11.1.1 Calculating the Mean of a Data Set
You have probably come across the mean before; it is the most commonly used type of average and takes into account all the data.
Let’s look at an example to explore this type of average.
Suppose eight students took an exam, with the following scores:
To get a feel for the problem and to help you check that your final answer is reasonable, look at the exam scores and decide what a typical value might be. Make a note of your estimate.
To calculate the mean value, we add all the data values together and then divide this sum by the number of values.
In this example, the sum of the data values is .
There are eight data values. Therefore, the mean value is .
The mean exam score for these students was 6.6 (rounded to 1 decimal place).
How did this compare with the typical value you estimated at the beginning? Did you decide that 6 or 7 might be a typical value?
The mean value will always lie between the smallest value and the largest value, and will often be somewhere towards the middle, though exactly where depends on the actual values in the data set.
Calculating the Mean
Calculating a mean can be summarized as follows:
- Add all the values together to find their sum.
- Count the number of values.
- Divide the sum by the number of values.
- Write down the conclusion and include the units.
Remember to make a note of this in your math notebook for easy reference later.
Take a look at this pencast to see how to calculate the mean.
Activity: Finding the Mean Time for a Trip
Here’s an example for you to try. The times for my trip to work during one week last month are shown in the table below:
|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 commute time using the method given above.
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.
To calculate the mean:
The sum of the values is .
There are five data values, so the mean is .
The mean commute time over that week was about 49 minutes.
Remember to include the units. In this case, the units are minutes.
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.