The Mean, Median and the Mode

The mean, median and the mode are summary statistics, they describe the central tendency of a set of numbers.

Related Topics

This page gives an overview of the statistics and how they might be interpreted, more detailed descriptions can be found by following these links:

The Mean
The Median
The Mode

Video Version

The Mean

The mean or average is the sum of all the numbers in a dataset divided by the count.  This is illustrated with this set of eight numbers:

8,5,3,5,10,7,6,4

These numbers add up to 48, thus the mean is:

48/6 = 6

The Median

In a dataset, half the numbers are less than or equal to the median and the other half are greater than or equal to it.  As its name implies, its the value in the middle.  Finding the median is a two stage process.  The example uses the same dataset which was used for the mean.

Step 1 - Sort the numbers into ascending order

3,4,5,5,6,7,8,10

Step 2 - Pick the value in the middle

In this case the dataset has an even number of values.  The middle numbers are 5 and 6.  Take the mean of these to get the median:

(5+6)/5 = 5.5

If the dataset has an odd number of values, the process is simpler as there will be single value.  This is illustrated by removing the largest value from the dataset, i.e.

3,4,5,5,6,7,8

The middle value and thus the median is 5.0

The Mode

The mode is the value which has the greatest frequency in the dataset.  This simplest way of finding the mode is to compile a frequency table and draw a histogram.

This allows us to identify the mode of our dataset as 5.0.

If the numbers not integers as in our dataset, but fractions (e.g. 3.3, 4.2 etc.), the frequency table is based on intervals.

In this case, the mode is expressed as the interval with the highest frequency or the mid point of that interval.

If all the values have the same count, then there is no mode.

Tools

Spreadsheets such as Microsoft's Excel and Google's Spreadsheet Doc. have functions for calculating the Mean and Median.  A spreadsheet can be used to compile frequency tables which can be used to identify a mode.

Bringing It All Together

The summary statistics for our sample dataset are shown on the same graphic

Interpretation

Imagine that our sample dataset represents the number of days it takes a business to process an order.  The mean (or average) is 6 days which is nice to know but does directly relate to experience.  The mode and median are more interesting.  The median value tells us that 50% of order will take longer than 5.5 days, or less than 5.5 days if we are talking to the salesman.  The most common experience is the modal value which is 5 days.

The mean is a very important statistic, but does not always have a value which reflects experience.  This can be illustrated by changing the maximum value in our dataset from 10 to 34.

The increase in the maximum to to process an order indicates that there is a greater risk of something going wrong.  However, this is not reflected in the magnitude of the increase in the value of the mean from 6 to 9 which is does not describe anyone's experience.  The median and the mode remain the same.

The mean value is influenced by the extreme values in a dataset to a much greater extent than the median and mode.

Datasets with the same Mean

The graph below shows two datasets with the same mean

However, it is clear that the experience from each would be significantly different.