Central Tendency: Median

The easiest way to find the center of a data set is just to put your finger down in the middle. That's just what the Median of a distribution or data set is - it's the "middle" value.

You can find the median by arranging your data from smallest to largest, then placing your finger on the bottom and top numbers and counting down one at a time till your fingers meet.

If they meet on a single data point, the median is that number. If they meet between two points, the median is halfway, or the average, between them.

How would we find the median height of the leaders at the 2014 G20 meeting?

The heights of the leaders (in order) at the G20 summit are: $$\begin{array}{|c|c|} \hline \text{Angela Merkel} & 5'5\\ \hline \text{Jacob Zuma} & 5'6\\ \hline \text{Vladimir Putin} & 5'7\\ \hline \text{François Hollande} & 5'7\\ \hline \text{Tony Abbott} & 5'9\\ \hline \text{Xi Linping} & 5'9\\ \hline \text{Shinzo Abe} & 5'10\\ \hline \text{King Abdullah of Saudi Arabia} & 5'10\\ \hline \text{David Cameron} & 6'1\\ \hline \text{Barack Obama} & 6'1\\ \hline \text{Stephen Harper} & 6'1\\ \hline \end{array}$$

When you count up and down, both of them meet at the 6th leader, Xi Linping

If you don't want to count both up and down, you can use a handy mathematical trick: Take the total number of data points, add one, and divide by two. If it is a whole number, count that many from the bottom - that data point is the number.

For example, there are $11$ G20 members in the table here- add one and divide by $2$, and you get $6$ - the $5'9$ Xi Linping, Chairman of the People's Republic of China.

If it is a whole number plus a half, the median is halfway between that number and the next one.

If you added one more person after former Canadian Prime Minister Stephen Harper, the number of leaders would be $12$. $12$ plus $1$ divided by $2$ is $6.5$.

This means that the median is between members $6$ and $7$ - Xi Linping and Prime Minister Shinzo Abe of Japan. Their heights are $5'9$ and $5'10$, so the median height would then be $5$ft $9.5$ inches.

The Median is usually similar to the Mean, as both attempt to give a sense of the centermost value of a distribution. Check out When to use Mean, Median, and Mode to see more about the differences between the two.

The median is the point where half the data is on the left and half the data is on the right.

Example: We are given data on the number of pages in 9 books: $$546\:389\:165\:478\:93\:1073\:848$$

To find the median, first we order this data from smallest to largest: $$93\:165\:389\:478\:546\:848\:1073$$

Then cross out the rightmost and leftmost data values: $$165\:389\:478\:546\:848$$ $$389\:478\:546$$

Thus, the median number of pages is $\boxed{478}$. The median is sometimes a good choice for measuring the center of data, if the data has outliers. The mean, which is a calculation that relies on all of the data values, would be greatly affected by these few extreme values. However, the median is more resistant to outliers.

If there were an even number of values, thus giving two values in the end, we would average the two values to find the median.

In Progress

Intuitively, the median value of a data set is the "middle value".