Succeed with maths – Part 2
Succeed with maths – Part 2

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Succeed with maths – Part 2

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:

Mean equation left hand side equals right hand side Sum of all the values divided by Number of values

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:

multiline equation row 1 Mean score equals 53 division eight row 2 equals 6.6 left parenthesis to one decimal place right parenthesis

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 _unit8.2.1 Activity 1 Finding the mean time for a trip

Timing: Allow approximately 5 minutes

Allow approximately 5 minutes

The times for my trip to work during one week last month are shown in the table below:

Table _unit8.2.1 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.

Answer

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.

Mean equation left hand side equals right hand side Sum of all the values divided by Number of values

multiline equation line 1 The sum of the values equation left hand side equals right hand side sum with, 5 , summands 42 plus 58 plus 45 plus 47 plus 52 line 2 equals 244 minutes

There are five data values.

multiline equation line 1 So the mean equals 244 minutes prefix division of five line 2 equals 48.8 minutes

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 Succeed with maths – Part 1 [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ). 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:

42 sum with, 4 , summands prefix plus of 58 plus 45 plus 47 plus 52 division five

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.

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