##### Different types of averages

An average is a single value used to represent a set of data. There are types of average used in statistics and the following shows how each can be calculated.

Tip! There is more than one “average” so you should never refer to the average.Always specify which average you are talking about: the mean, median or mode.

The shoe size of 19 students in a class are shown below:

4 7 6 6 7 4 8 3 8 11 6 8 6 3 5 6 7 6 4

How would you describe the shoe size in this class?

If you count how many size fours, how many size fives and so on, you will find that the most common( most frequent) show size in the class is six. This average is called mode.

What most people think as the average is the value you get when you add up all the shoe sizes and divide the answer by the number of students:

$ \displaystyle \frac{{total~~of~shoe~~sizes}}{{number~~of~~students}}=\frac{{115}}{{19}}=6.05$

If you take the mean of n items and multiply it by n, you get the total of all n values.

This average is called the mean. The mean value tells you that the shoe sizes appear to be spread in some way around the value 6.05. It also gives a good impression of the general `size` of the data. Notice that the value of the mean, in this case, is not a possible shoe size.

The mean is sometimes referred to as the measure of central tendency of the data. Another measure of central tendency is the middle value when the shoe sizes are arranged in ascending order.

If you now think of the first time and the last values as one pair, the second and second to last to another pair, and so on, you can cross these numbers off and you will be left with a single value in the middle.

This middle value, (in this case six), is known as the median.

Crossing of the numbers from each end can be cumbersome if you have a lot of data. You may have noticed that, counting from the left, the median is the 10th value. Adding one to the number of students and dividing the results by two, $ \displaystyle \frac{{(19+1)}}{2}$, also gives 10 as the median position. What if there had been 20 students in the class? For example, add extra student with a shoe size of 11. Crossing off pairs gives this result:

You are left with a middle pair rather than a single value. If this happens then you simply find the mean of this middle pair: $\displaystyle \frac{{(6+2)}}{2}=6$

Notice that the position of the first value in this middle pair is $\displaystyle \frac{{20}}{2}=10$

Adding an extra size 11 has not changed the median or mode in this example, but what will have happened to the mean?

In Summary:

Mode: The value that appears in your list more than any other. There can be more than one mode but if there are no values that accur more often than any other then there is no mode.

Mean: $\displaystyle \frac{{total~~of~all~~data}}{{number~~of~~values}}$ the mean may not be one of the actual data values.

Median: 1. Arrange the data into ascending numerical order.

2. If the number of data is n and n is odd, find $\displaystyle \frac{{(n+1)}}{2}$

### Dealing with extreme values

Sometimes you may find that your collection of data contains values that are extreme in some ways. For example, if you were to measure the speeds of cars as they pass a certain point you may find that some cars are moving unusually slowly or unusually quickly. It is also possible that you may have made a mistake and measured a speed incorrectly, or just written the wrong numbers down!

Suppose the following are speeds of cars passing a particular house over a five minute period (measured in kilometres per hour).

### One particular value will catch your eye immediately. 128.9 km/h seems somewhat faster than any other car. How does this extreme value affect your averages?

You can check yourself that the mean of the above data including the extreme value is 70.7 km/h.

This is larger than all but one of the values and is not representative. Under these circumstances the mean can be a poor choice of average. If you discover that the highest speed was a mistake, you can exclude it from the calculation and get the much more realistic value of 59.0 km/h.

If the extreme value is genuine and cannot be excluded, then the median will give you a better impression of the main body of data. Writing the data in rank order:

### The median is the mean of 58.3 and 65.0, which is 61.7. Notice that the median reduces to 58.3 if you remove the highest value, so this doesn’t change things a great deal.

There is no mode for this data.

### Note! As there is an even number of speeds the median is the mean of the

### 3rd and 4th data points.

Example 1: After six tests, Graham has a mean average score of 48. He takes a seventh test and scores 83 for the test.

a) What is Graham`s total score after six tests?

Since $\displaystyle mean=\frac{{total~of~all~data}}{{number~~of~values}}$ then, total of all data=mean x number of values= 48 x 6=288

b) What is Graham`s mean average score after seven tests?

Total of all seven scores=total of first six plus seventh=288+83=371

$\displaystyle mean=\frac{{371}}{7}=53$

Tip! This is a good example of where you need to think before you conclude that Graham is an average student(scoring 53%) He may have had extra tuition and will get above 80% for all future tests.