The median is a statistic that is commonly used to measure the center of a data set. However, it is still an unsung hero of statistics in the sense that it isn’t used nearly as often as it should be, although people are beginning to report it more nowadays.
The median of a data set is the value that lies exactly in the middle when the data have been ordered from the lowest value to the greatest value. It’s denoted in different ways, but most people use M.
Here are the steps for finding the median of a data set:
Order the values from smallest to largest.
If the data set contains an odd number of values , choose the one that is exactly in the middle. You’ve found the median.
If the data set contains an even number of values , take the two values that appear in the middle and average them to find the median.
The following table shows the salaries of each player for the Los Angeles Lakers during the 2009–2010 season. Can you find the median salary?
| Player | Salary ($) |
|---|---|
| Kobe Bryant | 23,034,375 |
| Pau Gasol | 16,452,000 |
| Andrew Bynum | 12,526,998 |
| Lamar Odom | 7,500,000 |
| Ron Artest | 5,854,000 |
| Adam Morrison | 5,257,229 |
| Derek Fisher | 5,048,000 |
| Sasha Vujacic | 5,000,000 |
| Luke Walton | 4,840,000 |
| Shannon Brown | 2,000,000 |
| Jordan Farmar | 1,947,240 |
| Didier Ilunga-Mbenga | 959,111 |
| Josh Powell | 959,111 |
| Total | 91,378,064 |
The salaries for the players are ordered from smallest (at the bottom) to largest (at the top). Because the list contains the names and salaries of 13 players, the middle salary is the seventh one from the bottom (and from the top): Derek Fisher, who earned $5,048,000 that season from the Lakers. Derek is at the median.
This median salary ($5.048 million) is well below the average of $7.029 million for the 2009–2010 Lakers team. Notice that only 4 players of the 13 earned more than the average Lakers salary. Because the average includes outliers (like the salary of Kobe Bryant), the median salary is more representative of center for the team salaries. The median isn’t affected by the salaries of those players who are way out there on the high end (or on the low end) the way the average is.
Numbers in a statistical data set that are extremely high or extremely low compared to the rest of the data are called outliers. Because of the way the average is calculated, high outliers tend to drive the average upward. Low outliers tend to drive the average downward. However, these outliers have no effect on the median.