlier is a data value that is either much larger or much smaller than the median. Data that are more than 1.5 times the value of the interquartile range beyond either quartile are called outliers. Before we begin, let’s go ahead and put all of our numbers in order from least to greatest. So here we have our numbers from least to greatest, and we’re trying to find all of the possible outliers.
Outliers are more than 1.5 times the interquartile range. Well, what is the interquartile range? It’s the upper quartile minus the lower quartile. So we need to find the upper and lower quartiles. Quartiles split data into four equal parts.
First, let’s begin by finding the median. Since there are fifteen total numbers, the eighth number would be our medium, the one in the middle. Now, we need to find the middle of each of these two halves, which would be 70 and 95. Those will be the numbers in the middle of each of the halves. So 70 is your lower quartile and 95 is your upper quartile.
Therefore, we can plug these in to the 1.5 times the upper quartile minus the lower quartile and find our outliers. So we have 1.5 times 95 minus 70. And 1.5 times 25 is 37.5. So this greater than symbol has stayed there because an outlier is a data value that is either much larger or much smaller than the median. Data that are more than 1.5 times the value of the interquartile range beyond either quartile are called outliers.
So if we would add 37.5 to our upper quartile, that gives us 132.5; any numbers greater than that would be considered an outlier. So 174 is an outlier. If you would take away 37.5 from the lower quartile, you get 32.5. So any numbers lower than 32.5 would be considered an outlier. That means 31 is an outlier. Therefore, 31 and 174 are outliers for this set of data.