The median is essentially the middle value of a batch when the values are placed in size order; it is found in the following way.
First, all the values in the batch are sorted into ascending order; that is, smallest first, then second smallest, and so on, ending with the largest.
Then, see if the batch size is odd or even. If there is an odd number of values in the batch, then the middle value in the list is the median. If there is an even number of values in the batch, then there are two middle numbers. Add up these two numbers and divide by two: this gives the median. In other words, you find the mean of the two middle numbers and that gives the median of the batch as a whole.
Find the median of the batch of seven peoples heights given previously.
Sorting the seven heights into ascending order produces the following list.
There is a single middle value, so this value is the median height: 1.69 m.
Suppose that one person is removed from the batch (the tallest, for example) leaving an even number of heights (six).
The remaining values (in ascending order) are as follows.
There are two middle values, namely 1.66 and 1.69. The median is found by calculating the mean of these two numbers, so
Of these two ways of finding a single number that is typical of a batch of numbers, which should you use: the mean or the median? And does it matter which is used? For the seven heights, the mean height (1.68 m) and the median height (1.69 m) are almost the same. So, in this case at least, it does not seem to make much difference which we use. However, to see that this is not always so, carry out the following activity.
(a) The weekly pocket money, in pounds, of each of four ten-year-old boys is given below.
Find the mean and the median weekly pocket money of these boys.
(b) The weekly pocket money, in pounds, of each of five eleven-year-old girls is given below.
Find the mean and the median weekly pocket money of these girls. Which ‘average? would you regard as the more representative of the pocket money these girls receive?
(a) The mean weekly pocket money for the boys is
The median is the average of the two middle values, so it is
(b) The mean weekly pocket money for the girls is
The median is the middle value: £4.50.
See also the comments in the text following the activity.
The mean weekly pocket money for the five girls in Activity 10(b) is £8.00 and the median is £4.50. This time, they differ substantially. On this evidence, the parents of another eleven-year-old might offer her £4.50 per week on the grounds that £4.50 (the median) is typical of what eleven-year-old girls receive in pocket money. On the other hand, the girl might argue that since the ‘average? (i.e. the mean) is £8.00, her parents should give her £8.00 per week!
In practice, the mean and the median are both widely used: which is chosen usually depends on the purpose for which an ‘average? is required. Here in this Unit, means are used throughout for finding average prices.
Now work through Section 2.1 of Chapter 2 of the Calculator Book.
This subsection has introduced two important statistical terms—mean and median. Make notes of how to calculate them by hand and using your calculator. Include the keys you need to press in your notes on the calculator techniques. Try to use your own words and examples rather than just copying out definitions from the Unit.
Think about how you have learned about these terms and what helped or hindered your learning.
Check your definitions against those given earlier.
Here are some developmental testing students' comments.
How I learned mean and median
‘For part of this study session I aimed to know what the mean and median are and how to calculate them. I feel confident that I can do this now as I've just finished working through Section 3, and feel I understand the terms.?
How I learned the terms
‘I worked through the explanation of mean and stopped after the worked example. I sensed I was understanding the term “mean” as I'm used to working out averages in this way. I carried on and read through the definition of “median” and stopped after Example 2. I read through this example again, as I think I missed the point the first time. I carried on reading and then did Activity 10. I had no trouble working this out.
I repeated each separate definition mentally to myself once or twice to test myself. Something told me I will remember these definitions for the next session—but I also feel sure I will remember them better when I have used them more.?
The second description above demonstrates how this student has gone about learning the terms ‘mean? and ‘median?. The strategies she has used include focusing on a particular aspect while reading, checking the meaning of each of the terms by doing examples and activities, repeating meanings for self-assessment, and so on. Sometimes it is useful to pause and think how you are going to approach learning a particular part of the course—for example, it may be unwise to try to learn how to use the calculator effectively by reading alone. Already you have a wide range of learning strategies to use—reading for learning, repeating to yourself, writing something down, using tables and graphs, using audio and video resources, using a calculator—to name just a few! Different strategies can be used to learn different things. Students who are aware of how they learn tend to learn both more effectively and more efficiently. This will not happen overnight and some activities may help you to do this better.
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