# How do you write down 5 test results that have their mean median and mode all equallying 6?

Apr 27, 2015

Let's start with definitions and exemplify them using the test results from $5$ experiments indexed in weak ascending order (that is, not descending, allowing the equal values):
${x}_{1} \le {x}_{2} \le {x}_{3} \le {x}_{4} \le {x}_{5}$

Mean is an arithmetic average of test results, that is their sum divided by their quantity:
$A = \frac{{x}_{1} + {x}_{2} + {x}_{3} + {x}_{4} + {x}_{5}}{5}$

Median is a particular result such that the number of results with values less than or equal to it equals to the number of results greater or equal to it. In our case of $5$ results indexed in the ascending order such an element is the third one because there are two that are smaller or equal and two - greater or equal:
$B = {x}_{3}$

Mode is a particular test result that occurs more often than others. Obviously, there is no formula for it, you just have to look at the results, find repetitive value and chose the one with most repetitions. If there is no value that repeats the most (for instance, two values repeat more than others, but the same as each other), the results have no mode.

A trivial example of a set of five values with mean, median and mode being the same and equal to $6$ is five identical values: $6 , 6 , 6 , 6 , 6$.
Their mean is $6$, as well as median and mode.

Less trivial example is three identical results that are equal to $6$, one smaller than $6$ by some value and one bigger than $6$ by the same value, like $2 , 6 , 6 , 6 , 10$.
Their mean, median and mode are all equal to $6$.