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Calculating an Average in Math


The average in Math is also often referred to as the mean. The average value that this page focuses on is the “arithmetic average” or “arithmetic mean”.


There are other averages that can be worked out when calculating an average, such as the weighted average and geometric average.

But the word &nbsp”average”&nbsp or &nbsp”mean”&nbsp when used on this page, will refer specifically to the arithmetic average, unless stated otherwise.




Arithmetic Average,
Calculating an Average

When calculating an average of a set/group of data values, what we do is we add up the data values together,
then divide this amount by the number of data values there are in the set/group.


So looking at a random list of &nbsp5&nbsp numbers: &nbsp&nbsp 11 , 14 , 6 , 21 , 8.

The &nbspaverage / mean&nbsp value of the group of &nbsp5&nbsp numbers is:

\bf{\frac{11 \space + \space 14 \space + \space 6 \space + \space 21 \space + \space 8}{5}} &nbsp=&nbsp \bf{\frac{60}{5}} &nbsp=&nbsp 12


Meaning if we picked out any number at random from this group of &nbsp4,&nbsp we would likely expect the value to probably be near to &nbsp14.
Although, &nbsp6&nbsp and &nbsp21&nbsp are a bit more apart from this mean value.






Calculating an Average Example

Shown below is a data frequency table showing values for how many minutes a group of &nbsp18&nbsp people spend preparing their lunch.


Minutes Number of people
6
4
7
5
8
2
9
4
10
3
Total
17

In order to find the mean/average amount of minutes a person spent preparing lunch.

We can help firstly by adding a third column, which will display the total amount of minutes spent preparing lunch by each individual group combined.
We can then add up all the minutes together, and then divide this amount by the number of people studied,&nbsp 18.



Minutes Number of people Total minutes amount
6
4
24 (6 × 4)
7
5
35 (7 × 5)
8
2
16 (8 × 2)
9
4
36 (9 × 4)
10
3
30 (10 × 3)
Total
18
141

Now to calculate the average, we divide &nbsp141&nbsp minutes by the &nbsp18&nbsp people.


\bf{\frac{141}{18}} &nbsp=&nbsp 7.83

The average or mean amount of time by the people preparing lunch is &nbsp7.83&nbsp minutes.







Calculating an Average, Outliers

With average values, something that we should often watch out for in a set/group of data is what can be referred to as “outliers”.

Outliers are certain values in a group of data that are seen to be fairly far away from most of the other values, these outliers can be either higher or lower.


For example we could have a list of the top speeds of different cars.


140mph &nbsp,&nbsp 144mph &nbsp,&nbsp 150cm &nbsp,&nbsp 154cm &nbsp,&nbsp 143cm &nbsp,&nbsp 190mph

190mph&nbsp can be seen to be an outlier.


The mean/average top speed is:

\bf{\frac{140 \space + \space 144 \space + \space 150 \space + \space 145 \space + \space 143 \space + \space 190}{6}} &nbsp=&nbsp \bf{\frac{912}{6}} &nbsp = &nbsp 152mph

The outlier value of &nbsp190mph&nbsp raises the mean/average of the group.




But without &nbsp190mph:

Mean/average &nbsp = &nbsp \bf{\frac{140 \space + \space 144 \space + \space 150 \space + \space 145 \space + \space 143}{5}} &nbsp = &nbsp \bf{\frac{720}{5}} &nbsp = &nbsp 144.4mph

144mph&nbsp is an average value that is a more reasonable mean/average top speed for the group.

Sometimes, it can be a useful idea to omit certain outliers in some situations, when trying to calculate an average.







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