# Average : Mean versus Median

Written by Ingmar Verheij on July 5th, 2011. Posted in Performance testing The average of a set of numbers is a commonly used, but the defintion of average is poorly understood and can raise the risk of being manipulated.

The average of a set of numbers can be determined with the mean or the median. To better understand the difference between mean and median I will explain the definition of both and illustrate it with examples.

#### The difference between mean and median

Understanding the difference between the mean and median allows you to take advantage of one another and prevent the risk of being manipulated. To illustrate the difference we’ll take this set of 5 numbers:

4  2  14  2  3

##### Mean

The mean is the also known as the arithmetic mean and is calculated by adding N numbers in a data set together and dividing it by N. The mean of the 5 numbers above is 5.0

4 + 2 + 14 + 2 + 3 / 5 = 5.0

##### Median

The median of a data set is the middle number when the set is sorted in numerical order. With an odd-numbered data set this is the number that is in the middle. When there is an even-numbered data set the mean of the two middle numbers is taken. The median of the 5 numbers above is 3.0

Odd-numbered : 2 + 2 + 3 + 4 + 14 = 3.0

Even-numbered: 2 + 2 + 3 + 4 + 5 + 14 = (3 + 4) / 2 = 3.5

#### When to use what

In most cases the arithmetic mean is used as the average of a data set since it will the take all numbers in the data set in the calculation. In other words, each number in the data set has influence on the outcome. If this outcome should not be influenced by spikes (high or low) the median will give a better result.

##### An example

A small town with 500 residents earn roughly  € 50,000.- per year. Both the median and mean are around € 50.000,-. Now a (super wealthy) family moves in to town, their income is around € 1 Billion a year.

The median income stays around € 50.000,- per year (since all 500 others stil earn around € 50.000,- per year) but the aritmethic mean is €2.025 million!

Although nothing has changed for the 500 citizens, the way the numbers are presented might affect them. For instance when the tax paid is based on the average income (…).

#### Service levels / performance metrics / ROI

Now what happens if we take this knowledge to our daily lives? There are numerous of examples where the average of a set of numbers is used. For instance “The average response time was 100 milliseconds, that’s great!” or “The average load on server X was 10% so the request to order a new server is denied.”

#### Conclusion

Juggling with numbers is easy, the result can be transformed into a better suiting result (for the presenter) just by using a different methodology of calculating the average. Knowing the difference in mean and median can help you prevent being misled or manipulated, it might even help you doing so.

Ingmar Verheij ### Ingmar Verheij

At the time Ingmar wrote this article he worked for PepperByte as a Senior Consultant (up to May 2014). His work consisted of designing, migrating and troubleshooting Microsoft and Citrix infrastructures. He was working with technologies like Microsoft RDS, user environment management and (performance) monitoring. Ingmar is User Group leader of the Dutch Citrix User Group (DuCUG). RES Software named Ingmar RES Software Valued Professional in 2014. 