1.1 Calculating the mean of a data set
You may well have come across the mean before, as it is the most commonly used type of average. However, it may not be clear that this is being used as the average, as many uses of averages (in the media for example), do not say which average is being employed. You’ll look at the difference to the average reported that this can make later in the week.
The important point to remember about the mean is that it takes all the data into account. It does this because to calculate the mean, all the values are added together and then divided by the number of values.
A word formula for this is:
Let’s look at an example now to explore this type of average:
Suppose eight students took an exam, with the following scores: 9, 7, 6, 7, 8, 4, 3 and 9.
Sum of all the values = 9 + 7 + 6 + 7 + 8 + 4 + 3 + 9 = 53
There are 8 values, so:
This feels right, as it lies between the smallest and the largest value, and is around the middle of the values. However, where the mean lies within the values will depend on the actual values in any data set. It may be close to the middle but equally, it could be closer to the smallest or largest value.
Here’s an example for you to try:
Activity 1 Finding the mean time for a trip
Allow approximately 5 minutes
The times for my trip to work during one week last month are shown in the table below:
Table 1 Time for trip to work over one week
|Day of week||Monday||Tuesday||Wednesday||Thursday||Friday|
|Time in minutes||42||58||45||47||52|
First, look at the data. What would you say is a typical length of time for the trip from this set of data? Write down your estimate.
Now calculate the mean commuting time.
The smallest time is 42 minutes and the largest is 58 minutes, so a typical time would lie between these, perhaps 50 minutes. Your estimate may be different from this, of course, because it is just a sensible guess at a typical value.
There are five data values.
The mean commute time over that week was about 49 minutes. (Remember to include the units with your answer!)
The mean is fairly close to the estimated typical value of 50 minutes, so it looks as if the calculated value for the mean is correct.
You probably used a calculator to help you arrive at the answer in the last activity and hopefully you got the same answer the first time. It is easy, however, to forget that your calculator probably knows the rules for the order of operations – Brackets, Exponents, Division, Multiplication, Addition, Subtraction, (this is covered in Week 2 of). So, if I had tried to calculate the mean in one step, without including any brackets, my calculator would have given me 202.4 as the answer. It would have calculated:
only dividing 52 by 5, rather than the total.
Fortunately, a quick comparison with the smallest and largest values in our data set would have immediately told me that something was not right!
So, always check you have a sensible answer when compared to the data you have, and work out the mean in two steps.
The mean, however, may not always give the best idea of what a truly typical value is, and in these situations it is best to turn to one of the other options available. Let’s explore this further in the next section.