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Mean

(a measures of position; average)

The mean of a set of numbers is the total of the set divided by the size of the set.

This can be written in summation notation as:

$$\underset{i=1}{\overset{n}{\Sigma}}=\frac{{x}_{i}}{n}$$
where ${x}_{i}$ is the ${i}^{\mathrm{th}}$ element of the set to be averaged,

and $n$ is the number of elements in the set.

Example:

The set $$\{5,6,9,11,11,12\}$$ has a total of $$5+6+9+11+11+12=54$$ and the set has size six, so the mean is $$54\xf76=9$$

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Median

(a measures of position; average)

The mean of a set of numbers is the middle member after the set is ordered.

Example:

The set $$\{33,40,52,45,37,50,61,38\}$$ can be ordered thus $$\{33,37,38,40,45,50,52,61\}$$ and the number of elements is eight. The middle members location is given by $$\frac{8+1}{2}=4.5$$ So the median is midway between two members? This has occurred because there are an even number of elements. If we had had an odd number the resulting location would have been an integer and therefore we just find the value of that member. Here we need to do a further calculation to determine the median $$\frac{40+45}{2}=42.5$$ So the median of the set is 42.5

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Mode

(a measure of position; average)

The mode is a useful type of average as it tells you the number that occurs the most in a set of numbers. Not every set of numbers has a mode. Furthermore, some sets have more than one mode.

Example:

The mode of $$\{1,2,4,4,5,8,9\}$$ is 4 as it occurs the most.

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Geometric Mean

(a measure of position; average)

The arithmetic mean is given by

$$\stackrel{-}{x}=\frac{{x}_{1}+{x}_{2}+{x}_{3}+...}{n}=\frac{\underset{i=1}{\overset{n}{\Sigma}}{x}_{i}}{n}$$where $n$ is the number of numbers ${x}_{1},{x}_{2},{x}_{3},...$

If we multiply through by $n$ we get:

$$n\stackrel{-}{x}=\underset{i=1}{\overset{n}{\Sigma}}{x}_{i}$$Here knowing the mean and the number of elements we can find the sum of all the elements, the total.

We can do something similar with the product of all the elements.

$${\left(\stackrel{-}{x}\right)}^{n}={x}_{1}\times {x}_{2}\times {x}_{3}\times ...=\underset{i=1}{\overset{n}{\Pi}}{x}_{i}$$So the geometric mean is

$$\stackrel{-}{x}=\sqrt[n]{\underset{i=1}{\overset{n}{\Pi}}{x}_{i}}$$