Calculating Averages (the Mean, Median and Mode)
Calculating averages is a remarkably important skill. It is taught to and is essential for a wide range of people; ranging from students, business people and obviously statisticians. Calculating an average such as the mean, median or mode allows a person to give a fair representation to a set of data.
For the examples which I am about to conduct I will use the below figures:
1, 5, 6, 7, 2, 3, 4, 9, 10, 8, 10
Calculating the mean is a very quick, effective and widely used method of gaining an average from a set of data. It can be done in 3 very easy steps.
"Mean = (x)/n"
Step 1: Add up all of the numbers (in the formula these will be our "x" values.)
1+5+6+7+2+3+4+9+10+8+9+10= 55 (Therefore 55 is x)
Step 2: Count the total amount of numbers.
There are 11 separate values in our example (thus n=11 in our formula).
Step 3: Divide the answer from Step 1 by Step 2.
55 / 11 = 5 (Therefore our mean is 5!)
a) The mean does not always have to be an integer. There may be decimals and recurring decimals in your final answer. The significant figures or decimal places to which you give your answers are generally personal preference unless stated otherwise (in an exam question for example).
b) Be careful with zero-values. They do not contribute to the x, although they still count when tallying up the value of "n".
Calculating the mode is by far the simplest of all three averages to calculate and is exceptionally easy to get a grip on. The mode is simply the most commonly occurring value within the set of values.
In our set of values this would be 10.
As a more effective way to calculate the mode a person may wish to place his/her numbers in order. Our list of values would now look like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10
This makes it a bit easier to find which of our values is the most common and if using computer programmes such as Excel the hard work can be done for you when calculating this average.
a) When handling very small or very large sets of data ordering the numbers may not be time effective.
b) Mode will require you to browse through and keep a tally of the times a value has occurred, thus it may be unsuitable for a person who has a large piece of data to manage.
c) The mode must always be a value that has occurred within your data, and thus can only be a decimal should one appear within the set of data.
In my personal experience median has always been my least favourite average. This is because it is not only the most fiddly to calculate but also the hardest to teach to a learner. Once broken down into steps though this is another average which a person can come to grips with quickly.
Step 1: Put your numbers in ascending or descending order.
Our set of data would now look like this: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10
Step 2: Cross out the first and last number on the list.
Our set of data would now look as follows: 2, 3, 4, 5, 6, 7, 8, 9, 10
Step 3: Repeat "Step 2" until there one or two number remaining.
Our set of data would now appear as follows: 6
This is means that our median = 6. As you may have guessed the median essentially just calculates the "middle" value within a set of numbers.
a) Should there be two numbers remaining after repeating "Step 2" then we calculate the Mean of these two numbers and present this as the median.
b) Should there be an odd number of values to begin with we will be left with one value, if there is an even number then we will be left with two. If this is not the case then you have done something wrong.
c) You MUST order the numbers unlike the other two averages.
In conclusion these three averages are very simple to calculate and hopefully can easily be picked up with the use of this guide. Personally I would advise that you use the mean for your calculations and it is usually the least time consuming and most representative of the set of data. One thing that should have come across through this guide is how easily averages are manipulated for the benefit of the calculator. If I wanted to promote my business then I would perhaps use the Mode, because it offers a larger number than the others; giving a more impressive figure to state in meetings etc. This also works conversely, as the Mean provides a lower value than all the others. You should always be careful when deploying this trick however, as you never know who else has read this guide!